Proceedings of the

32nd International
Workshopon
Statistical Modelling
Volume I

Groningen, Netherlands
3-7 July, 2017
Editors: Marco Grzegorczyk, Giacomo Ceoldo

iiProceedings of the 32nd International Workshop on Statistical Modelling
Volume I,
Groningen, July 3-7, 2017,
Marco Grzegorczyk, Giacomo Ceoldo (editors),
Groningen, 2017.

Editors:
Marco Grzegorczyk, m.a.grzegorczyk@rug.nl
University of Groningen
Nijenborgh 99747 AG Groningen
The Netherlands

Giacomo Ceoldo, g.ceoldo@student.rug.nl
University of Groningen
Nijenborgh 99747 AG Groningen
The Netherlands

iii

Preface
Dear Participants,
First of all, I would like to welcome you to the 32nd International Workshop on
Statistical Modelling (IWSM 2017) in the Netherlands, and I wish you a very
pleasant stay in Groningen. With around 200,000 inhabitants Groningen is the
7th largest town in the Netherlands, and Groningen has two large universities:
The Hanzehogeschool for Applied Sciences (about 20,000 students) and the Rijksuniversiteit Groningen (RUG) with about 30,000 students. The venue of the
IWSM 2017 is the Academy Building of RUG and located in the town center ofGroningen.
Before you lies one of the two proceedings volumes of IWSM 2017. It is a uniquefeature within the statistical community that all speakers at this workshop also
provide an extended abstract of their talk. This not only provides the participants
with a compact written account of interesting contributions, but it also improves
the quality of the talks.
Like every year, there was a huge amount of excellent paper submissions, andit was a really challenging task to select from 138 abstracts 56 (41%) for oralpresentations. Each paper had to be reviewed and scored by three members of
the scientific programme committee. This was a very time-consuming task for thereviewers, and for their valuable efforts I thank all members of the scientific committee: Ernst Wit (RUG), Marijtje van Duijn (RUG), Kenan Matawie (IWSM
2005), Arnošt Komárek (IWSM 2012), Vito Muggeo (IWSM 2013), Thomas
Kneib (IWSM 2014), Helga Wagner (IWSM 2015), Jean-François Dupuy (IWSM
2016), Simon Wood (IWSM 2017), Paul Eilers, John Hinde, Dirk Husmeier, Sonja
Greven, Edwin van den Heuvel, Jörg Rahnenführer and Korbinian Strimmer.
Some of the abstracts that could not be selected for oral presentation have been
given the opportunity to be presented on a poster. On Tuesday (17:15-19:30h)
there will be a poster presentation where everybody is invited to meet the researchers and to discuss their ongoing work one-on-one. It will be taken care ofdrinks and some ‘fingerfood’ in order not to be distracted by a thirsty throat
or a hungry belly. Although it would have been possible to extend the numberof presentations, it is an important feature of the IWSM workshops that there
are no parallel sessions. This means that each presentation, whether by a PhD
student or by a famous statistician, are awarded the same amount of attention.
This means that the IWSM is a very coherent meeting, whereby the emphasis lies
precisely on the word: ‘meeting’. It is a place where junior and senior researchers
mix and mingle.
Not coincidently, also this year you will get ample opportunities to meet yourfellow participants. On Sunday evening, after the excellent short course by Tom
Snijders and before the official start of the conference on Monday, there wasalready the informal welcome drink gathering in the Pool Restaurant of the Student Hotel. On Monday evening, there will be the official welcome reception in
the Spiegelzaal of the Academy Building. On Tuesday, the Pizza & Beer Poster

iv
session in the Academia Restaurant of the Academy Building will encourage lively
scientific interactions. On Wednesday, after the excursions, we will reconvene altogether in the Ni Hao restaurant (Gedempte Kattendiep 122). On Thursday
evening, if you are hungry again, there will be the official conference dinner in‘t Feithhuis (Martinikerkhof 10). Even the conference dinner will be kept quite
informal, as the emphasis should be on meeting your fellow participants. It would
be great if, long after this conference is over, you could look back on IWSM 2017
and say: ‘Groningen was the place where I met ’em all! ’.
I thank the Statistical Modelling Society for trusting in my proposal and for giving me this great opportunity to chair IWSM 2017. Many colleagues from RUG
helped me planing and realizing this workshop. I would like to thank all membersof my Local Organizing Committee, in particular, Ernst Wit, Ineke Schelhaas,
Martijn Wieling, Casper Albers, Wendy Post, Marijtje van Duijn and Hans Burgerhof for their contributions. Also Mariska Pater and Sharon de Puijselaar fromthe Groningen Congres Bureau have helped me tremendously.
My special acknowledgements go to all sponsors of the IWSM 2017. Without
those sponsorships certain things could not have been realized and the programme
would certainly have been much sparser. A list of the sponsors of IWSM 2017
can be found on the last pages of this volume.
Last but not least, I would like to thank all authors for the excellent scientificcontributions, and I hope that every participant of IWSM 2017 will have a greatand especially research-stimulating week in Groningen,
Marco Grzegorczyk
Groningen, 16 June 2017

Scientific committee• Chair: Marco Grzegorczyk (Groningen University, Netherlands)• Ernst Wit (Groningen University, Netherlands)• Marijtje van Duijn (Groningen University, Netherlands)• Arnošt Komárek (Charles University Prague, Czech Republic)• Dirk Husmeier (Glasgow University, Scotland)• Edwin van den Heuvel (Eindhoven University, Netherlands)• Helga Wagner (Linz University, Austria)• Jean-François Dupuy (INSA of Rennes, France)• John Hinde (NUI Galway, Ireland)• Jörg Rahnenfüehrer (TU Dortmund University, Germany)• Kenan Matawie (University of Western Sydney, Australia)• Korbinian Strimmer (Imperial College London, England)• Paul Eilers (Erasmus Rotterdam, Netherlands)• Simon Wood (Bath University, England)• Sonja Greven (LMU Munich, Germany)• Thomas Kneib (Goettingen University, Germany)• Vito Muggeo (Palermo University, Italy)

vi

Local Organizing Committee• Chair: Marco Grzegorczyk• Ernst Wit• Ineke Schelhaas• Martijn Wieling• Casper Albers• Wendy Post
• Marijtje van Duijn• Hans Burgerhof• Sacha la Bastide• Christine zu Eulenburg• Mark Huisman• Ruud Koning• Mariska Pater• Laura Spierdijk• Christian Steglich• Marieke Timmerman

Contents
Part I – Invited Papers
Roland Langrock: Nonparametric inference in hidden Markov and related models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Laura M. Sangalli: Functional Data Analysis, Spatial Data Analysis
and Partial Differential Equations: A fruitful union. . . . . . . . . . . . . . . . . . .

3
4Jelle Goeman, Aldo Solari: Minimal Adequate Models: Assessing Uncertainty in Variable Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Emmanuel Lesaffre, Kris Bogaerts, Arnošt Komárek: Why bothering about Interval Censoring? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

R. Harald Baayen, Tino Sering, Cyrus Shaoul, Petar Milin: Language comprehension as a multiple label classification problem . . . . . . .

21Part II – Contributed Papers
Paola Stolfi, Mauro Bernardi, Lea Petrella: The Multivariate
Method of Simulated Quantiles for Portfolio Optimisation . . . . . . . . . . . .
Irene Mariñas, Adrian Bowman, Vincent Macaulay: Gaussian Process model for evolving 3D lip curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35
41Pariya Behrouzi, Ernst C. Wit: Construction of High-resolution Linkage Maps Using Discrete Graphical Models . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Alan Lazarus, Dirk Husmeier, Theodore Papamarkou: Inferencein complex systems using multi-phase MCMC sampling with gradientmatching burn-in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Jennifer Pohle, Roland Langrock: Pragmatic order selection in hidden Markov models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Vicente Núñez-Antón, Edilberto Cepeda-Cuervo: Spatial Conditional Overdispersed Models: Application to count area data . . . . . . . . .

64

Elaine A. Ferguson, Jason Matthiopoulos, Dirk Husmeier: Constructing wildebeest density distributions by spatio-temporal smoothing of ordinal categorical data using GAMs . . . . . . . . . . . . . . . . . . . . . . . . . .

70Ashwini Venkatasubramaniam, Ludger Evers, Konstantinos Ampountolas: Spatio-temporal clustering of traffic networks . . . . . . . . . . .

76

Rianne Jacobs, Emmanuel Lesaffre, Peter Teunis, Jan van deKassteele: Bayesian variable selection for identifying the source offood-borne disease outbreaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

viii

Contents

Hassan Pazira and Ernst Wit: Using extended dgLARS to study Diabetes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Christian Staerk, Maria Kateri: Stable Variable Selection with AdaSub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

Andreas Mayr, Janek Thomas, Matthias Schmid, Florian
Faschingbauer, Nadja Klein: Boosting distributional regression
models for multivariate responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

Andreas Groll, Thomas Kneib, Nikolaus Umlauf: LASSO-type penalization in the framework of Generalized Additive Models for Location, Scale and Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

Almond Stöcker, Sarah Brockhaus, Sophia Schaffer, Benedikt
von Bronk, Madeleine Opitz, Sonja Greven: Boosting Generalized Additive Models for Location, Scale and Shape for Functional Data

109

Julien Hambuckers, Thomas Kneib, Roland Langrock, Alexander
Sohn: Markov-Switching GAMLSS with an Application to Operational
Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

Mariana Melo, Cristiano Fernandes, Eduardo Melo: Aggregate
Claims Modelling via Dynamic Score Driven Models . . . . . . . . . . . . . . . . .

119

Maike Hohberg, Katja Landau, Thomas Kneib, Stephan Klasen,
Walter Zucchini : Enhancing predictive performance of vulnerability
to poverty estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125

Annette Möller , Jürgen Groß : A heteroscedastic probabilistic temperature forecasting model incorporating spread-error correlation andhigh-resolution forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131

Manuel Gebetsberger, Georg J. Mayr, Reto Stauffer, Achim
Zeileis: Probabilistic temperature post-processing using a skewed response distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137

Thorsten Simon, Nikolaus Umlauf, Georg J. Mayr, Achim Zeileis:
Boosting multivariate Gaussian models for probabilistic temperature
forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

Claudia Kirch, Matthew Edwards, Alexander Meier, Renate
Meyer: Generalization of the Whittle likelihood for nonparametric
spectral density estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149

Mirko Signorelli1,2 , Ernst C. Wit: Model-based clustering for populations of networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155

Tom A. B. Snijders, Viviana Amati, Felix Schönenberger: Generalized Method of Moments for Estimating the Parameters of Stochastic
Actor-oriented Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161

Contents

ix

Francesco Bartolucci, Maria Francesca Marino, Silvia Pandolfi: Stochastic block models for social network data: inferential developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167

Mahdi Shafiee Kamalabad, Marco Grzegorczyk: A sequentially coupled non-homogeneous dynamic Bayesian network model with segmentspecific coupling strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173

Faisal Maqbool Zahid, Christian Heumann: Multiple imputation for
high-dimensional data using sequential penalized regression . . . . . . . . . .

179José Clelto Barros Gomes, Cibele Maria Russo: REML in nonlinear
mixed-effects models with heavy-tailed distributions . . . . . . . . . . . . . . . . . .

184

L. Mihaela Paun, M. Umar Qureshi, Mitchel Colebank, Mansoor
A. Haider, Mette S. Olufsen, Nicholas A. Hill, Dirk Husmeier:Parameter Inference in the Pulmonary Circulation of Mice . . . . . . . . . . .

190

Xanthi Pedeli, Cristiano Varin: The Pairwise Expectation Maximization Algorithm for Fitting Parameter-Driven Models . . . . . . . . . . . . . . . . .

196

Salvatore Fasola, Stefania La Grutta, Vito M.R. Muggeo: Estimating abrupt change models with covariate-dependent changepoint .

200

Šárka Rusá, Arnošt Komárek, Emmanuel Lesaffre, Luk
Bruyneel: Identifying influential observations in complex Bayesian
mediation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

206

Massimo Ventrucci, Maria Franco-Villoria, Håvard Rue: Penalized complexity priors for varying coefficient models . . . . . . . . . . . . . . . . . .

212

Jochen Einbeck, Elizabeth Gray, Nick Sofroniou, Antonio Hermes Marques da Silva Junior, Jacob Gledhill: Confidence intervals for posterior intercepts, with application to the PIAAC literacy
survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

217

Rosaria Simone, Gerhard Tutz: Dealing with response styles in finite
mixture models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

223

Sujit Sahu, Mark Bass, Carla Sabariego, Alarcos Cieza, Carolina Fellinghauer, Somnath Chatterji: Extending the inferential
capability of a generalised partial credit model using Bayesian computation: An application to an international disability survey developedby WHO and the World Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229

Gunther Schauberger, Gerhard Tutz, Moritz Berger: Response
Styles in the Partial Credit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

231

Maurizio Manuguerra, Gillian Heller, Jun Ma: Semi-parametricordinal regression models for continuous scales . . . . . . . . . . . . . . . . . . . . . . .

236

Sophie Vanbelle, Emmanuel Lesaffre: Modeling agreement on continuous recordings in the presence of a binary scale. . . . . . . . . . . . . . . . . . .

242

x

Contents

Helga Wagner, Sylvia Frühwirth-Schnatter: Bayesian Inference for
Mixed Effects Multinomial Logit Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

246

Alan Agresti, Claudia Tarantola: Simple Effect Measures for Interpreting Models for Ordinal Categorical Data . . . . . . . . . . . . . . . . . . . . . . . . .

252

Paul H. C. Eilers: The truth about the effective dimension . . . . . . . . . . . .

258

Giovanna Cilluffo, Gianluca Sottile, Stefania La Grutta, Vito
M.R. Muggeo: Score inference in LASSO regression. . . . . . . . . . . . . . . . .

264

Craig Alexander, Jane Stuart-Smith, Tereza Neocleous, Ludger
Evers : Using chain graph models for structural inference with anapplication to linguistic data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

270

Paul Wilson, Jochen Einbeck: Sample quantiles corresponding to mid
p-values for zero–modification tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

275

Betsabe Blas, Scott J. Cook, Raymond J. Carroll, SamiranSinha: Two wrongs make a right: addressing underreporting in binary
data from multiple sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

280

Luı́s Meira-Machado, Beatriz Sampaio: Estimation of multivariate distributions for recurrent event data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

284

Kevin Burke and Gilbert MacKenzie: Multi-Parameter Regression
and Frailty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

290

Il Do Ha, Jong-Min Kim: Comparison of Semiparametric Copula andFrailty Models for Clustered Survival Data . . . . . . . . . . . . . . . . . . . . . . . . . . .

294

Wei Fu, Jeffrey S. Simonoff: Conditional inference survival trees for
nonstandard data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299

Elisabeth Waldmann, Nadja Klein, David Taylor-Robinson:
Bayesian Joint Modelling of Distributional Regression . . . . . . . . . . . . . . . .

305

Thomas Husken, Maarten Cruyff, Peter van der Heijden: Incorporating cyclical effects and time-varying covariates in models for singlesource capture-recapture data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311

Diana Giurghita, Dirk Husmeier: Statistical modelling of cell movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

317

Jocelyn Chauvet, Catherine Trottier, Xavier Bry: Regularisationof Generalised Linear Mixed Models with autoregressive random effect323

K. M. Matawie, A. Mehar, A. Maeder: Optimal Number of Clusters
Based on Inter Cluster Elements Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . .

329

Jan van de Kassteele, Paul Eilers, Jacco Wallinga: Nowcasting
infectious disease outbreaks using constrained P-spline smoothing . . . .

335

Francisco Richter, Rampal Etienne, Ernst Wit: A general statistical framework to study the diversification of species . . . . . . . . . . . . . . . . . .

339

Contents

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

345

Part I – Invited Papers

Nonparametric inference in hidden Markov
and related models
Roland Langrock1

Department of Business Administration and Economics, Bielefeld University,
Germany

E-mail for correspondence: roland.langrock@uni-bielefeld.de
Abstract: Hidden Markov models (HMMs) have been successfully applied in various disciplines, including biology, speech recognition, economics/finance, climatology, psychology and medicine. They combine immense flexibility with relative
mathematical simplicity and computational tractability, and as a consequence
have become increasingly popular as general-purpose models for time series data.
In this talk, I will first introduce the basic HMM machinery and showcase thepractical application of HMMs using intuitive examples. I will then demonstrate
how the HMM machinery can be combined with penalized splines to allow for
flexible nonparametric inference in general-purpose HMM-type classes of models. The focus of the presentation will lie on practical aspects of nonparametricmodelling in these frameworks, with the methods being illustrated in economicand ecological real data examples, featuring, inter alia, the famous wild haggis
animal, blue whales and the well-known Lydia Pinkham sales data.
Keywords: Animal behaviour; Markov-switching regression; P-splines
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Functional Data Analysis, Spatial Data
Analysis and Partial Differential Equations:
A fruitful union
Laura M. Sangalli
1

MOX - Dipartimento di Matematica, Politecnico di Milano, Italy

E-mail for correspondence: laura.sangalli@polimi.it
Abstract: I will discuss an innovative class of regularized regression models
for the analysis of spatially distributed data, that merges advanced statisticaland numerical analysis techniques. Based on these regression models, I will thenpresent a principal component analysis method that can handle functional signals
distributed over complex domains.
Keywords: Penalized regression; functional principal component analysis; data
distributed over two-dimensional manifold domains; finite elements.

1

Spatial regression with differential regularization

I will present a novel class of models for the analysis of spatially (or space-time)
distributed data, based on the idea of regression with differential regularizations.
The models merge statistical methodology, specifically from functional data analysis, and advanced numerical analysis techniques. Thanks to the combination ofpotentialities from these different scientific areas, the proposed method has important advantages with respect to classical spatial data analysis techniques. Spatial
regression with differential regularizations is able to efficiently deal with data distributed over irregularly shaped domains, with complex boundaries, strong concavities and interior holes [Sangalli et al. (2013)]. Moreover, it can comply withspecific conditions at the boundaries of the problem domain [Sangalli et al. (2013),
Azzimonti et al. (2014, 2015)], which is fundamental in many applications to obtain meaningful estimates. The proposed models have the capacity to incorporate
problem-specific priori information about the spatial structure of the phenomenon
under study, formalized in terms of a governing partial differential equation [Azzimonti et al. (2014, 2015)]; this very flexible modeling of space-variation allows
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Sangalli

5

to naturally account for anisotropy and non-stationarity. Space-varying covariate information is accounted for via a semiparametric framework. The models
can also be extended to space-time data [Bernardi et al. (2017)]. Furthermore,
spatial regression with differential regularizations can deal with data scatteredover non-planar domains, specifically over two-dimensional Riemannian manifold
domains, including surface domains with non-trivial geometries [Ettinger et al.
(2016), Dassi et al. (2015), Wilhelm et al. (2016)]. This has fascinating applications in the earth-sciences, life-sciences and engineering. The use of advanced
numerical analysis techniques, and in particular of the finite element method orof isogeometric analysis, makes the models computationally very efficient. The
models are implemented in the R package fdaPDE [Lila et al. (2016)].

2

Smooth principal component analysis for functionalsignals over complex domains

Based on the regularized regression models outlined above, I will present a regularized method for principal component analysis of functional signals observed
over two-dimensional Riemannian manifold domains [Lila et al. (2016)]. This
will be illustrated with an application in the neurosciences, studying neuronalconnectivity on the cerebral cortex, starting from functional magnetic resonanceimaging scans on about 500 healthy volunteers.

FIGURE 1. Study of high-dimensional neuroimaging signals (dataavailable
from
The
Human
Connectome
Project
Consortium,
www.humanconnectomeproject.org). Left: Triangulated surface approximating the left hemisphere of a cerebral cortex. Right: functional connectivity
map obtained from fMRI signal. Figure adapted from Lila et al. (2016).

Acknowledgments: This talk is based on joint works with John A.D. Aston,
Laura Azzimonti, Mara Bernardi, Bree Ettinger, Michelle Carey, Eardi Lila, Fabio
Nobile, Simona Perotto, Jim Ramsay, Piercesare Secchi.

6Functional Data Analysis, Spatial Data Analysis and Partial Differential Equations

References
Azzimonti, L., Nobile, F., Sangalli, L.M., Secchi, P. (2014). Mixed Finite Elements
for spatial regression with PDE penalization. SIAM/ASA Journal on Uncertainty Quantification, 2, 1, pp. 305 – 335.
Azzimonti, L., Sangalli, L.M., Secchi, P., Domanin, M., Nobile, F. (2015). Bloodflow velocity field estimation via spatial regression with PDE penalization.
Journal of the American Statistical Association, Theory and Methods, 110,
511, pp. 1057 – 1071.
Bernardi, M.S., Sangalli, L.M., Mazza, G., Ramsay, J.O. (2017). A penalized regression model for spatial functional data with application to the analysisof the production of waste in Venice province. Stochastic Environmental
Research and Risk Assessment, 31, 1, pp. 23 – 38.
Dassi, F., Ettinger, B., Perotto, S., Sangalli, L.M. (2015). A mesh simplification
strategy for a spatial regression analysis over the cortical surface of thebrain. Applied Numerical Mathematics, 90, 1, pp. 111 – 131.
Ettinger, B., Perotto, S., Sangalli, L.M. (2015). Spatial regression models overtwo-dimensional manifolds. Biometrika, 103, 1, pp. 71 – 88.
Lila, E., Aston, J.A.D., Sangalli, L.M. (2016). Smooth Principal Component Analysis over two-dimensional manifolds with an application to Neuroimaging.
Annals of Applied Statistics, 10, 4, pp. 1854 – 1879.
Lila, E., Sangalli, L.M., Ramsay, J.O., Formaggia, L. (2016). fdaPDE: functional
data analysis and Partial Differential Equations; statistical analysis of functional and spatial data, based on regression with partial differential regularizations, R package version 0.1-4,, http://CRAN.R-project.org/package=fdaPDE.
Sangalli, L.M., Ramsay, J.O., Ramsay, T.O. (2013). Spatial spline regression models. Journal of the Royal Statistical Society Ser. B, Statistical Methodology,
75, 4, pp. 681 – 703.
Wilhelm, M., Dede’, L., Sangalli, L.M., Wilhelm, P. (2016). IGS: an IsoGeometric approach for Smoothing on surfaces. Computer Methods in Applied Mechanics and Engineering, 302, pp. 70 – 89.
Wilhelm, M., Sangalli, L.M. (2016). Generalized Spatial Regression with Differential Regularization. Journal of Statistical Computation and Simulation,
86, 13, pp. 2497 – 2518.

Minimal Adequate Models: Assessing
Uncertainty in Variable Selection
Jelle Goeman1 , Aldo Solari2
1
2

Leiden University Medical Center, The Netherlands
University of Milano-Bicocca, Italy

E-mail for correspondence: j.j.goeman@lumc.nl
Abstract: We present an alternative approach to variable selection that does
not select a single “best” model but attempts to find a collection of models
that is “good enough”. We call models adequate if they are not significantly
worse than the true model. The collection of all adequate models is spanned bya smaller collection of minimal adequate models: the smallest of the adequate
models. These minimal adequate models give great insight in model selection
uncertainty as well as in collinearity, and are therefore a very practical modelbuilding tool. We illustrate the approach with several classical data sets.
Keywords: Model selection; Closed testing; Model misspecification.

1

Variable selection

The goal of variable selection methods in regression is to discard a subset of thecovariates without reducing the predictive potential of the remaining variables.
Typical variable selection methods find a single “best” model according to achosen criterion, e.g. AIC or BIC. Variable selection is done to reduce overfit but
also for reasons of interpretation. Selected variables are interpreted as important,and discarded variables as irrelevant. Different criteria and different methods,
however, can yield very different selected models. Especially when collinearity ispresent, variable selection methods tend to differ greatly both in which variables
are selected and how many.
Clearly, there is uncertainty about the selected model. If we see the single selectedmodel as a point estimate of the true model, then we can say that typical variableselection methods neglect to give standard errors or confidence intervals aroundthe statements they make. Interpreting the selected model in terms of important
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

8Minimal Adequate Models

and irrelevant variables is like interpreting point estimates without an associatedmeasure of uncertainty.
We present a different approach to variable selection that, in contrast, emphasizes
the uncertainty in the variable selection process. We adopt a hypothesis testing
framework to construct a confidence interval around the true model. Thus, weselect not a single model, but a range of models. A model is in the confidence
interval if its likelihood is not significantly worse than that of the true model. By
construction, the confidence interval contains the true model with probability atleast 1 − α.
Our work builds upon work laid out by Mallows (1973), Aitkon (1974) andSpjotvoll (1977). We show that their work can be seen as a special case of closed
testing, which allows their results to be extended outside the scope of linear
models e.g. to generalized linear models.

2

Minimal adequate models

The construction of the confidence intervals will be such that if a model is in
the confidence set all supersets of the model are also in the confidence set. The
confidence set is therefore spanned by its smallest members, the minimal adequate
models.
The minimal adequate models give great insight in the reliability of inferential
statements made as a result of variable selection methods. They can be used to
distinguish between variables that must always be selected by a variable selection
method and variables that can take each other’s roles in the model because they
contain the same information, e.g. because of collinearity.
For example, two minimal adequate models can be {A, B} and {A, C, D}. In thiscase covariate A is necessary for any adequate model, but the role of B can betaken over by the combination of C and D, which together contain the same
information. A user may have a preference for model {A, B} because it is moresparse or for {A, C, D} because it may have a better fit or be more interpretable.
In either case the presence of the other minimal adequate model functions as aprotection against overinterpretation of the selected model.

3

Model misspecification

A confidence interval for the true model supposes the existence of the true model,
which in turn implies that the full model is true. Since this is quite a strong
assumption, we will investigate how to relax it. We do this by refining the null
hypothesis to be tested for each model: instead of testing whether the reduced
model is as good as the true model, we test whether it is as good as the fullmodel. We show that this hypothesis can be conservatively tested even when thefull model is not the true model.

Goeman and Solari

49Application

The use of minimal adequate models will be illustrated with several classicalregression data sets, such as Hald’s cement data and the famous prostate cancer
data set (Hastie et al. 2001).

References
Aitkin, M.A. (1974). Simultaneous inference and the choice of variable subsets
in multiple regression. Technometrics, 16, 221 – 227.
Hastie, T., Tibshirani, R., and Fiedman, J. (2001). The Elements of Statistical
Learning. New York: Springer.
Mallows, E. (1973). Some comments on CP . Technometrics, 15, 661 – 675.
Spjotvoll, E. (1977). Alternatives to plotting CP in multiple regression. Biometrika,
64, 1 – 8.

Why bothering about Interval Censoring?
Emmanuel Lesaffre1 , Kris Bogaerts1 , Arnošt Komárek2
1
2

I-BioStat, KU Leuven, Leuven and UHasselt, Hasselt, Belgium
Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

E-mail for correspondence: emmanuel.lesaffre@kuleuven.be
Abstract: We consider here methods to analyze interval-censored survival times.
Interval censoring occurs when it is only known that the event happened inbetween two examinations. Well-known examples of an interval-censored time
are the time until HIV, AIDS, the emergence of a tooth, etc. Most often interval
censoring is not appropriately addressed in a statistical analysis and dealt withby methods that handle right censoring of data, e.g. by replacing the intervalby the mid-point. Despite several published results it is still too often believedthat ignoring the interval-censored character of the data has a minimal impact
on the results and conclusions of the statistical analysis. In this contribution wesummarize the literature on interval censoring largely from a practical point ofview under the frequentist and a Bayesian paradigm. It will be also discussed
when it is important to take interval censoring into account.
Keywords: Bayesian inference; Interval censoring; Survival analysis.

1

Introduction

In survival studies, right censoring is most prevalent and generally dealt withappropriately. Occasionally also left censoring occurs, but in randomized controlled trials and epidemiological studies interval censoring occurs frequently.
Left censoring occurs, e.g., in a dental study on emergence of permanent teeth
when a tooth emerged prior to the start of the study. An emergence time is theninterval-censored when it is only known that the tooth emerged in-between two
examinations. Interval censoring also occurs often in HIV/AIDS studies, wheretime to HIV seroconversion and AIDS are usually determined at planned visits to the clinical researcher. In fact many developments on interval censoring
find their origin in HIV/AIDS research. Finally, in cancer trials progression freesurvival can only be established in the hospital at planned visits. Despite theThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Lesaffre et al.

11frequent occurrence of interval censoring, this interval censoring is often treatedinappropriately in practice. Note that left and right censoring can be consideredas special cases of interval censoring. Also, in practice most often a mix of thecensoring types is encountered.
Often interval censoring is bypassed with a single imputation technique, with
mid-point imputation being most popular. That is, while the interval-censoredsurvival time is replaced by the mid-point of the interval, the data are analyzed
using methods for right-censored data. The effect of inappropriately dealing withinterval censoring depends on the size of the intervals, on whether covariates
impact the size of the interval and the type of statistical analysis. In the past,absence of statistical software was the main reason for avoiding interval censoring. This is not the case anymore nowadays. This is clearly illustrated in theforthcoming book on interval censoring (Bogaerts, Komárek and Lesaffre, 2017),
hereafter referred to as BKL.
One distinguishes case I interval-censored data, also called current status data.
This occurs in practice when it is only known whether the event has happend
or not at the time of examination. We concentrate in this contribution on case
II interval censoring. Namely, we assume that an independent sample of survival
times T1 , . . . , Tn is only observed to lie in intervals bli , ui c (i = 1, . . . , n), where
bli means that either li is included or not in the interval and the same for ui . By
allowing li to be zero, interval censoring reduces to left censoring. On the other
hand right censoring is a special case of interval censoring when ui = ∞ (in practice this implies a large value). Further we assume that the censoring mechanism
is independent of the true survival times. We will return to this assumption in
Section 5.
A popular data set in the statistical literature on interval-censored data comesfrom a breast cancer study. It consists of the subset of 96 patients who weretreated at the Joint Center for Radiation Therapy in Boston between 1976 and1980. Forty-six patients were randomized to radiation therapy only regimen, while
48 patients to the radiation therapy and adjuvant chemotherapy regimen. Theintervals represent the time period during which breast retraction occurred. A
graphical representation of the data is shown in Figure 1. Illustrations will also betaken from the Signal Tandmobielr study, which is a longitudinal dental study
that examined, e.g., the emergence distributions of several permanent teeth.

2

Univariate models

2.1

Frequentist approaches

Let the true survival times T1 , . . . , Tn be i.i.d. with survival distribution S(·).
When the survival times are interval-censored with intervals bli , ui c (i = 1, . . . , n),
the likelihood to maximize is:L=nY{S(li ) − S(ui )},

(1)

i=1

with S(t) the unknown but true survival distribution. Peto (1973) was the first
to note that the nonparametric maximum likelihood solution of S results in a set

12Interval Censoringof intervals {[pj , qj ]}m
j=1 with the following properties: outside these intervals, theestimated survival function is constant. Further, the mass assigned to each of theintervals is well determined but within each interval there is no information as
to how that mass is assigned. The intervals are called regions of possible mass or
support because the maximum likelihood procedure can only tell in which regions
there is probability of events to occur. Peto (1973) and Turnbull (1976) suggested
a simple reduction algorithm to identify the intervals of possible mass from thedata. Further, Turnbull (1976) suggested the self-consistency algorithm, a version of the EM algorithm, to determine the nonparametric maximum likelihood
estimator (NPMLE) of S. Thus, in contrast to the Kaplan-Meier estimator, the
NPMLE of the survival function for interval-censored data has no closed solution
and must be obtained by an iterative algorithm.
Two versions of the NPMLE are given in Figure 2 obtained from the patientstreated with radiotherapy alone in the breast cancer study. The left panel is the
NPMLE of the cumulative distribution function of the time to cosmetic deterioration of the breast. Fourteen regions of possible support were found but only
to eight regions mass > 0 has been attributed. In Table 1, these intervals areshown. The gray areas indicate that the distribution of probability within theregions of support is not determined. In the right panel the corresponding estimated survival distribution is given but assuming a linear behavior of Sb in theintervals.
Since the seminal papers of Peto and Turnbull, the classical significance tests insurvival analysis to compare two or more groups (logrank test, Gehan-Wilcoxon
test, Peto-Prentice-Wilcoxon test, etc.) have been extended to interval-censoredobservations.
Because for a long time Turnbull’s algorithm was not available in statistical software, it was standard to show the Kaplan-Meier estimate based on singly imputed
survival times. Alternatively and depending on the application area, also a para-

Radiotherapy onlyRadiotherapy + Chemotherapy

50

Number of patients

40

30

20

10

0
0

20

40

60

0

20

40

60

Time to breast retraction (months)

FIGURE 1. Breast cancer study. Observed intervals in months for time to breast
retraction of early breast cancer patients per treatment group.

Lesaffre et al.

13

TABLE 1. Breast cancer study. Regions of possible support and NPMLE equivalence classes for the radiotherapy-only group.
(pj , qj ]mass

(4, 5]
0.046(6, 7]
0.033(pj , qj ]mass

(25, 26]
0(7, 8]
0.089(33, 34]
0.082(11, 12]0.071(34, 35]
0(15, 16]0(36, 37]
0(17, 18]
0(38, 40]
0.121(24, 25]0.093(40, 44]0(46, 48]0.466

metric estimate was computed. In medical applications the most popular choices
are the Weibull and the log-normal distribution. Computations and inference
are simpler in the parametric case relying often on Newton-Raphson type of algorithms and standard asymptotic likelihood theory. In-between nonparametricand parametric approaches are flexible estimation methods. Numerous techniques
have been proposed that either smooth the hazard, the cumulative hazard or thesurvival distribution. Popular in this sense is spline smoothing based on cubic
splines, B-splines or penalized B-splines adapted to interval-censored data. A
smooth solution can also be obtained from, say, a mixture of Gaussian densities
for the survival density. Examples of these approaches with software applicationsin R and SAS software can be found in BKL.

2.2

Bayesian approaches

1.0
0.8
0.6
0.0

0.2

0.4

^
S(t)

0.6
0.4
0.0

0.2

^
F(t)

0.8

1.0

Parametric analysis of interval-censored observations is fairly standard in classical Bayesian statistical software, such as Win/OpenBUGS or SAS, as long as

0

10 20 30 40 50
Time0

10 20 30 40 50
Time

FIGURE 2. NPMLE of the cumulative distribution function (left panel) and
NPMLE of the survival function with the additional assumption of a piecewise
linear survival curve (right panel).

14Interval Censoring

the chosen survival distribution is supported by the package. More complicatedis to perform a Bayesian nonparametric (BNP) analysis. BNP estimation of acumulative distribution function (and thus of the survival distribution) started
with the seminal paper of Ferguson (1973), who introduced the Dirichlet process(DP) prior. The DP prior D(cS ∗ ) is a prior on the survival distributions S definedaround a guess survival distribution S ∗ with variability ruled by a scalar c. Basedon a DP prior, Susarla and Van Ryzin (1976) proposed a nonparametric Bayesian
approach to estimate the survival function for right-censored survival times. Calle
and Gómez (2001) further extended the procedure to interval-censored data. Limiting cases of the posteriors are the Kaplan-Meier for right-censored observations
and Turnbull’s estimate for interval-censored observations. Until recently no generally available software was available for the BNP approach, this changed with
the R package DPpackage (Jara, 2007). In the supplementary materials of BKL,
some self-written R programs can be found for fitting survival distributions ina nonparametric way as well as illustrations of the use of DPpackage.

3

Regression models

Of more interest are survival models that allow for covariates, say X1 , . . . , Xn .
In that case the likelihood becomes:
L=nY{S(li | Xi ) − S(ui | Xi )}.

(2)

i=1

While for right-censored data the Cox proportional hazards (PH) model takes acentral position because the partial likelihood approach renders the estimation of
the baseline hazard obsolete, with interval-censored survival times the baseline
hazard/distribution needs to be estimated together with the regression coefficients. We consider here the PH model and the accelerated failure time (AFT)model for interval-censored survival times. Again a variety of approaches weresuggested from semiparametric to parametric.

3.1

Frequentist approaches

The likelihood to maximize for the PH model is given byL(β, S0 ) =

n n
Y

o
>
>
S0 (li )exp(Xi β) − S0 (ui )exp(Xi β) ,

(3)

i=1

with β a p-vector of regression parameters and S0 (t) the baseline survival function.
Finkelstein (1986) extended the nonparametric approach of Turnbull to the proportional hazards model with interval-censored data. For this she assumed themodel expressed in (3). Note that likelihood (3) depends only on the baseline hazard through its values at the different observation time points. Let s0 = 0 < s1 <. . . < sK+1 = ∞ denote the ordered distinct time points of all observed time intervals bli , ui c (i = 1, . . . , n). Further, let αij = I{sj ∈ bli , ui c} (j = 1, . . . , K +1,

Lesaffre et al.

15

i = 1, . . . , n). To remove the range restrictions on the parameters for S0 , the likelihood is parameterized by γk = log[− log S0 (sk )] (k = 1, . . . , K + 1). Note that
because S0 (s0 ) = 1 and S0 (sK+1 ) = 0, γ0 = −∞ and γK+1 = ∞. In terms of βand γ = (γ1 , . . . , γK )> the log-likelihood function `(β, S0 ) can be written as
(K+1
)
n
h
i
X
X
−ζk−1 exp(Xi> β)−ζk exp(Xi> β)`(β, γ) =
log
αik e
−e
,
(4)
i=1

where ζk =

k=1

Pk

m=0 exp(γm ).

To estimate the mass of the regions of possible support and the regression parameters, Finkelstein proposed the Newton-Raphson algorithm. It turned out thatthe score equations provide a generalization of the self-consistency algorithm
suggested by Turnbull when β = 0. For the appropriateness of the asymptotic
χ2 -distribution for testing β = 0, it is assumed that K does not increase with thesample size. Farrington’s approach (1996) allows to fit the approach of Finkelsteinwith generalized linear model software. He also provided a technique to select a
subset of L < K significant time points sl (l = 1, . . . , L). Other approaches suggested for the PH model are: the piecewise exponential model (available in SAS
procedure ICPHREG) and a variety of smooth approaches some based on splinesmoothing.
Another popular semiparametric approach is to apply the partial likelihood approach on multiple imputed (MI) data sets. In the MI approach finite intervalcensored survival times are regarded as missing and replaced by a possible survival
time given an assumed model. Standard methodology can then be used to analyze
the (often between 3 and 10) imputed data sets of right-censored survival times.
The results of the multiple analyses are then combined. Pan (2000) proposed twosuch multiple imputation schemes assuming a particular distribution within theregions of support, but no other assumptions are made.
Finally, here again a parametric approach is easiest to handle, but could be too
restrictive in practice.
For the AFT model, the true survival times T1 , . . . , Tn are assumed to satisfy
Yi = log(Ti ) = Xi> β + εi(i = 1, . . . , n),(5)where εi are independent and identically distributed with density g(e). In caseof interval-censoredsurvival times the likelihood is given by (2) but now withS(t | X) = S0 exp(−X > β) t .
Several distributions have been suggested for g, but there exists no semiparametric version of the AFT model. While the parametric AFT model is the simplest tohandle it is often too restrictive. More flexible approaches are based on a smootherror density. One option is the Penalized Gaussian Mixture (PGM) model, whichassumes that the error density is a mixture of a (large) number of Gaussian densities with fixed means (knots) and with weights that are constrained by a penalty
term to produce a smooth density. This approach has been implemented in the R
package smoothSurv. In Figure 3 the solution from smoothSurv is comparedto the NPMLE for the emergence distribution of tooth 44 from boys.

16Interval Censoring

3.2

Bayesian approaches

For a long time, the Bayesian PH model could only be fit parametrically tointerval-censored observations making use of the statistical packages Win/OpenBUGS
and later with the SAS procedure MCMC. Unfortunately, a pure semiparametric approach does not seem to be possible here, but recently at least two flexible
modelling approaches have been proposed and implemented in software. Wanget al. (2013) proposed a Bayesian PH model with a piecewise constant baseline
hazard via a reversible jump MCMC procedure in combination with data augmentation. Their approach fits a dynamic survival model to the data, thereby
providing a check for the PH assumption. The method is implemented in the
R package dynsurv. The recently developed R package ICBayes is based on
fitting the baseline hazard in a smooth manner using integrated I-splines, see Lin
et al. (2015). To this end the relationship of the PH model with a latent nonhomogeneous Poisson process was used in combination with data augmentation.

1.0

^
h(t)

0.15

0.0

0.00

0.05

0.5

0.10

^
f (t)

0.20

0.25

1.5

0.30

0.35

The parametric Bayesian AFT
BUGS-like and SAS softDensitymodel can be fitted with
Hazardware in very much the same manner as the PH model. We are only aware of
the R package bayesSurv to fit a smooth AFT model in a Bayesian way. The
package is based on the reversible jump MCMC technique, but also a PGM as
an error distribution can be fitted. With the package DPpackage practitioners
have several programs at their disposal for fitting Bayesian AFT models in asemiparametric manner. In the package a Mixture of Dirichlet process prior isused to fit an AFT model for interval-censored observations, which is the basis
for the R function DPsurvint. This function was applied to examine the dependence of the emergence distribution of a permanent tooth (tooth 44) on genderof a child and the history of caries status of the predecessor deciduous tooth
84 expressed by its dichotomized DMF score (DMF=1, caries on the deciduous
tooth, 0 otherwise).6 Figure
shows
7
8
94 10
11 12the13 posterior
6 predictive
7
8
9 10 survival
11 12 13 function forthe different gender and DMF
combinations.
Age (years)
Age (years)

0.8
0.6
0.4
0.2
0.0

Proportion not yet emerged

1.0

Survival function

6

7

8

9

10

11

12

13

Age (years)

FIGURE 3. Signal Tandmobielr study (boys). Estimated survival function compared to NPMLE (dashed line) for the time to emergence of tooth 44 estimatedusing the penalized Gaussian mixture with R package smoothSurv.

Lesaffre et al.

417Multivariate models

When several outcomes are measured on a subject who is examined at regulartime intervals, we obtain multivariate interval-censored observations. With multivariate outcomes, it is natural to ask for the association between the outcomes.
Most of developments have been done for the bivariate case, i.e., when there are
two related survival times T1 and T2 measured in an interval censored manner.
A special case of bivariate interval-censored data are doubly interval-censoredtimes. In that case the T1 measures the onset of the time-at-risk and T2 ≥ T1
measures the time of the event, and again both T1 and T2 are interval-censored.

4.1

Frequentist approaches

Betensky and Finkelstein (1999a) generalized Peto’s and Turnbull’s argument tobivariate interval censored data. That is, information on the bivariate nonparametric survival function is limited to a number of rectangles bearing (possibly)
non-zero mass, again called the regions of possible support. The trigger to develop
the bivariate NPMLE of S for interval-censored observations, was the computation of the association between the two true survival times. However, it turned
out that the bivariate NPMLE is not a good basis for this because too dependent
on the amount right censoring in the data, see Betensky and Finkelstein (1999b).

0.6
0.4

Boy, DMF 0
Boy, DMF 1
Girl, DMF 0
Girl, DMF 1

0.0

0.2

Survival distribution

0.8

1.0

The absence of statistical software for fitting a rich class of (multi/bi)variate
models (for interval-censored data), restricts the use of parametric modelling for
multivariate responses. Instead one could use copula models, which disentangle the specification of the association structure and the marginal distributions.
The three popular copulas: the Clayton copula, the Gaussian copula and the
Plackett copula have been extended to bivariate interval-censored survival times
and are implemented in the function fit.copula from the R package icensBKL

6

7

8

9

10

11

12

13

Age at emergence tooth 44 (years)

FIGURE 4. Signal Tandmobielr study. Survival functions for emergence of permanent tooth 44 in gender by DMF groups based on a semiparametric Bayesianmodel with MDP priors, obtained with R function DPsurvint.

18

Interval Censoring(will accompany the BKL book). Even more flexible are the bivariate smoothing
techniques, such as the bivariate PGM model implemented in the SAS macro%smooth. This macro produced Figure 5 that shows a smooth approximationof the distribution of the true emergence times of the contralateral (left andright) maxillary first premolars (teeth 14 and 24) for boys, collected in the Signal
Tandmobielr study. One can observe that the two emergence times are highly
correlated.
For survival outcomes the association measures Spearman’s rank correlation,
Kendall’s tau and the global and local cross-ratio are in use. These measures
can be estimated by plugging in sample values in the population versions of theassociations. This can be done for parametric models, but when based on the
PGM approach a goodness-of-fit check for the parametric models is obtained.
To graphically explore the association structure of multivariate observations one
can use a biplot. On a biplot the original p-dimensional outcome is projected onto
2 (or 3) dimensions displaying individuals as points and variables as vectors. If the
2-dimensional plot captures most of the original variability, then the projections of
the points on the vectors provide useful visual information on the characteristicsof (groups of) individuals. The biplot has been extended to multivariate intervalcensored observations (Cecere et al., 2013) and implemented in the function ICBiplot of the package icensBKL.
Hierarchical models, called frailty models in the survival context, provide yet
another way to model multivariate interval-censored outcomes. Conditional ona random intercept, the outcomes are then assumed independent. This class ofmodels has been also extended to the interval-censored case. Again various illustrations of methodologies and software can be found in BKL.

f
0.2

0.1

0.1

0.0
14.9
13.5
12.1
10.8
Emergence time of tooth 14 (yrs)
9.4
8.1
6.7 6.7

8.1

9.4

14.9
13.5
12.1
10.8
Emergence time of tooth 24 (yrs)

FIGURE 5. Signal Tandmobielr study. Density of penalized normal mixture
model for emergence of permanent teeth 14 and 24 obtained from SAS using
macro %smooth.

Lesaffre et al.

4.2

19

Bayesian approaches

Parametric frailty models can be fit with standard Bayesian software such Win/OpenBUGS
and the SAS procedure MCMC. More challenging is to fit multivariate models
for interval-censored data in a semiparametric manner. A few approaches havebeen suggested to fit the frailty distribution in a flexible manner. The approachof Komárek and Lesaffre (2007) builds on the penalized Gaussian mixture idea.
Let (Ti1 , . . . , Tini )> be independent random vectors representing times-to-eventof the i-th cluster which are observed as intervals blil , uil c and Xil be the covariate vector for the lth observation in the ith cluster (i = 1, . . . , n; , l = 1, . . . , ni ).
In the random-effects AFT model, the (i, l)-th event time is expressed aslog(Til ) = Xil> β + bi + εil(i = 1, . . . , n; l = 1, . . . , ni ),(6)where εil are (univariately) i.i.d. random errors with a density gε and b1 , . . . , bn
are cluster-specific i.i.d. random-effects with a density gb . The approach thenconsists in expressing either of densities gε and gb as a univariate PGM. The
approach was implemented in the R package bayesSurv and illustrated withdata from the Signal Tandmobielr study. More specifically the software was used
to examine the impact of caries (now or in the past) of deciduous teeth and theirsuccessors. Another option is to use the package DPpackage, which provides
functions that allow for a multivariate semiparametric approach.

5

Discussion

The list of statistical approaches extended to deal with interval-censored data isendless. In fact, each statistical approach developed for fully observed or rightcensored data can be extended to interval-censored data. Additional topics thathave been investigated with interval-censored data: competing risks, multi-statemodels, interval-censored covariates, etc. We also omitted here the discussion of
doubly interval-censored observations, important for HIV/AIDS research.
Finally, the majority of the developments (if not all) have been done under the assumption of non-informative independent censoring. This assumption is violated
when the censoring intervals are associated with the actual and unobserved timeto-event. This may happen more often in practice than assumed, and may affectthe conclusions considerably. Developments that deal with informative censoring
are therefore desirable.
To conclude, there is no reason anymore to bypass interval censoring since there
is ample software available for a great variety of problems.

References
Betensky, R. A. and Finkelstein, D. M. (1999a). A non-parametric maximum likelihood estimator for bivariate interval censored data. Statistics in Medicine,18, 3089 – 3100.
Betensky, R. A. and Finkelstein, D. M. (1999b). An extension of Kendall’s coefficient of concordance to bivariate interval censored data. Statistics in
Medicine, 18, 3101 – 3109.

20Interval Censoring

Bogaerts, K., Komárek, A. Lesaffre, E. (2017). Survival Analysis with IntervalCensored Data: A Practical Approach with examples in R, SAS and BUGS.
Boca Raton: CRC/Chapman and Hall
Calle, M. L. and Gómez, G. (2001). Nonparametric Bayesian estimation frominterval-censored data using Monte Carlo methods. Journal of Statistical
Planning and Inference, 98, 73 – 87.
Cecere, S. and Groenen, P. J. F. and Lesaffre, E. (2013). The interval-censoredbiplot. Journal of Computational and Graphical Statistics, 22, 123 – 134.
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health. Lifetime Data Analysis, 19, 297 – 316.

Language comprehension as a multiple label
classification problem
R. Harald Baayen1 , Tino Sering1 , Cyrus Shaoul2 , Petar Milin3
1

Eberhard Karls University of Tübingen, Germany
Landmark College, USA
3
University of Sheffield, UK
2E-mail for correspondence: harald.baayen@uni-tuebingen.de
Abstract: The initial stage of language comprehension is a multi-label classification problem. Listeners or readers, presented with an utterance, need to discriminate between the intended words and the tens of thousands of other words
they know. We propose to address this problem by pairing a network trained withthe learning rule of Rescorla and Wagner (1972) with a second network trained
independently with the learning rule of Widrow and Hoff (1960). The first network has to recover from sublexical input features the meanings encoded in thelanguage signal, resulting in a vector of activations over all meanings. The second
network takes this vector as input and further reduces uncertainty about the intended meanings. Classification performance for a lexicon with 52,000 entries isgood. The model also correctly predicts several aspects of human language comprehension. By rejecting the traditional linguistic assumption that language is a(de)compositional system, and by instead espousing a discriminative approach(Ramscar, 2013), a more parsimonious yet highly effective functional characterization of the initial stage of language comprehension is obtained.
Keywords: multi-label classification, language comprehension, error-driven learning, Rescorla-Wagner, Widrow-Hoff
Table 1 presents 10 simple sentences. When reading these sentences, the lettersand their combinations succeed in bringing to the fore a small number meanings
while dismissing thousands of others as irrelevant. Sentences present the reader
with a multi-label classification problem.
We address this problem as follows. First, we represent the orthographic input bymeans of letter trigrams. For the first sentence, these are #Ma Mar ary ry# y#p#pa pas ass sse sed ed# d#a #aw awa way ay#(the#symbol represents the space character). Letter trigrams provide a much richer
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

22Language as multi-label classification

representation of the visual input than do orthographic words. For the data in
Table 1, there are n = 104 distinct letter trigrams, to which we refer as cues.
The second column lists the lexical meanings (lexomes) that are the targets ofclassification. Lexomes are pointers to locations in a high-dimensional semantic
vector space (defined below). Note that past-tense word forms such as passed(regular) and ate (irregular) are coupled with the lexomes pass and eat as wellas with past tense (past). Likewise, the two word forms apple and pie are coupledwith one lexome applepie, and the three expressions with the word forms kicked
the bucket, passed away, and died, are all linked with the same lexome die.
TABLE 1. Sentences, lexomes in the message, and frequency of occurrence (F ).
The total number of learning events is k = 771.

1
2
3
4
5
6
7
8
9
10

Sentence

Lexomes in the message

Mary passed away
Bill kicked the ballJohn kicked the ball away
Mary died
Mary bought clothes
for the ball
Ann bought a ballJohn filled the bucket
John kicked the bucket
Bill ate the apple pie
Ann tasted an apple

mary die pastbill kick past def balljohn kick past def ball away
mary die pastmary buy past clothes
for danceparty
ann buy past indef balljohn fill past def bucket
john die pastbill eat def applepie
ann taste past indef apple

F
40
100
120
300
20
45
100
10
3
33

Is it possible to discriminate between the targeted lexomes given the letter trigrams in the sentences? We will show that considerable headway can be made by
an error-driven incremental multi-label classifier that comprises two simple networks, each with only an input layer and an output layer. In what follows, we firstprovide a formal definition of the algorithm, and illustrate it for the sentences inTable 1. We then turn to a more realistic example in which lexomes targeted inaround a million of utterances have to be discriminated from some 52,000 otherlexomes.

1

An algorithm for multiple label classification

The problem of incremental learning of multi-label classification is defined by asequence of events at which a set of features (henceforth cues) are present andgenerate predictions about classes (henceforth outcomes), only some of which are
actually present in the learning event. The mismatch between predicted outcomes
and the outcomes actually present in a learning event provides the error driving
learning.
From a total of n distinct cues and m possible outcomes, only small subsets will
be present in a given learning event. Let k denote the number of unique learning
events (learning events may repeat, cf. good morning and tickets please). We index
a specific learning event in the sequence t (of length K ≥ k) of learning events by
t. The classification problem is defined by t, a sparse n × k cue matrix C which

Baayen et al.

23

is 1 whenever a given cue is present in a specific event and zero otherwise, and asparse m × k target matrix T that is 1 whenever an outcome is present and zerootherwise.
Classification proceeds in two steps, using two networks. The first network hascues as inputs and outcomes as outputs. It is defined by an m × n matrix W
of connection weights from cues (columns) to outcomes (rows). Given W, thepredicted support (henceforth activation) for a specific outcome given the cuesin the learning event is obtained by summation of the weights on the connections
from these cues to that outcome. The m × k activation matrix A specifies these
activations for all outcomes across all unique learning events:
A = WC.
The classification performance of this first network is assessed by checking whether
the outcomes with the highest activations are those of the targeted lexomes.
As shown by Danks (2003), if over a sequence of learning events no further changes
in the weight matrix take place other than the tiny increments and decrementsthat come with individual updates, i.e., when the weight matrix has entered astate of equilibrium, then, given the incremental learning rule of Rescorla andWagner (1972) (see below), W can be estimated straight from conditional probabilities characterizing the input. Let E specify pairwise conditional probabilitiesof cues given cues,
Pr(c0 |c0 ) Pr(c1 |c0 ) . . . Pr(cn |c0 ) Pr(c0 |c1 ) Pr(c1 |c1 ) . . . Pr(cn |c1 ) 
,
E=


...
...
...
...
Pr(c0 |cn ) Pr(c1 |cn ) . . . Pr(cn |cn )and let F denote a matrix specifying conditional probabilities of outcomes given
cues,


Pr(o0 |c0 ) Pr(o1 |c0 ) . . . Pr(on |c0 ) Pr(o0 |c1 ) Pr(o1 |c1 ) . . . Pr(on |c1 ) 
.
F=


...
...
...
...
Pr(o0 |cn ) Pr(o1 |cn ) . . . Pr(on |cn )Danks’ equilibrium equations state thatF = EWT ,which can be solved using the generalized inverse.
When a weight matrix is calculated in this way, the effect of the exact order oflearning events is lost. Furthermore, a Danks weight matrix dampens the consequences of the frequencies of occurrence of cues and outcomes in the input
space, while highlighting the contrasts that allow cues to discriminate between
outcomes. Thus, the Danks weight matrix is useful when there is no informationon the sequence of learning events (e.g., when only the frequency of learning
events is available but not their order) and when interest is directed specificallyto an idealised endstate of learning.
Preferably, the weight matrix W is estimated by repeated application of thelearning rule of Rescorla & Wagner (1972) to the learning events t. The update
at learning event t,
Wt = Wt−1 + ∆rw

24

Language as multi-label classification

depends on the learning rate η (typically set at 0.001) regulating the magnitudeof the changes to the weight matrix, on the predictions for the outcomes asgauged by the activations of these outcomes given the cues, and on whether theoutcomes are actually present in the learning event. Specifically, let c denote thetranspose of that column vector of C specifying which cues are present at thecurrent learning event t, and let o denote the transpose of that column vector of
T detailing which outcomes are present at t, and let J denote an m × n all-ones
matrix. Let the (row) vector a1 to specify the activations of those outcomes thatare present in the learning event while setting to zero the activations for all otheroutcomes:
a1 = (((J · o)T · c)T · W)i.
Here, i is a row unit vector of length n. Note that ((J · o)T · c)T is 1 for all cueoutcome combinations that are present in the learning event, and zero elsewhere.
Next, let the (row) vector a0 represent the activations of those outcomes notpresent in the learning event, again given the cues in that learning event, and let
it be zero for all other outcomes:
a0 = (((J · [1 − o])T · c)T · W)i.
((J · [1 − o])T · c)T is 1 for all cue-outcome pairs where the cue is present but theoutcome not, and zero elsewhere. The update to the weight matrix, ∆rw , cannow be defined as follows:∆rw = η{((J · o)T · c)T · (1 − a1 ) − ((J · [1 − o])T · c)T · a0 }.
For cue-outcome pairs that are both in the learning event, the update of theirweight is given by the difference from the maximal activation, 1 by definition.
As the summed activations a1 tend to be less than 1, weights will be strengthened. For cue-outcome pairs where the cue is present but the outcome is not,the corresponding connection weight is decreased by the summed activations a0 .
Estimation of W using incremental updating over the sequence of learning events
is fast, first because only parts of the weight matrix require updating (efferent
weights from cues not present in the learning event are left untouched), and also
because the updates to individual outcomes are independent and hence allow forparallelization.
The activation matrix A = WC specifies, for each unique learning event and foreach outcome, the joint support provided by the cues in that learning event forthat outcome.
Although class predictions based on A can do well for small constructed data sets,
they lack precision for large real data sets. Prediction accuracy can be further
improved by a second network that is given the task to predict the target T fromthe activation matrix A:
T = DA.
The prediction matrix
P = DA
is the resulting approximation of T. Although D can be calculated using thegeneralized inverse of A, computation costs can be prohibitive for large numbers

Baayen et al.

25of learning events. It is therefore preferable to estimate D as follows:
T

=

DA

TAT

=

DAAT

Y

=

DX,

which leads to D = YX−1 . Since X is m × m, and since generally m  k, computational costs are much lower when calculating X+ as compared to calculating
A+ .
The prediction matrix can also be estimated iteratively by means of the update
rule of Widrow and Hoff (1960). This update rule, which specifies the update∆wh to the m × m second weight matrix D, is important, first, as it allows us toassess the consequences of how the order of learning events affects classification,and second, because for large numbers of training events (in the order of hundredsof millions), it is not feasible to actually calculate A (and P).
Let Z denote an m × m matrix initialized with zeroes, let a denote the column
vector of the activation matrix A giving the predicted activations for the current
learning event, and let o denote the transpose of the corresponding column vectorof the target matrix T. The Widrow-Hoff update to Z is:∆wh = η{a(o − aT Z)}.
We take the transpose to obtain D = ZT .
The weights for the two networks (m × n for the Rescorla-Wagner network, andm×m for the Widrow-Hoff network) can be estimated in two ways. One possibility
is to first estimate W and then estimate D. Alternatively, one can update both
networks in tandem for each successive learning event. In this case, it is notnecessary to calculate A. Note that when estimating
P = (WC)+ TWC
we ‘inject’ error twice: once during the estimation of W and again during theestimation of P.
The equilibrium equations are implemented in the ndl package for R on cran.
An efficient Python implementation for incremental learning of W is available atgithub.com/quantling/pyndl. An implementation of incremental learning for R
is available (for linux only) upon request from the authors. Software for efficient
updating of D by Widrow-Hoff is currently under development.
Returning to the example of Table 1, first consider classification performance
when W and D are estimated independently, using incremental updating for theformer, and the generalized inverse for the latter. In this case, for each of the 10
sentences, the lexomes in that sentence have the highest prediction values in P.
When the two networks are updated in tandem, with at each learning event firstan update of W and then an update of D, accuracy varies with the (random)order in which the 771 learning events are made available to the model. For one
such random order, the proper lexomes had the highest ranks in A for 9 out of
10 sentences. The one sentence with an error is John kicked the bucket, wheredef (the lexome for the definite article) intrudes with a higher activation before
die, which is found at the next rank (4).

Language as multi-label classification

200

400

t

600

800

1.0
0.8
0.2
0.0

0.0

0.0
0

KICK
BUCKETDIE
APPLE
APPLEPIE

0.6

prediction

0.6
0.4

prediction

0.8

KICK
BUCKETDIE
APPLE
APPLEPIE

0.2

0.4

0.6

0.8

KICK
BUCKETDIE
APPLE
APPLEPIE

0.2

prediction

P

1.0

P

1.0

A

0.4

26

0

200

400

t

600

800

0

2000

4000

6000

8000

t

FIGURE 1. Prediction strengths for selected lexomes in the learning events ofsentences 8 and 9 in Table 1, using incremented coupled Rescorla-Wagner andWidrow-Hoff. Left and center panels: frequencies as in the table; right panel:frequencies increased tenfold.
Figure 1 illustrates this incremental training regime. The left and center panels
show the predictions based on A and P when training proceeds on a random
order of 771 learning events, and the right panel when training proceeds on 7710
learning events. Solid lines represent key lexomes from sentence 8 in Table 1: kickand bucket for the unintended literal reading and die for the intended idiomatic
reading. Dashed lines represent the competitors apple and applepie in sentence
9. The spiky behavior in the left and center panels reflects the learning andunlearning that unfolds as outcomes competing for the same cues are encountered.
Comparison of the left and center panels shows that the Rescorla-Wagner network
learns much faster than the Widrow-Hoff network. By the end of the learning
sequence, the former, but not the latter network succeeds in giving the intended
lexomes higher prediction scores. The rightmost panel shows that with sufficientexperience, the model learns that kick the bucket means die, and that an apple
pie is not an apple but a particular kind of pie.
An important property of this approach to language comprehension is that thecorrect lexomes are selected without any worries about regular or irregular verbs,
literal versus idiomatic expressions, finding boundaries between words, decomposing words into parts, or disambiguating homographs. Given the assumptionthat understanding drives the recalibration of weights, the rich information available in the combinatorics of sublexical cues and lexomes is sufficient for multiple
label classification to be effective.

2

Multiple label classification with 52,000 classes

To clarify whether this approach scales up, we applied our algorithm to the tasa
corpus (Zeno, 1995), a collection of texts comprising a total of 10,807,146 words
representing 52,401 word types. Lemmatization was carried out with TreeTagger(www.cis.uni-muenchen.de/∼schmid/tools/ TreeTagger/), which distinguished
90,339 lemmata, of which 37,938 occurred once. To keep computations tractable,
the model was trained on all words occurring at least twice and 351 hapax legomena that occurred in a precompiled list of words. Hapax legomena that were notincluded were replaced by the dummy word HAPAX, resulting in a total of 52,401

Baayen et al.

27

lexomes. Learning events were sentences in the tasa corpus. Sequences of morethan 8 words were split at the next available occurrence of and or or. This resulted
in a total of 992,752 learning events. The multi-label classification challenge is topredict the appropriate lexomes (out of 52,401) given the letter trigrams of the(possibly inflected) words in the learning events.
Using the ndl2 package for R, W (52,401 lexomes × 11,724 letter trigrams)
was estimated using all learning events. To keep computations tractable for thesecond network, two learning events were selected randomly from a precompiled
list of 8866 targeted lexomes, resulting in a total of 17,455 learning events (in 276
cases there was overlap with two or more lexomes in the same event, and for one
word, there was only 1 learning event available). The total number of outcomesin this subset of learning events was 19,020. With these restrictions, the matrices
A (19,020 lexomes × 17,455 learning events), D (19,020 × 19,020 lexomes) andP (19,020 lexomes × 17,455 events) could be estimated straightforwardly.

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0.2

0.8
0.6
0.4

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0.2

ratio intruders/targets

0.8 1.0

isolated words

1.0

utterances

●

0.0

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●

0.2

●

●

0.4

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0.8

deciles 0 through 0.9

1.0

1

10

100

1000

10000

rank

FIGURE 2. Left: Quantiles of the ratio of intruders (false positives) to targets(correct identifications), full utterances. Right: Rank and corresponding cumulative proportion based on A (red) and P (blue), isolated words.
The left panel of Figure 2 presents the ratio of intruders (lexomes with an activation exceeding that of the least activated target lexome) to the number oftargeted lexomes. The median number of intruders is zero, at the 8th decile theratio is 0.17, and at the 9th, it is 0.33. At the 10th decile, we find cases with vast
numbers of intruders, leading to a maximal ratio of 1208.9. Examples of intruders are down for the sentence The aleuts were housed in abandoned rundown gold
mines or fish canneries, and field and success for the sentence He is an ecologist
who studied succession in abandoned cornfields.
We also tested identification performance when target lexomes were presented inisolation. The right panel of Figure 2 plots in blue cumulative proportion (out of a
total of 7179) against rank based on P: 34% of lexomes had the highest prediction
value, 88% of the targeted lexomes had at most a rank of 16 (indicating 15
intruders with higher activations). As show by the red curve, performance basedon A instead of P is substantially worse. Human lexical decision performance, as
gauged using the British Lexicon Project (blp, Keuleers et al. 2012) was for thepresent data at 90% correct. As the lexical decision task does not require actual
identification, but only sufficient evidence for lexicality, it appears that humansubjects tolerate around 16 intruders.
As shown in Figure 3, the model also predicts power-transformed lexical decision

28

Language as multi-label classification

response times (t0 = −1000t−1 ). For all but the first decile, log activation ai =Wci (with c the vector specifying the present and absent cues in the input, andi indexing a specific lexome) shows a nearly linear effect with negative slope. Log
rank prediction (the log rank of pi = DWci ) has a smaller effect that is again
negative and nearly linear, but now for the first nine deciles. The 90% decile of
the rank is at 18, which is close to the cut-off at rank 17 for lexicality decisions in
the right panel of Figure 2. Apparently, the same range of ranks influences bothdecisions and reaction times.

4
2
0
−4

−2

partial effect (−1000/t)

4
2
0
−2
−4

partial effect (−1000/t)6

edf 4.2, p < 0.0001

6

edf 6.9, p < 0.0001

−12

−10

−8

−6

−4

log activation

−2

0

0

2

4

6

8

log P rank

FIGURE 3. Partial effects in a gam fitted to power-transformed (−1000t−1) reaction times. Left: log activation; Right: log prediction rank. Vertical lines denotedeciles. The 90% decile of log prediction rank is at rank 18 (red lines indicate
deciles). Regression analyses were carried out with gams (Wood, 2006).
PT defines a semantic vector space (cf. Landauer & Dumais, 1997), and lexomes
are indices or pointers for locations in this space. By way of illustration of thesemantic nature of PT , the left panel of Figure 4 presents partial effects forhuman semantic similarity ratings for word pairs (Bruni et al., 2014) as predictedfrom correlations of the corresponding column vectors of PT (left). For 90% of
the data points, a nearly linear relation is observed. Clearly, extreme values areunreliable as predictors. Similarity in PT -space, i.e., similar prediction values
across events and thus greater similarity of experiences communicated, correctlypredicts greater perceived semantic similarity.
The column vectors of D also define a lexomic space, but similarities in this spaceturn out to be positively correlated with the Levenshtein distance between theorthographic forms of the two words. As shown in the right panel of Figure 4, themore different two word forms are, the lower their perceived semantic similarity.

3

Concluding remarks

Multi-label classification is a hard problem, not only for statistics, but also forhumans. For instance, in auditory word recognition, isolated words taken fromconversational speech have recognition rates between 20% and 40% (Arnold et al.,2017). In the visual lexical decision task, undergraduate students perform nearchance on the lower-frequency words (Baayen et al., 2017). From this perspective,
the model’s performance, with training on a mere 10 million words, is too good
to be true. This is, of course, due to the model being given perfect feedback,Baayen et al.

29

whereas human learning tends to proceed under uncertainty and lack of fullunderstanding.
Given that the model presents a simplified perspective on the first stage of comprehension — understanding the words — several of its features are remarkable.
First, the traditional linguistic assumption that language is a (de)compositional
system is replaced by a perspective in which the language signal is a code that
discriminates between possible messages (Ramscar 2013, Shannon, 1956).
Second, the model is parsimonious with only one free parameter, the learning
rate η. And although W and D can be very large, most of the weights are close
to zero. E.g., for W, only 5,885 weights exceed 0.1 (0.00058% of the total number
of weights), and only 195 weights are greater than 0.5. Arnold et al. (2017) show
for auditory comprehension that W can be pruned down to a fraction of theoriginal weights without noticeable loss of accuracy.
Third, the classifier implements a three-layer network that differs from backpropagation networks in that there is direct error injection twice, once for W
using the Rescorla-Wagner equations, and once for D, using Widrow-Hoff (or thegeneralized inverse). Importantly, the power of the first network should not be
underestimated. Although ever since the criticism of the perceptron by Minsky& Papert (1972), two-layer networks have been regarded as far too restricted
for any classification tasks requiring more than the simplest linear separation,
it turns out that actually, with an appropriate choice of cues, Rescorla-Wagner
networks can solve much more interesting problems. Figure 5 illustrates this fora simple example with two classes (represented by gray and red points) that in
R × R are not linearly separable (left panel). When the data are re-represented
by identifiers for rows and columns (right panel), a Rescorla-Wagner network
correctly predicts the highest activations for around 210 of the 260 elements of
the red class (see Baayen and Hendrix, 2017, for detailed comparison with othermachine learning classifiers, and also Ghirlanda, 2005).
Fourth, more sophisticated features than letter trigrams can be used as cues,
such as the frequency band summary features used by Arnold et al. (2017) for
modeling auditory word recognition, and for reading the histogram of oriented

20
−10

0

10

partial effect

20
10
−10

0

partial effect30

edf = 5.9, p < 0.0001

30

edf = 8.0, p < 0.0001

−0.1

0.0

0.1

0.2

0.3

cor(t(P))

0.4

0.5

0.6

−0.8

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

cor(D)FIGURE 4. Partial effects of the correlations of row vectors of P (left) andcolumn vectors of D as predictors of human similarity ratings for 2,369 wordpairs. Red vertical lines indicate 5% and 95% percentiles. Regression analyses
were carried out with gams (Wood, 2006).

30

Language as multi-label classification

FIGURE 5. A non-linearly separable classification problem with a majority classin gray (2240) and a minority class in red (260). Left: data points in a Cartesian
grid (x = 1, 2, . . . , 50; y = 1, 2, . . . , 50). Right: rerepresentation with row andcolumn identifiers as cues for a Rescorla-Wagner network: hits in blue, misses
and false alarms in red.
gradients feature descriptor proposed by Dalal and Triggs (2005).
Finally, the model is transparent to interpretation. W specifies the support provided by sublexical features for lexomes. D transforms activation vectors thatare still strongly influenced by form similarity into vectors closer to the targeted
lexomes, which in turn results in a semantic vector space, PT .

References
Arnold, D., Tomaschek, F., Sering, K., Lopez, F., and Baayen, R.H. (2017). Words
from spontaneous conversational speech can be recognized with human-likeaccuracy by an error-driven learning algorithm that discriminates between
meanings straight from smart acoustic features, bypassing the phoneme asrecognition unit. PLOS-ONE, 12, e0174623.
Baayen, R. H., and Hendrix, P. (2017). Two-layer networks, non-linear separation, and human learning. In Wieling, M., Kroon, M., van Noord, G., andBouma, G. (Eds.) From Semantics to Dialectometry. Festschrift in honorof John Nerbonne. London, College Publications, 13–22.
Baayen, R. H., Tomaschek, F., Gahl, S., and Ramscar, M. (2017). The Ecclesiastes principle in language change. In Hundt, M., Mollin, S., and Pfenninger, S. (Eds.) The changing English language: Psycholinguistic perspectives. Cambridge, Cambridge University Press.
Bruni, E. and Tran, N.K. and Baroni, M. (2014). Multimodal distributional semantics, Journal of Artificial Intelligence Research, 49, 1–47.
Dalal, N. and Triggs, B. (2005). Histograms of oriented gradients for human detection. In CVPR’05, volume 1, 886–893.

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Danks, D. (2003). Equilibria of the RescorlaWagner model. Journal of Mathematical Psychology, 47, 109–121.
Ghirlanda, S. (2005). Retrospective revaluation as simple associative learning.
Journal of Experimental Psychology: Animal Behavior Processes, 31, 107–
111.
Keuleers, E. and Lacey, P. and Rastle, K. and Brysbaert, M. (2012). The British
Lexicon Project: Lexical decision data for 28,730 monosyllabic and disyllabic English words, Behavior Research Methods, 44, 287–304.
Landauer, T.K. and Dumais, S.T. (1997). A Solution to Plato’s Problem: The
Latent Semantic Analysis Theory of Acquisition, Induction and Representation of Knowledge. Psychological Review, 104, 211–240.
Minsky, M. and Papert, S. (1969). Perceptrons: An introduction to computational
geometry. Cambridge, MA: The MIT Press.
Ramscar, M. (2013). Suffixing, prefixing, and the functional order of regularities
in meaningful strings. Psihologija, 46, 377–396.
Rescorla, R. A. and Wagner, A. R. (1972). A theory of pavlovian conditioning:
Variations in the effectiveness of reinforcement and nonreinforcement. In:
A. H. Black and Prokasy, W. F. (Eds.), Classical conditioning II: Current
research and theory. New York: Appleton Century Crofts.
Shannon, C. E. (1956). The bandwagon, IRE Transactions on Information Theory, 2, 3.
Widrow, B. and Hoff, M. E. (1960). Adaptive switching circuits. 1960 WESCON
Convention Record Part IV, 96–104.
Wood, S. (2016). Generalized additive models. New York: Chapman & Hall.
Zeno, S.M. and Ivens, S.H. and Millard, R.T. and Duvvuri, R. (1995). The educator’s word frequency guide. New York: Touchstone Applied Science.

Part II – Contributed Papers

The Multivariate Method of Simulated
Quantiles for Portfolio Optimisation
Paola Stolfi1 , Mauro Bernardi2 , Lea Petrella3
1

Department of Economics, Roma Tre University, Italy
Department of Statistical Sciences, University of Padova, Italy
3
MEMOTEF Department, Sapienza University of Rome, Italy2E-mail for correspondence: paola.stolfi@uniroma3.it
Abstract: The sparse multivariate method of simulated quantiles is proposed
as a likelihood–free alternative to indirect inference procedures that does notrely on an auxiliary model specification. We extend the asymptotic theory andwe show that the sparse–MMSQ estimator enjoys the oracle properties under
mild regularity conditions. The method is applied to estimate the parameters of
the Skew Elliptical Stable distribution with the aim of finding optimal portfolios
within a Mean–Value–at–Risk objective function.
Keywords: Projectional quantiles; Sparse regularisation; Skew Elliptical Stable
distribution; Portfolio optimisation.

1

Introduction

In this paper we extend the method of simulated quantiles (MSQ) of Dominicy
and Veredas (2013) to a multivariate framework (MMSQ). The MMSQ is thenapplied to estimate the parameters of the distribution of a portfolio of asset returns with the purpose of allocating a fixed amount of preexisting wealth amongalternative risky assets. The asset allocation problem requires the prior selectionof an appropriate distribution for modelling the multivariate structure of financial returns being characterised by the presence of skewness, heavy–tails andpositive tail–dependence which may prevent the existence of the moments. The
Skew–Elliptical Stable distribution (SESD) introduced by Branco and Dey (2001)
extends the elliptical Stable distribution using the skewing mechanism of Azzaliniand Dalla Valle (1996) and it is particularly useful to model skewed and heavy–
tailed continuous data. Despite their prominent role in the financial literature,
multivariate Stable distributions have not been massively employed because ofThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

36

MMSQ for Portfolio Optimisation

the inferential issues that can arise when fitting these distributions to data. The
lack of a closed form expression for the density function as well as moments finiteness prevent either the use of valid likelihood–based inferential procedures andthe moment–matching estimators. Furthermore, the inferential problem becomeseven more relevant as the dimension of the distribution increases. The method of
simulated quantiles (MSQ), like alternative likelihood–free procedures, is basedon the minimisation of the distance between appropriate quantile–based statistics
evaluated on the true and simulated data. Specifically, the MSQ estimates thevector of unknown parameters ϑ by solving the following minimisation problem0ϑ̂ = arg min Φ̂y − Φ̃RWϑ Φ̂y − Φ̃R
(1)
ϑ
ϑ ,
ϑ

where Φ̂y is the vector of quantile–based statistics evaluatedavailable samPRon ther1
ple of observations y = (y1 , y2 , . . . , yT ), while Φ̃R
ϑ = R
t=1 Φ̃ϑ is the averageof the vector of quantile–based statistics Φ̃rϑ based on the r–th vector of simulations from the postulated model yr = (y1r , y2r , . . . , yTr ) with r = 1, 2, . . . , R, andWϑ is a positive definite weighting matrix. The success of the method strongly
depends on a careful selection of the vector of quantile–based measures Φ evenin the univariate setting where quantiles are well defined. The lack of a naturalordering in the multivariate setting instead requires an accurate definition of theconcept of quantile. Here, we rely on the notion of projectional quantile recently
introduced by Hallin et al. (2010) and Kong and Mizera (2012). This notion ofmultivariate quantile makes the estimator flexible and it allows us to deal with
non–elliptically contoured distributions.
An issue frequently observed in high dimensions is the curse of dimensionality,i.e., the situation where the number of parameters grows quadratically or exponentially with the dimension of the problem. In those circumstances, the right
identification of the sparsity pattern becomes crucial since it reduces the numberof parameters to be estimated. Those reasonings motivate the use of sparse estimators that automatically shrink to zero some parameters, such as, for example,
the off diagonal elements of the variance–covariance matrix. In this paper, we
penalise the quadratic objective function of the MMSQ by adding a smoothly
clipped absolute deviation (SCAD) `1 –penalisation term that shrinks to zero theoff–diagonal elements of the scale matrix of the postulated distribution. We extend the asymptotic theory in order to accommodate sparse estimators, and we
prove that the resulting sparse–MMSQ estimator enjoys the oracle properties of
Fan and Li (2001) under mild regularity conditions.
The remainder of the paper is organised as follows. Section 2 briefly describes
the MMSQ and its sparse extension. Section 3 introduces the multivariate Skew–
Elliptical distribution and discusses some of its properties which are useful forthe application to the portfolio optimisation problem. We apply the MMSQ tothe portfolio optimisation problem in Section 4. Portfolio optimisation has a long
tradition in finance initiated by the Markowitz’s (1952) seminal paper that introduced the mean–variance (MV) approach. The MV approach relies on quite
restrictive conditions about the underlying distribution of asset returns that are
relaxed here by assuming that financial returns follow a multivariate SESD. Moreover, since these distributions do not admit finite second moment, we consider
a portfolio allocation problem where the expected return is traded–off against
higher Value–at–Risk (VaR) profiles that make investment less attractive.

Stolfi et al.

2

37

The multivariate method of simulated quantiles

In this section we briefly describe the MMSQ. To this aim, we first introduce
the following definition of projectional quantiles. Let Y ∈ Rd be random vector,u ∈ Sd−1 be a vector in the unit sphere and τ ∈ (0, 1), then the τ u projectional
quantile of Y is defined asq τ u ∈ arg min Ψτ u (q) ,
(2)
q∈R

where
h
i
Ψτ u (q) = E ρτ u0 Y − q ,(3)and ρτ (z) = z τ − I(−∞,0) (z) . Clearly the τ u–projectional quantile is the τ –quantile of the univariate random variable u0 Y. This feature makes the definitionof projectional quantile particularly appealing in order to extend the MSQ to themultivariate setting because, once the direction is properly chosen, it reduces tothe univariate quantile. The multivariate method of simulated quantiles is now
introduced. Let FY (·, ϑ) be the distribution function of Y which depend on avector of unknown parameters ϑ ⊂ Θ ∈ Rk and let:(i) qτϑ u = (qϑτ1 u , qϑτ2 u , . . . , qϑτs u ) be a s × 1 vector of projectional quantiles atgiven confidence levels τi ∈ (0, 1) with i = 1, 2, . . . , s, and u ∈ Sd−1 ;(ii) Φu,ϑ = Φ (qτϑ u ) be a b × 1 vector of quantile functions assumed to becontinuously differentiable with respect to ϑ for all Y and measurable for
Y and for all ϑ ⊂ Θ;(iii) q̂τ u = (q̂ τ1 u , q̂ τ2 u , . . . , q̂ τs u ) and Φ̂u = Φ (q̂τ u ) be the corresponding sample counterparts;
and assume that Φu,ϑ cannot be computed analytically but it can be empiricallyestimated on
data. At each iteration j = 1, 2, . . . the MMSQ compute
Psimulated
R
r
1
Φ̃R
Φ̃
,
where Φ̃ru,ϑj is the function Φu,ϑ computed at the r–thu,ϑj = R
u,ϑ
r=1
j


simulation path from FY ·, ϑ(j) . The parameters are subsequently updated byminimising the distance between the vector of quantile measures calculated on
the true observations Φ̂u and that calculated on simulated realisations Φ̃Ru,ϑj .
The subscript u denotes that those quantities depend on a set of directions, thatshould be selected in such a way that they fully characterise the feature thecorresponding parameter of the distribution identifies. We establish consistency
and asymptotic normality of the proposed estimator. The MMSQ estimator isthen extended in order to achieve sparse estimation of the scaling matrix. Specifically, the SCAD `1 –penalty of Fan end Li (2001) is introduced into the MMSQ
objective function as follows

0


X
ϑ̂ = arg min Φ̂u − Φ̃RWϑ Φ̂u − Φ̃Rpλ (|σij |) ,(4)u,ϑu,ϑ + nϑi<jwhere Wϑ is a b × b symmetric positive definite weighting matrix, Σ = (σij )di,j=1
is the scale matrix and pλ (·) is the SCAD `1 –penalty:(aλ − γ)p0λ (γ) = λ I(γ≤λ) +I(γ>λ)(5)(a − 1) λ38

3MMSQ for Portfolio Optimisation

Multivariate Skew Elliptical Stable distribution

Here, we introduce the definition of Skew–Elliptical Stable distribution introduced by Branco and Dey (2001). Specifically, we consider a slightly different
parameterisation from Branco and Dey (2001), having the interesting property
that the diagonal elements of the scale matrix do not affect the overall skewness
0of the distribution. Let (X0 , Y ) be a Normal random vector of dimension (d + 1)
conditional on the latent factor ζ ∼ S α2 (ω̄α , 1, 0) where S α2 (ω̄α , 1, 0) denotes the2αunivariate totally right–skewed α–Stable distribution with scale ω̄α = cos πα2and shape parameter α ∈ (0, 2), i.e.
X| ζ ∼ Nd+1 (0, ζΩδ ) ,
(6)
Y


Ω̄ δ
where Ωδ =and Ω̄ is a proper correlation matrix, symmetric and0δ 1
positive definite with |σij | < 1, for i, j = 1, 2, . . . , d and i 6= j. Then the
random variable Z = (X | Y > 0) is Skew Elliptical Stable distributed, i.e.,
Z ∼ SESDd α, 0, Ω̄, δ with densityfZ z, α, 0, Ω̄, δ = 2

Z +∞
0


φd z, 0, ζ Ω̄ Φ1



λ0 z
√
ζ


h (ζ) dζ,(7)

where φd (·) and Φ1 (·) denote the density of the multivariate Normal distribution
and the cumulative density function of the univariate Normal distribution, respec− 1tively, h (ζ) is the density of the mixing variable and λ = 1 − δ 0 Ω̄−1 δ 2 Ω̄−1 δ ∈Rd denotes the vector of skewness parameters. Furthermore, the transformation
Y = ξ + ωX where ω = diag {ω1 , ω2 , . . . , ωd } is Y ∼ SESDd (α, ξ, Ω, δ). Amongthe attractive properties of the SESD, the closure under linear combinations isof relevance for the portfolio optimisation problem discussed in the next Section.

4

Portfolio application

We consider a portfolio allocation problem, where, at each time t = 1, 2, . . . , T ,the investor’s wealth allocation is based on the choice of the vector of optimal
portfolio weights wt > 0 by minimising the following objective function
arg minwt −Et (wt0 Yt+1 ) − κVaRλt (wt0 Yt+1 ) , s.t. wt0 1 = 1,(8)where Yt ∼ SESD (α, ξ, Ω, δ), Et (wt0 Yt+1 ) and VaRλt (wt0 Yt+1 ) denote the portfolio’s expected return and the portfolio Value–at–Risk at level λ ∈ (0, 1) evaluated at time t for the period (t, t + 1], respectively. Here, κ ≥ 0 denotes theinvestor’s risk aversion parameter: the larger κ, the higher is the penalisation for
the risk profile of the selected portfolio. The empirical application is structured
as follows. We consider a basket of weekly returns of seventeen MSCI European
indexes, covering the period from January 6th, 1995 to November 25th, 2016.
Then, for each week, from April 23, 2010 to the end of the sample period, we
estimate the SESD parameters using a rolling windows of n = 800 observations

Stolfi et al.

39

1.0
0.10
0.0

0.50

−0.10
0.0

−0.20
−0.30
04/10 01/11 09/11 05/12 02/13 10/13 06/14 02/15 11/15 07/16−0.50
04/10 01/11 09/11 05/12 02/13 10/13 06/14 02/15 11/15 07/16
1

0.1
0

0.05
0
04/10 01/11 09/11 05/12 02/13 10/13 06/14 02/15 11/15 07/16−1
04/10 01/11 09/11 05/12 02/13 10/13 06/14 02/15 11/15 07/16

FIGURE 1. Mean–VaR0.95 optimal portfolio results over the period from April
23th, 2010 to the November 25, 2016. Figures plot the optimal portfolios cumulative returns for different values of the risk aversion parameter κ in the top panel
and the optimal portfolio weights for κ = 10 in the bottom panel. The dottedthinned brown line represents the equally weighted portfolio cumulative returns
which has been added for comparison.
for the MMSQ and of n = 200 for the Sparse–MMSQ. The optimal tuning parameters of the SCAD penalty are selected by K–fold cross validation, with K = 5.
As regards the portfolio allocation exercise, for each window, we solve equation(8) for the vector of optimal allocations wt , where the portfolio expected returns
and VaR are calculated exploiting the closure property with respect of the linear−1 0
wt ωδ ,
combination of the SESD: Zt+1 = wt0 Yt+1 ∼ SESD α, wt0 ξ, wt0 Ωwt , ωZ
1where ωZ = (ΩZ Ih ) 2 and denotes the Hadamart multiplication. Moreover,the VaR confidence level is fixed at λ = 0.99 and several levels of investors’ risk
aversion are considered κ = {0.10, 0.5, 1.0, 2.0, 5.0, 10.0}. We present the resultsof the empirical portfolio performance evaluation. To this end, we forecast theone–step ahead conditional returns’ distribution over the whole sample period.
The sequence of predictive distributions delivered by the competing models, arethen used to build the mean–VaR optimal portfolios with and without the shortselling constraint in equation (8). Figure 1, reporting the cumulative returns for
the MMSQ (on the left) and the Sparse–MMSQ (on the right), evidence that,as the risk aversion coefficient κ increases from κ = 0.1 to κ = 10, cumulative
returns increase as well. This evidence is stronger for the Sparse–MMSQ meaning that the shrinkage effect induced by the estimation method have a positive
impact on the estimation of the scale matrix and, as a consequence, the portfolio results greatly benefited from a better estimate of the dependence structure
among assets. The bottom panels of Figure 1 plot the evolution over time of the
optimal weights. Optimal weighs for the Sparse–MMSQ are characterised by amarked heterogenous behaviour while those implied by the MMSQ are flat anddisplay lower levels of diversification.

References
Azzalini, A. and Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika, 83(4):715–726.
Branco, M.D. and Dey, D.K. (2001). A general class of multivariate skewelliptical distributions. J. Multivariate Anal., 79(1):99–113.

40

MMSQ for Portfolio Optimisation

Dominicy, Y. and Veredas, D. (2013). The method of simulated quantiles. J.
Econometrics, 172(2):235–247.
Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihoodand its oracle properties. J. Amer. Statist. Assoc., 96(456):1348–1360.
Hallin, M., Paindaveine, D., and Šiman, M. (2010). Multivariate quantiles andmultiple-output regression quantiles: from L1 optimization to halfspace
depth. Ann. Statist., 38(2):635–669.
Kong, L. and Mizera, I. (2012). Quantile tomography: using quantiles withmultivariate data. Statist. Sinica, 22(4):1589–1610.
Markowitz, H.M. (1952). Portfolio selection. Journal of Finance, 7:77–91.

Gaussian Process model for evolving 3D lip
curves
Irene Mariñas1 , Adrian Bowman1 , Vincent Macaulay1
1

School of Mathematics and Statistics, University of Glasgow, UK

E-mail for correspondence: i.marinas.1@research.gla.ac.uk
Abstract: This paper addresses the problem of estimating a ridge curve embedded in a three-dimensional surface that changes over time. The main challenge is
to exploit the details of surface shape, while maintaining computational feasibility. A Gaussian Process approach is adopted.
Keywords: Gaussian Processes; Shape analysis; 3D curves; Lip morphology

1Introduction and motivation

FIGURE 1. Three-dimensional curves representing an upper lip, embedded in athree-dimensional facial image, during the emotion Disgust, at time points 3, 13
and 37.
This study is motivated by the shape of the lips in a three-dimensional facial
image (surface) and their variation over the expression of different emotions suchas disgust, fear, anger, happiness, et cetera (Figure 1). To record the expressions,a large number of pictures of a person producing the emotion are taken witha stereophotogrammetric camera system, which leads to a set of data in four
dimensions (the three spatial dimensions plus time). The statistical analysis of
information on shape has been a research topic of considerable interest since theearliest part of the twentieth century, but it has developed substantially in thepresent century especially thanks to advances in computational tools. Interest in
shape analysis of the human face began because of its applications in biology,
medicine and psychology.
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

422

GPs for 3D curves evolving over time

Gaussian Process model for three-dimensional
curves evolving over time

A ridge curve embedded in a three-dimensional surface can be expressed in termsof a continuous index, i.e. its arc-length (rescaled to be from 0 to 1), s ∈ [0, 1],
and the discrete label (i.e. coordinate), c ∈ {x, y, z} (Mariñas et al. 2016). If thesurface is, moreover, changing over time, a time component t ∈ R can be added
to the model. A GP can be then specified as:w(t, c, s) ∼ GP m(t, c, s), k(t, t0 , c, c0 , s, s0 ) .
(1)
Let s = (s1 , ..., sn )T for a choice of n values of s. Each coordinate can be
represented as a function of the arc-length and the time: w(t, x, s) = x(t, s),w(t, y, s) = y(t, s) and w(t, z, s) = z(t, s) (Figure 2).

FIGURE 2. 3D upper lip curves, from the emotion Disgust. Each coordinate
represented as function of the arc-length, at time points 3, 13 and 37.
Then a three dimensional curve at time t can then be notated as:TW(t) = x(t) y(t) z(t) ,(2)where x(t) = (x(t, s1 ) · · · x(t, sn ))T , and similarly for y(t) and z(t). The sequence
for a choice of T values of t, t = (t1 , ..., tT )T : (W(t1 ) · · · W(tT ))T = W ∼N3T n (m, K), where m is the mean, assumed to be zero, and K is the covariancematrix. Separability is assumed such that: k(t, t0 , c, c0 , s, s0 ) = kt (t, t0 )kc (c, c0 )ks (s, s0 )(Rasmussen 2006), i.e. K = Kt ⊗ Kc ⊗ Ks .
• If the process is assumed Markovian, the Ornstein-Uhlenbeck (OU) covariance function can be used, i.e., kt (t, t0 ) = exp(−|t − t0 |/µ), with hyperparameter µ, the time-scale. Hence, Kt represents the covariance of curves atdifferent time points, with (i, j)th element equal to kt (ti , tj ).
• For the 3 × 3 matrix Kc , two hyperparameters were specified: κ1 , thecorrelation between x and y or z, and κ2 , between y and z:1 κ1 κ1
Kc = κ1 1 κ2  .
(3)
κ1 κ2 1• The space-covariance function used
 is the Squared-Exponential (SE), i.e.,ks (s, s0 ) = σf2 exp − 21 (s − s0 )2 /λ2 , with hyperparameters: σf2 , the signal
variance and λ, the length-scale. Therefore, Ks represents the covariancematrix for the n arc-length inputs, with (i, j)th element equal to ks (si , sj ).

Mariñas et al.

3

43

Conditional dependencies, likelihood and predictive
distributions

The distribution for the first three-dimensional curve at time t = 1, W(1) ∼N3n (0, Kc ⊗ Ks ), can be factorised as:x(1)∼Nn (0, Ks ),y(1) | x(1)∼z(1) | x(1), y(1)∼Nn (κ1 x(1), (1 − κ21 )Ks ),

 


Nn {κ1 − κ1 κ2 } x(1) + κ2 − κ21 y(1) / 1 − κ21 ,

 2



1 − κ1 + κ22 − 2κ21 κ2 / 1 − κ21 Ks .

(4)
Subsequent three-dimensional curves in the sequence can be conditioned on theprevious time-point (by the Markov property):W(t) | W(t − 1) ∼ N3n κW(t − 1), (1 − κ2 )Kc ⊗ Ks ,(5)
where κ = exp(−1/µ), assuming the time difference between curves is one.
Analogous conditional dependencies between coordinates can be used:x(t) | W(t − 1) ∼ Nn κx(t − 1), (1 − κ2 )Ks ,y(t) | x(t), W(t − 1)∼z(t) | x(t), y(t), W(t − 1)∼Nn κy(t − 1) + κ1 [x(t) − κx(t − 1)] ,(1 − κ2 )Ks (1 − κ21 ) ,(6)1Nn κz(t − 1) +[{κ1 − κ1 κ2 }{x(t)−1 − κ21κx(t − 1)} + {κ2 − κ21 }{y(t) − κy(t − 1)} ,κ2 + κ22 − 2κ21 κ2(1 − κ2 ) 1 − 1K.
s
1 − κ21Given the hyperparameters θ = (σf , λ, µ, κ1 , κ2 ), the total log-likelihood of thesequence can be calculated as:log p(W | θ) = log p(W(1) | θ) +TXlog p(W(t) | W(t − 1), θ).

(7)

i=2

At each time point t, log p(W(t) | W(t − 1), θ) = log p(x(t) | W(t − 1), θ) +log p(y(t) | x(t), W(t − 1), θ) + log p(z(t) | x(t), y(t), W(t − 1), θ).
Marginal predictions at time q ∈ R can be done at a set of test points s∗ =(s∗1 , . . . , s∗n ) for each coordinate x, y and z, using:W∗ (q) | W ∼ [L ⊗ Kc ⊗ Ks∗ s ][M ⊗ Kc ⊗ Ks ]−1 W,Kc ⊗ Ks∗ − [L ⊗ Kc ⊗ Ks∗ s ][M ⊗ Kc ⊗ Ks ]−1 [L ⊗ Kc ⊗ Ks∗ s ]T ,(8)∗

where Ks∗ s denotes the n × n matrix of the covariances evaluated at all pairsof training and test points, M has (i, j)th element equal to κ|i−j| and
L = exp(−|q − 1|/µ) exp(−|q − 2|/µ) · · · exp(−|q − T |/µ) .
(9)
The matrix L will change depending on the value of q. Analogous conditional
dependences between coordinates can be calculated.

44

4

GPs for 3D curves evolving over time

Fitting the model for the emotion Disgust andclassification of emotions

FIGURE 3. Observations, posterior means and predicted values for 3-dimensional
lip curves of the emotion Disgust.
The model was fitted for the 3D upper lip curves from the emotion Disgust (Figure 2), of 61 pictures, i.e. t = (1, . . . , 61)T . Optimal hyperparameters were foundby maximum likelihood. To ease the optimization process, the number of hyperparameters can be reduced by finding the signal variance, σf , that maximises
the log-likelihood function analytically. The small difference between time points
makes the hyperparameter µ very large. Through experimentation it was observedthat the larger the value of µ, the smaller its effect on the likelihood function.
Moreover, it was found that µ and κ2 , the correlation between coordinates y andz, were negatively correlated. For this reason, the hyperparameters κ1 and κ2
were optimised for a series of individual time points, using a GP model for threedimensional curves (Mariñas et al. 2016), and then fixed to the mean of thesevalues. Optimization is then carried out for the remaining hyperparameters. An
additive normal error of standard deviation 0.5 mm was added to the model of observations to accommodate errors in the observed facial surface. The optimal values found are: θ̂ = (σ̂f , λ̂, µ̂, κ̂1 , κ̂2 ) = (5.3383, 0.1553, 25.5058, −0.0164, 0.7850),
with respective se: 0.0565, 0.0027, 2.5851, 0.0059, 0.0286. The same time-points(t = (1, . . . , 61)T ) were considered to make predictions at 25 spatial-points (Figure 3). Retrodiction was done at time q = −1, using the data at time point 1.
Prediction at time q = 63, conditioned on the data from the last curve available,
i.e. at time 61, was also made. Marginal prediction was done at time q = 30,
using the observed values at that time, and interpolation at q = 40.75, using theobserved values at times 40 and 41. The posterior means are displayed with 2
standard deviations bands (shown dotted). Note how the error bands expand as
predictions are further away from the observations.
Methods of classification are being studied to categorise each emotion in terms ofthe correlation parameters. An initial approach was to perform Principal Com-

Mariñas et al.

45

FIGURE 4. Biplot of the (scaled) first two principal components.
ponent Analysis (PCA) (Figure 4). Data consists of six different emotions: anger,
disgust, fear, happy, sadness and surprise. There are available sequences from 60
images, in the case of the expression of disgust, to sequences of about 180 images,
in the case of happiness. Data were collected by Oliver Garrod and colleagues
from the School of Psychology at the University of Glasgow from a 25 year old
actress performing each expression between three and five times. There was no
stimulation: she based the expressions on the Facial Action Coding System proposed by Paul Ekman (Ekman et al. 1997).

5

Conclusions and further lines of investigation

The use of shape information, expressed in a continuous and multivariate scaleraises a number of very interesting issues from a methodological perspective. The
lip curve data represent a peculiar scenario due to their high smoothness both
spatial and temporally, therefore special adjustments had to be made, which are
not needed in other sequences of three-dimensional curves. Nonetheless, the modelinterpolates the data well and produces accurate predictions. When it comes tothe classification of the different emotions using PCA, it can be seen that thisis limited by the small number of replicates available. The first two principal
components explain together 72.4% of the variability. It is clear from the biplotthat Surprise is the most different from the rest of the emotions, and Disgust theone with less variability between replicates, which can be interpreted as having
a series of strong unique characteristics that do not allow much change. For abetter understanding of the differences between the emotions, in terms of theircorrelation parameters, it would be necessary to increase the number of replicates,
as well as adding more subjects to the study, to account for the variability across

46

GPs for 3D curves evolving over time

people. The notion of a shape evolving in time can be extended to a phylogenetic
setting, where branching points in the evolution can occur. The aim is to develop
statistical methods by which shape information on organisms can be used toreconstruct a phylogenetic tree.

Acknowledgments: IM is grateful to the School of Mathematics and Statistics at the University of Glasgow for her research studentship.

References
Rasmussen, C.E. and Williams, C.K.I. (2006). Gaussian Processes for Machine
Learning. Cambridge, Massachusetts: The MIT Press.
Mariñas, I., Bowman, A. and Macaulay, V. (2016). Modelling the shape of emotions. In Proceedings of the 31st International Workshop on Statistical Modelling, Dupy, J-F. and Josse, J. (Eds.), pp. 201-206.
Ekman, P. and Rosenberg, E. (1997). What the face reveals: Basic and apliedstudies of spontaneous expressions using the Facial Action Coding System(FACS). Oxford University Press, USA.

Construction of High-resolution Linkage
Maps Using Discrete Graphical Models
Pariya Behrouzi1 , Ernst C. Wit1
1

Johann Bernoulli Institute, University of Groningen, Netherlands

E-mail for correspondence: P.Behrouzi@rug.nl
Abstract: Linkage maps are important for fundamental and applied genetic
research. In this article, we introduce an algorithm to construct high-quality andhigh-density linkage maps for diploid and polyploid species. We employ a sparse
Gaussian copula graphical model and the nonparanormal skeptic approach toconstruct linkage maps. We compare our method with other available method
when the data are clean and contain no missing observations and when data arenoisy and incomplete. In addition, we implement the method on real genotype
data of barley. We have implemented the method in the R package ”qploidMap”
which is freely available at CRAN.
Keywords: Graphical models; Gaussian Copula; Linkage map; Polyploids.

1

Genetic background on linkage map

A linkage map provides a fundamental resources to understand the order of markers for the vast majority of species whose genome are yet to be sequenced. Furthermore, it is an essential ingredient to identify genes associated with different
traits such as, disease resistance in plants or animals. Diploid organisms containtwo sets of chromosomes, whereas polyploids contain more than two sets of chromosomes. Here, we refer to diploids and polyploids as q-ploid q ≥ 2, where in
diploids q = 2, triploids q = 3, and so on. The genotype of any q-ploid organism
at each single locus on the genome can be either homozygous if all q allele copies
of an organism are identical, or heterozygous otherwise.

1.1

Meiosis and Markov dependence.

Linkage mapping is possible because of a biological crossover process, which occurs during meiosis. Assume a sequence of ordered SNP markers X1c , . . . , Xpcc
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

48Discrete graphical models and its application in genetics

along chromosome c. Due to the genetic linkage, neighboring markers across achromosome are linked. The key biological fact during meiosis is that markers
at different chromosomes segregate independently. Given the above mentioned
fundamental genetic concepts, Xjc can be defined as

Xjc =



1
0

parental marker at locus j,otherwise.

The sequence X1c , X2c , . . . , Xpcc forms a Markov chain with the state space S
where the genotype state of a genetic marker on a genome depends only on the
genotype state in the previous genetic marker. Thus, genotype state of markercc
Xj in following set {X1c , . . . , Xj−1
, Xjc , Xj+1
, . . . Xpcc } can be written as follow
c
c
c
c
c
c
P (Xj |Xj−1 , Xj−2 , . . . , X1 ) = P (Xj |Xj−1 ). This property implies that neighboring markers have conditional dependence relationship. Two complications arise
when (i) a population contains heterozygous genotypes, in this case we define
q
P
c
Yjc =
Xjk
, (ii) we do not know what ”parental” is, in that case the conditional
k=1

independence relationships are more complicated. Treating the SNP markers according to the Markovian assumption yields the graphical model. The variables
vector X = {X1c , . . . , Xpc }, where p is the total number of given markers in agenome, is a discrete graphical model with a joint distribution P (X) which canQ
Qpc (c)(c)(c)
be factorized as P (X) = C
c=1
j=1 fj,j+1 (xj , xj+1 ). Here, C defines the number of chromosomes in a genome, and pc stands for the number of markers inchromosome c.

2

Algorithm to construct linkage map

We propose to build a linkage map in two steps. First, reconstruct an undirectedconditional independence graph between SNP markers in a genome. Second, determine the correct order of markers in the obtained linkage groups.

2.1

Construct undirected graphical model

To reconstruct the conditional independence graph between SNP markers in a
q-ploid species we propose two methods, latent graphical model, and the nonparanormal skeptic. The former method can deals with missing values, whereas
the latter is computationally faster.

Latent graphical model
A relatively straightforward approach to discover the conditional independence
relationships among markers is to assume underlying continuous variables Z1 , . . . , Zp
for the markers Y1 , . . . , Yp such that Zj = Φ−1 (Fbj (Yj )), where Φ−1 (.) is the standard Gaussian quantile function. We assume Z ∼ Np (0, Θ) where Θ = Σ−1
contains all the conditional independence relationship between SNP markers.
Furthermore, we implement the EM algorithm which iteratively finds penalizedBehrouzi et al.

49

b Using the extended rank likelimaximum likelihood estimation of parameter Θ.
hood in the E-step we compute the expected complete penalized log-likelihood
as followb | Θ(m) ) = n [log |Θ| − tr(R̄Θ) − p log(2π)] − λ||Θ||1
Qλ (Θ
2
P
n
b (m) ), and λ is a nonnegative tuning
where R̄ = n1 i=1 EZ (i) (Z (i) Z (i)t |y (i) , Θ
parameter. To calculate R̄ we propose two different approaches, namely Gibbs
sampling and approximation method (Behrouzi et al. (2017)). The M-step isthe maximization problem which can be solved efficiently using graphical lasso(Fridman et al., 2008).
TABLE 1. Comparison the two methods over 50 independent run, where q = 2
Grouping Accuracy
Ordering Accuracy
Missing rate Error rate DGMmap
p=500 & n=200
0
0
0.05
0.05
0.10
0.10
0.15
0.15
p=1000 & n=200
0
0
0.05
0.05
0.10
0.10
0.15
0.15

MSTMap

DGMmap

MSTMap1.00 (0.00) 0.55 (0.34)
1.00 (0.00) 0.10 (0.07)
1.00 (0.00) 0.01 (0.00)
1.00 (0.00) 0.01 (0.00)

1.00 (0.00) 0.90 (0.09)
0.77 (0.04) 0.61 (0.12)
0.60 (0.03) 0.18 (0.23)
0.56 (0.01) 0.00 (0.00)

1.00 (0.00) 0.61 (0.36)
1.00 (0.00) 0.04 (0.03)
1.00 (0.00) 0.44 (0.16)
1.00 (0.01) 0.05 (0.00)

1.00 (0.00) 0.91 (0.06)
0.56 (0.00) 0.51 (0.09)
0.52 (0.00) 0.78(0.02)
0.52 (0.00) 0.60 (0.13)

Nonparanormal SKEPTIC
Alternatively, we use the nonparanormal skeptic approach (Liu et al., 2012) tocompute the correlation matrix. In this approach, a sample correlation matrix Γ
can be computed from pairwise rank correlations, namely Kendall’s tau τbjl , andSpearman’s rho ρbjl .
2 sin( π6 ρbjl ) j 6= lsin( π2 τbjl ) j 6= l,b jl =
Γ1
j=l
1
j = l.
To estimate the graph we use the graphical lasso. To determine the number oflinkage groups we use the EBIC model selection which picks the penalty termthat minimizes the EBIC value over λ > 0.


b jl =
Γ

2.2

Ordering markers in each linkage group

Assume that a set of d markers have been assigned to the same linkage group.
Let G(d, Ed ) be a sub-graph. To order d markers, the algorithm uses multidimentional scaling (MDS) method to post processing of each linkage group. The
goal of using the MDS is to find an one-dimensional map such that the distances
between markers fit a given set of measured partial correlation that indicate how
far markers are.

50

Discrete graphical models and its application in genetics

3

Data analysis

3.1

Simulations

We set up simulations to generate q-ploid genotype data. We compare the performance of our proposed method with MSTMAP(Wu et al. (2008)) for different
ranges of missing rates and genotyping errors when q = 2. The results of comparisons are provided in Table 1. We note that high values of the grouping andordering accuracy scores indicate good performance. These results suggest thatthe proposed approach performs well compare with the other method.

3.2

Construct linkage map in Barley

A barley genotyping dataset is used in the literature to compare different map
construction methods for real-world diploid data. This genotyping dataset is generated from a doubled haploid population which allows to achieve homozygous
individuals, Y = {0, 1}. Barley genotype data is the result of crossing Oregon
Wolfe Barley Dominant with Oregon Wolfe Barley Recessive. The Oregon Wolfe
Barley (OWB) data includes p = 1328 markers that were genotyped on n = 175
individuals which 0.02% genotypes are missing. The barley dataset is expected
to yield 7 linkage groups, one for each of the 7 barley chromosomes.
Table 2 shows that our method estimated correctly the number of chromosomes
for the OWB dataset. Whereas, The MSTMAP grouped all 1328 markers as one
chromosome. The ordering accuracy scores are higher in qploidMap compare with
the MSTMAP, except in chromosomes one and three. Also, the size of the markers
within each linkage group is consistent with the number of markers that has beenreported in Cistue et al. (2011).

Behrouzi et al.

451

Conclusion

Construction of linkage map is the most fundamental step required for a detailed
genetic study in any species. We propose to build a linkage map in two steps:
First, reconstructing an undirected conditional independence graph between SNP
markers in a genome, Second determining the order of markers in the obtained
chromosomes from the first step. Our simulations show that the proposed method
outperforms the alternative method in terms of linkage map quality. In the application of our method in Barley, the proposed method construct more accurate
linkage map compare with the alternative method.

References
Behrouzi, P. and Wit, E. C (2017). Detecting Epistatic Selection with PartiallyObserved Genotype Data using Copula Graphical Models. Submitted.
Cistu, L., et al. (2011). Comparative mapping of the Oregon Wolfe Barley using
doubled haploid lines derived from female and male gametes. Theoreticaland applied genetics, 1399-1410.
Liu, H., F. Han, M. Yuan, J. Lafferty, and L. Wasserman (2012). High-dimensionalsemiparametric Gaussian copula graphical models. The Annals of Statistics, 2293-2326.
Wu, Y., et al. (2008). Efficient and accurate construction of genetic linkage maps
from the minimum spanning tree of a graph. PLoS Genet, 4(10), e1000212.

Inference in complex systems using
multi-phase MCMC sampling with gradientmatching burn-in
Alan Lazarus1 , Dirk Husmeier1 , Theodore Papamarkou1
1

School of Mathematics and Statistics, University of Glasgow, UK

E-mail for correspondence: a.lazarus.1@research.gla.ac.uk
Abstract: We propose a novel method for parameter inference that builds onthe current research in gradient matching surrogate likelihood spaces. Adoptinga three phase technique, we demonstrate that it is possible to obtain parameter
estimates of limited bias whilst still adopting the paradigm of the computationallycheap surrogate approximation.
Keywords: Parameter inference; Delayed Rejection Adaptive Metropolis; Surrogate likelihood; Markov Chain Monte Carlo; Gradient matching1

Introduction

Statistical inference in nonlinear differential equations (DE) is challenging. The
log-likelihood landscape is typically multimodal and every parameter adaptation,e.g. in an MCMC simulation, requires a computationally expensive numerical integration of the DEs. Using numerical methods to solve the equations results
in prohibitive computational cost; particularly when one adopts a Bayesian approach in sampling parameters from a posterior distribution. Alternatively, one
can try to reduce this computational complexity by obtaining an interpolant
to the data from which one can obtain a comparative objective function thatmatches the gradients of the interpolant and the DEs. By sampling on this cheap
representative likelihood surface, bias is introduced to the modelling problem.
Current research focuses on reducing this bias by introducing a regularising feedback mechanism from the DEs back to the interpolation scheme (e.g. Niu et al.
2016). The idea is to make the interpolant maximally consistent with the DEs.
Although this paradigm has proved to improve performance over naı̈ve gradientmatching, the feedback loop fails to fully eradicate bias in the final estimate.
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Lazarus et al.

53

For this reason, a natural progression would be to sample from the true likelihood space whilst reducing computational complexity in the discarded burnin
steps. Assuming this hypothesis, we postulate the use of a surrogate likelihood in
the burnin phase alone. Through an example possessing multimodal likelihood,we will show the ability of the algorithm to avoid any local entrapment whilst
obtaining accurate parameter estimates.

2

MethodConsider time-dependent observations y(t) = x(t) + —where x(t) denotes thesignal and  independent additive zero mean Gaussian noise with variance parameter σ 2 —whose signals are governed by a system of differential equations:dx(t)= f (x(t), θ)
dt(1)dependent on some (partially) unknown parameters θ. Assuming Gaussian noise,
we place a GP prior on the latent variable x
x(t) ∼ GP(0, k(t, t0 )),(2)

leading us to a Gaussian distribution, p(xi |φi ) = N (xi |0, Ki ) for an arbitrary
set of time points T = {t1 , ..., tn } with entries of Ki given by evaluating kernel
function k at each element of T × T (Rasmussen and Williams, 2006). Under ourassumption of Gaussian noise, we consider the joint distribution, p(y, x|φ, σ).
Marginalising over latent variables x provides a zero mean distribution for theobservations:y ∼ N (0, K + σ 2 I)

(3)(see Dondelinger et al. (2013) for details). Considering the joint distribution between our signal and observed values, we may implement an elementary transformation of a Gaussian distribution to obtain the posterior distribution for oursignal with mean given by:µ(τ ) = k(τ, T )(K + σ 2 I)−1 y,(4)where k(τ, T ) denotes evaluation of the kernel function at τ over T . Subsequently,
firstly estimating the hyperparameters via ML, we adopt the mean of the posterior
as a representation of our signal x. This allows us to proceed under the supposition
that we have a fixed interpolant for the true signal. Given that the derivative ofeq. 4,∂µ(τ )= k0 (τ, T )(K + σ 2 I)−1 y,∂τ(5)is the mean of the Gaussian distributed DE derivative (see section 7.5 of Vanhatalo et al., 2015), we may consider:f (x̂(t), θ) ∼ N dx̂(t)dt, γ2I ,

(6)

54

MCMC parameter inference with surrogate burnin

where γ 2 is a fictitious noise term (assumed equal for both gradients) and x̂ isgiven by eq. 4. Contrary to the work done by Dondelinger et al. (2013), fixing
the GP hyperparmeters φ and the interpolant x̂ allows us to abandon the Gibbs
sampling routine at this stage of the algorithm, further reducing the overall computational burden. The corresponding negative log-likelihood term provides a
gradient matching objective function:π(θ) = n logγ 2 +1
2γ 2dx̂(t)− f (x̂(t), θ)
dt

2

(7)

which gives a representative computationally tractable surface as a surrogate forthe log-likelihood. This involves the term γ 2 representing the mismatch between
the gradient obtained from the differential equation and that from explicit differentiation of the GP posterior mean. This parameter will be sampled throughout
the surrogate burnin phase of the algorithm. The proposed sampling scheme involves three phases. In the initial burnin phase, samples are drawn from thesurrogate distribution in eq. 7 using a Delayed Rejection Adaptive Metropolis1(DRAM) scheme (Haario et al. (2006)). Assuming a degree of similarity between
the surrogate and true likelihood surfaces, this drives the sampler towards theglobal minimum of the true likelihood function until a PSRF2 value of 1.1 has
been achieved. From here, we initialise a corrective phase in the true likelihood
space, correcting for any bias introduced by the inconsistencies between the surrogate and true likelihood spaces. Sampling with DRAM, this phase is concluded
upon obtainment of a PSRF equal to 1.1. The proceeding sampling phase replicates this corrective phase with sampling steps recorded until we achieve a PSRF
value of 1.05. The stepwise decrease in target PSRF values allows time for theadaptive component of AM to learn the topology and adjust the covariance accordingly. We adopt an uninformative Inv-Gamma(0.001,0.001) prior for σ 2 andγ 2 and a Ga(4, 0.5) prior for the parameters of the DE. All parameters are sampled on the log scale to account for the positivity constraint.

3

Results

We assess performance on the following DE model of circadian oscillation3 :dp1
k1=− k3 ,
dt
36 + k2 p2

dp2= k4 p 1 − k5dt(8)This is a notoriously challenging problem due to the extreme multimodality of
the likelihood. Following Girolami et al. (2010), we focus on the inference of two
parameters (k3 and k4 ), setting the other parameters and initial conditions to
1
Obtained using the adaptive Metropolis component of DRAM with modMCMC function in the FME package in R.
2
Obtained at intervals of 20 steps using the gelman.diag function from thecoda package in R.
3
We used the same differential equations as in Girolami et al 2010. The actual
Goodwin oscillator is of a slightly different form, where the terms k3 and k5 are
replaced by k3 p1 and k5 p2 , respectively.

Lazarus et al.

55

5
4
3

k4
2

4

k4
2

1

1

1000

3

−206

5.1

●
−295
398.6

−206

−206

806.6

●

4

−2

06

80
6.6

5

1

−1

18

21

4.7

2

k3

806.6

806.6

−177

276

−20

x
●
−88684.05
−14
774
5.3
−177276

●

2

6.6

.3
37

680
6.6

●

x
●

80

6.6

●

500

1

06

3

36

70

90

−5

1000

●

80
6.6

−2

●

● −2

2000

●

−2
06

●

●

80

1500●

06

●

●
●

−2

3

31000

30500
29500 29000 30000
28000 27000 28500 27500
26000 25500 26500
24500 23500 25000 24000
23000 22500
22000
21500
21000
20500
20000 19500
19000
18500
18000
17500
17000 16500
16000 15500● 15000
14500 14000
13500 13000
12500
12000
11500
11000
10500
10000
9000 9500
8000 8500
7000 7500
6000 6500
5500
5000
●
4500
4000
3500
3000
2500

−76565013 −99652917 −120964828−48149132−25573634−32165199 −39269169 −46373139−13407
007−22591−14411049 −17718481−71760038
40−15798−7500877 −8
−111
989
6267
1800
−472
−3661893
7
82
4996
−30
−54337−9006940
41−209
31−4990772 −56768 749−3425647
−141
0793
7563
−2864565 −
64−94−197864
30
−37
6
507
−1
12
209
890
2.9
21
−2
54
−1092726 − −1
054
273
8
97 24
952
46 037−502113.1
03−2
9 −14470.5
94 −16 126299
5380
−115
8
1787 −1
−88
269
601
−6
910
1.6
79
29
−4
−100
7
4134
−3 725−708827. −
83
82
7 76
99
.5 −56117
−2
78
4.4
0.5
89
−6
65
20
86
−4
23
−
7.9
43
5
−3
05 5316 .7
54
1.8
43
45
.8
9.9
−4
13
5
21
−3
.2
24
92
9.2

5

the same fixed values as in Girolami et al. (2010). Five sets of initial parameter values for k3 and k4 were obtained using a Sobol sequence over the domain[0, 5]2 . Figure 1 shows the chain moving through the k3 -k4 parameter domain.
Comparing with the traditional method, we observe the ability of the proposed
method to evade the various local minima. PSRF values of 17.1, 10.4 and 12.3
were obtained for k3 , k4 and σ 2 simulations respectively after 10000 steps using the traditional DRAM method in true likelihood space. Comparatively, theproposed method required 1690 steps in surrogate space, 1690 in the corrective
phase and 1010 in the sampling phase to achieve a PSRF of 1.05. The numberof numerical integration steps required are given in Table 1 for each of the ten
DRAM chains. In Figure 2, boxplots are given that provide the distribution of−206806.6●−236337.3●−265867.9−295398.6

3

●

4

5

−2

4
3

k4

●
68

06

06

6.6

−2
0

●

●

80

06

−2

−76565013 −99652917 −120964828−48149132−25573634−32165199 −39269169 −46373139−13407
007−22591−14411049 −17718481−71760038
40−15798−7500877 −8
−111
989
6267
1800
−472
−3661893
7
82
49
96
−30
−54337−9006940
41−209
31−4990772 −56768 749−3425647
−141
0793
7563
−2864565 −
64−94−1978646
30
−37
507
−1
12
2095
890
2.9
21
−2
−1092726 − −1
4
054
273
8
97 24
952
46 037−502113.1
03−2
9 −14470.5
94 −16 126299
5380
−115
8
1787 −1
−88
269
601
−6
910
1.6
79
29
−4
−100
7
4134
−3 725−708827. −
83
82
7 76
99
.5 −56117
−2
78
4.4
0.5
89
−6
65
20
86
−4
23
−
7.9
43
5
−3
05 5316 .7
54
1.8
43
45
.8
9.9
−4
13
52
−3
1.2
24
92
9.2

5

k3

80

6.6

.6

−2

06
80

6.6

●

.3
37
63

3

● −2

−206

806.6

2

.1

05

7
90

−5

●

●
−295
398.6

−206

−20

1

680

1

−177

276

−206806.6
●

6.6

x
●
−88684.05
−14
774
5.3
−177276

−1

18

21

4.7

2

−206
806.6

x

806.6

−236337.3●−265867.9−295398.6

3

4

●

5

k3

FIGURE 1. Ten chains generated with the proposed method shown in the parameter domain of the negative surrogate log-likelihood space (topleft) and negative
log-likelihood space (topright). The bottom plot shows simulations generated using the traditional method. The true parameter value is given by a point at (2,1).
The black crosses denote the final point of each chain.

TABLE 1. Number of numerical integration steps (N) for the traditional method.
The number required in the proposed scheme is 2700.
Chain Index
N

1

2

3

4

5

6

7

8

9

10

25583 29778 29835 28452 29708 29672 29768 29530 29859 29847

bias in our sampled parameter estimates for each of the five chains. Figure 3
provides RMS deviation in function space obtained using eq. 9,
r
1
2
RM Sf unction =x − x̂
(9)n

56

MCMC parameter inference with surrogate burnin

where x denotes the true signal and x̂ denotes the numerical solution of the DE
for one parameter sample from the sample phase of the multiphase approach andthe post burnin
period of the traditional method. This provides a measure of the●predictive accuracy of the● MCMC samples.

k3

k●4

−0.01

●
●
●
●
●

k3

●
●
●

0

●
●
●

●
●
●
●
●
●
●
●
●
●
●

●

4

0

Bias

●
●
●

3

●
●
●
●

2

0.005

●
●
●
●
●
●
●

●
●
●
●

●

1

0.01

●

−0.005

Bias

●
●

k4

●
●

●
●
●
●
●
●
●
●
●
●

●
●
●
●
●
●
●
●
●
●
●
●
●
●

●
●
●
●
●
●
●
●
●
●
●
●
●
●

●
●
●
●

●

k3

k4

FIGURE 2. Boxplots showing the distribution of bias for both methods (left)
where red boxes give the bias of the standard DRAM samples (without outliers)
and the blackboxes give the bias of the proposed method. The plot on the right●gives the bias in both parameters for the DRAM method with outliers included.
●

●

0.4

●
●
●
●
●
●
●
●
●
●
●

30

0.3

●

0.2

●
●
●

●

●
●
●
●
●
●
●

Novel Method

Standard Method

00.1

RMS Value

20

●
●
●
●
●

10
0

RMS Value●

Novel Method

Standard Method

FIGURE 3. Functional RMS comparison between the proposed method (black)and DRAM (red). On the left, we include outliers (from DRAM) and, on theright, these are removed to enable better scalability of the plots for comparison.
The red dotted line denotes a functional RMS equal to zero.

4

Conclusion

Our work considers the sampling in DE parameter inference as a computationallyefficient three-phase scheme that achieves low levels of bias in sampled parameter estimates (Figure 2). Achieving a PSRF of 1.05, we observe the ability of thealgorithm to converge in the parameter space of the circadian oscillator systemof equations; a model for which the standard DRAM procedure fails to replicate
this success (bottom of Figure 1). Considering the results in function space, we
observe the superior performance of the proposed method compared with the traditional DRAM method, showing that the performance improvement is witnessedin both domains of study. These features, along with the vast improvement incomputational efficiency, demonstrate improved parameter inference comparedwith the traditional method.

Lazarus et al.

57

Acknowledgments: This project was supported by a grant from the Engineering and Physical Sciences Research Council (EPSRC) of the UK, grant reference number EP/L020319/1.

References
Dondelinger, F. et al. (2013). ODE parameter inference using adaptive gradientmatching with Gaussian processes. Proceedings of Machine Learning Research, Volume 31, pp 216-228.
Girolami, M. et al. (2010). System identification and model ranking: the Bayesian
perspective. Learning and Inference in Computational Systems Biology.
MIT Press, pp. 201-230.
Haario, H. et al. (2006). DRAM: Efficient adaptive MCMC. Statistics and Computing, Volume 16, Issue 4, pp 339-354.
Niu, M. et al. (2016). Fast Parameter Inference in Nonlinear Dynamical Systems
using Iterative Gradient Matching. Proceedings of Machine Learning Research, Volume 48, pp 1699-1707.
Rasmussen, C.E. and Williams, C.K.I. (2006). Gaussian Processes for Machine
Learning. MIT Press.
Vanhatalo, J. et al. (2015). Bayesian Modeling with Gaussian Processes using
the GPstuff Toolbox. MIT Press.

Pragmatic order selection in hidden Markov
modelsJennifer Pohle1 , Roland Langrock1
1

University Bielefeld, Germany

E-mail for correspondence: jennifer.pohle@uni-bielefeld.de
Abstract: We discuss the notorious problem of order selection in hidden Markov
models, i.e. of selecting an adequate number of states, highlighting typical challenges arising when analyzing complex real data. Extensive simulations are used
to demonstrate why standard information criteria tend to favor models with
undesirably large numbers of states. We also propose a pragmatic step-by-step
approach to implement order selection.
Keywords: Information Criteria; Model Selection; Time Series

1Introduction

Hidden Markov models (HMMs) are time series models for observations thatare driven by an underlying finite-state Markov chain. Despite their popularity,
identifying the most appropriate number of states has proven to be notoriously
difficult in practice (when the focus lies on inference on the data-generating process, i.e. in an unsupervised learning situation). In particular, it has been demonstrated that information criteria, when applied in empirical settings, often leadto the selection of much larger numbers of states than seem desirable (see, e.g.,Langrock et al., 2015). This behavior can partly be explained by the complexityof many real data sets. In addition to the features that actually motivate theuse of state-switching models (such as multimodality and autocorrelation), realtime series data often exhibit additional patterns, e.g. outliers, seasonal fluctuations or non-trivial dependence structures. When neglecting these features in thespecification of an HMM, then additional states within the model can “mop up”
the neglected structure in the data, therefore providing an improved model fit,
however at the price of reduced interpretability.
In this paper, we demonstrate the above points using simulations, and suggest
a pragmatic approach to choose the number of hidden states, which takes into
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Pohle et al.

59

account formal criteria for guidance, but also stresses the importance of the study
aim, expert knowledge and model checking procedures.

2

Basic formulation of hidden Markov models

An HMM comprises an observed time series {Xt }Tt=1 which is assumed to be
driven by an unobservable state process {St }Tt=1 . In the basic model formulation,{St } is assumed to be a first-order Markov chain, characterized by the state transition probabilities γij = Pr(St = j | St−1 = i), i, j = 1, . . . , N . In addition, it
is usually assumed that the observations are conditionally independent of eachother, and of past states, given the current state: p(Xt |Xt−1 , .., X1 , St , ..., S1 ) =p(Xt |St ). Thus, the distribution of each observed variable Xt is completely determined by the current state St . These two assumptions complete the basic model
formulation.

3

Simulation Studies

We use simulations to investigate the performance primarily of the AIC andthe BIC when it comes to selecting a suitable number of hidden states. We also
investigate the performance of the integrated completed likelihood (ICL) criterion(Biernacki et al., 2001), which has a stronger focus on the model’s ability topartition the data. The integrated completed likelihood is obtained using themost probable (Viterbi-decoded) state sequence ŝ:
ICL = −2 log Lc (x , ŝ) + p log(T ),where Lc (x , ŝ) denotes the (approximate) complete-data likelihood function given
the observed time series x = (x1 , . . . , xT ), and p is the number of model parameters (see Zucchini et al. (2016), for details on how to evaluate the likelihood of
an HMM).
We showcase six scenarios where there is additional structure in the data that is
not accommodated within the basic HMM formulation detailed above. Each typeof additional structure considered may be found in real data, where the assumptions made with the basic HMM formulation typically are overly simplistic.
Scenario 1 (benchmark, correct model specification): We simulate data using a
two-state gamma HMM without additional structure, primarily as a benchmark
for the subsequent scenarios (see Fig. 1).
Scenario 2 (outliers): The data are generated using the benchmark model, adding
uniformly distributed errors to 0.5% of the observations generated.
Scenario 3 (inadequate emission distribution): Modified emission distribution
within state 2, using a nonparametrically constructed density with a form similar
to the gamma distribution, but exhibiting a heavy tail.
Scenario 4 (temporal variation): Transitions probabilities dependent on the timeof the day, i.e. state occupancy exhibits within-day variation.
Scenario 5 (semi-Markov state process): Replacing the geometric dwell-time distribution within state 2 of the benchmark model by a Poisson.

60

a)

Pragmatic order selection in HMMs
0.8

state 1state 2b)state 1state 2state 30.6
densitydensity

0.6

0.8

0.4

0.4

0.2

0.2

0.0

0.0
0

2

4

6

observation

8

10

0

2

4

6

8

10

observation

FIGURE 1. a) Densities specified in Scenario 1; b) Densities used in Scenario 7.

Scenario 6 (violation of conditional independence assumption): Time-varying mean
parameters of the state-dependent gamma distributions, generated using autoregressive processes of order 1.
For each scenario, 100 data sets were generated (T = 5000). In each run, stationary gamma HMMs with 2–5 states were fitted using maximum likelihood
estimation (neglecting the additional structure in scenarios 2–6).
Table 1 shows, for each simulation scenario, the percentages of runs in which themodels with 2–5 states are chosen by AIC, BIC and ICL, respectively. Both AIC
and BIC mostly failed to detect the true number of states, in all five scenarioswith model misspecifications, with AIC doing worse than the BIC, due to thehigher penalty on model complexity in the latter.
The ICL did fairly well in our simulations, which can be explained by the tendency of the ICL to favor non-overlapping solutions, i.e. HMMs where the statedependent distributions are clearly distinct. This tendency is also pointed out in
Biernacki et al. (2001). However, crucially, this behavior will not always be desirable. To demonstrate this point, we carried out an additional simulation using
more overlapping states.
Scenario 7 (stronger overlapping emission distributions): We simulate data usinga three-state gamma HMM without additional structure, where the states overlap
more strongly than in the previous scenarios (see Fig. 1). Furthermore, in this
scenario transition probabilities are specified as γ11 = γ22 = 0.8.
For this scenario, we generated 100 data sets, with T = 2000 observations each,and fitted stationary gamma HMMs with 2–4 states to each data set.
The results obtained for Scenario 7 are displayed in Table 2. We find that the
ICL here does indeed tend to favor models with too few states. While the AIC
performed equally poorly as the ICL — often selecting four and hence too manystates — the BIC always chose the correct number of three states in this scenario.
In complex real data, we will usually find several violations of the assumptions
involved in the basic HMM formulation. While the deviations may be relatively
minor, they may effectively accumulate, such that order selection can in fact evenbe more problematic than with just a single, yet stronger assumption violation.
While conceptually it would seem natural to simply overcome the limitations of
HMM formulations that cause criteria-based order selection to fail, this is usuallyPohle et al.

61

TABLE 1. Percentages of runs in which the models with 2–5 states are chosen
by AIC, BIC and ICL, for all simulation scenarios.

simul.
scenario

number of hidden states selected
criterion

2 (%)

3 (%)

4 (%)

5 (%)

1
(benchmark)

AIC
BIC
ICL

37
100
100

43
–
–

20
–
–

–
–
–

2
(outliers)

AIC
BIC
ICL

–
30
58

47
70
42

49
–
–

4
–
–

3
(inadequate emission
distribution)

AIC
BIC
ICL

–
–
26

27
100
71

60
–
3

13
–
–

4
(temporal variation)

AIC
BIC
ICL

–
14
100

–
84
–

57
2
–

43
–
–

5
(semi-Markov stateprocess)

AIC
BIC
ICL

–
–
100

14
100
–

74
–
–

12
–
–

6
(violation of cond.
indep. assumption)

AIC
BIC
ICL

–
5
100

–
95
–

28
–
–

72
–
–

TABLE 2. Percentages of runs in which the models with 2–4 states are chosen
by AIC, BIC and ICL, for the additional simulation scenario 7.

simul.
scenario
7(stronger overlapping
emission distributions)

number of hidden states selected
criterion

2 (%)

3 (%)

4 (%)

AIC
BIC
ICL

–
–
62

54
100
36

46
–
–

not a useful strategy in practice, both because of computational considerations
and as corresponding highly parameterized models may distract from the actual
study aim.

62

Pragmatic order selection in HMMs

4

Pragmatic Oder Selection

Given the difficulties outlined above, we suggest the following pragmatic stepby-step approach to selecting the number of states of an HMM. The proposed
strategy applies only to the unsupervised learning case, where inference related
to the state process is of primary interest.
Step 1 decide a priori on the candidate models, in particular the minimum and the
maximum number of states that seem plausible, and fit the corresponding
range of models;
Step 2 closely inspect each of the fitted models, in particular plotting their estimated state-dependent distributions and considering their Viterbi-decodedstate sequences;
Step 3 use model checking, in particular residuals, to obtain a more detailed picture of the fitted models;
Step 4 use model selection criteria for guidance as to how much improvement, if
any, is obtained for each increment in the number of states;
Step 5 make a pragmatic choice of the number of states taking into account findings from Steps 2-4, but also the study aim, expert knowledge and computational considerations;
Step 6 in cases where there seems to be no strong reason to prefer one particularmodel over another (or several other) candidate model(s), results for eachof these models should be reported.
The resulting choice of the number of states will necessarily be somewhat subjective. However, a corresponding analysis nevertheless will be as scientific, if
not more scientific, than any allegedly objective choice of the number of states.
Furthermore, we have made the experience that a thorough implementation as
detailed in Steps 1-4 will usually make it fairly easy to pick a suitable N .

5Conclusion

Model selection criteria are problematic with respect to choosing the number ofstates of an HMM applied to complex real data within an unsupervised learning
framework. In particular, any structure in the data that is neglected in the model
formulation will, to some extent, be mopped up by additional model states that do
not have a clear interpretation anymore. The ICL criterion appears to overcome
several of the problems associated with the more established AIC and BIC, yet
it does not come without its own limitations, namely a sensitivity to overlappingstate-dependent distributions.
We proposed a pragmatic step-by-step approach to order selection which, while
lacking objectivity, we believe is the best possible practical solution. As pointed
out in Hennig (2015) in the context of cluster analysis, it is crucial that the individual researcher’s modeling decisions, and in particular the rationale underlying
the selection of the number of states, need to be made transparent.
Overall, the selection of the number of states clearly is an important yet challenging issue, which requires statistical expertise (when applying model selection andmodel checking tools) and modeling experience, but also a good understanding
and intuition of the data and research question at hand (in order to arrive at asensible choice of the number of states).

Pohle et al.

63

References
Biernacki, C., Celeux, G. and Govaert, G. (2001). Assessing a mixture model forclustering with the integrated completed likelihood. IEEE Transactions on
pattern analysis and machine intelligence, 22, 719 – 725.
Hennig, C. (2015). What are the true clusters? Pattern Recognition Letters, 64,
53 – 62.
Langrock, R., Kneib, T., Sohn, A. and DeRuiter, S.L. (2017). Nonparametric inference in hidden Markov models using P-splines. Biometrics, 71, 520–528.
Zucchini, W., MacDonald, I.L. and Langrock, R. (2016). Hidden Markov Models
for Time Series: An Introduction using R. Boca Raton: Chapman & Hall.

Spatial Conditional Overdispersed Models:Application to count area data
Vicente Núñez-Antón1 , Edilberto Cepeda-Cuervo2
1

Department of Econometrics and Statistics, University of the Basque Country
UPV/EHU, Bilbao, Spain
2
Department of Statistics, National University of Colombia, Bogotá, Colombia
E-mail for correspondence: vicente.nunezanton@ehu.eus
Abstract: We propose alternative models for the analysis of count data featuring a given spatial structure. We assume that the overdispersion data structure
partially results from the existing and well justified spatial correlation between
geographical adjacent regions, so an extension of existing overdispersion models
that include spatial neighborhood structures within a Bayesian framework is proposed. Finally, their usefulness is illustrated by fitting them to infant mortality
rates and to data including the proportion of mothers who, after giving birth totheir last child, underwent a postnatal screening period in Colombia.
Keywords: Count area data; Infant mortality; Overdispersion; Postnatal screening period rates; Spatial statistics.

1

Introduction and motivating examples

Generalized linear models are commonly used in medical data analysis, speciallyfor their flexibility in the modelling of the distribution of the response variable.
One of the most relevant contributions in the development of the generalized
linear models theory centers on count data, such as infant mortality rates or
the proportion of mothers who underwent a postnatal screening period. Whenthe data show some type of overdispersion, this phenomenon can be generated
from very different sources, which should be appropriately assessed in order tobe able to reduce the impact wrong model formulations not able to capture allof the variability in the data have on the results from the statistical analysis.
Several models have been proposed following the theory originally developed foroverdispersed binomial and Poisson models. Therefore, taking into account theexisting spatial association between observations on the variable under study isThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Núñez-Antón and Cepeda-Cuervo

65

very relevant in the statistical analysis of this type of data. One would desire thatthe data provide spatially ordered information for the variable under study, sothat the general idea that observational units closer to a specific observation may
have some influence on its distributional behavior may be assessed and includedin the model specification. Several techniques have been proposed to model spatial correlation, but interest in the proposals here lies in trying to quantify thiscorrelation with the use of a parametric model that includes a set of parameters able to model this effect. Therefore, we propose alternative overdispersion
models that include basic neighborhood structures in spatial models, which are
based on a self-constructed covariate, defined by using a series of non stochastic
spatial weights that somehow establish the strength of the spatial dependence between the geographical units being considered in the study. The proposed models
include spatial neighborhood structures both for the mean and for the overdispersion parameter specification, so that the spatial association can be quantified
both in the mean and overdispersion behavior, by using neighborhood structure
assumed by the researcher performing the analysis. Thus, our models are able toquantify the spatial association related to the neighborhood structures proposedor assumed by the researcher. In addition, they include regression structures associated to the overdispersion parameter, to the different factors that may berelated to the existing heteroscedasticity, and to the spatial association not explained by the assumed neighborhood structures that could be related to othercovariates or to model structures specified in the proposed dispersion model. In
order to illustrate the usefulness of the proposed models we apply them to tworeal data sets. The first one corresponds to the analysis of the number of children under 5 years who died in the 5-year period 2000-2005 in Colombia. The
second one corresponds to the study of the proportion of mothers who had given
birth to their last child between 1999 and 2005 and who underwent a postnatal
screening period. In both data sets, we have variables available at the departmental level, such as the percentage of the population that has basic services not
being satisfactorily attended to, the percentage of young people who had accessto a higher academic achievement level, and the resources provided for academic
achievement or education and integral attention for young children per household
for families coming from the general participation system. All of these variables
will be used as covariates for the proposed models.

2(Generalized) spatial conditional overdispersion
modelsGeneralized overdispersion models for count data, in which regression structures
are assumed for both the mean and overdispersion parameters, may be an appropriate alternative to analyze count area data, where spatial association between
observations is present. However, given that these models do not provide information related to the strength of the spatial association between observations of
the variable under study, some more general models should be specified. In thespatial overdispersion models proposed here, this effect can be easily estimated,
an issue that is addressed by proposing the use of a weight matrix in the model,
where its parameter estimates are associated to the corresponding lag variable
modelling this neighborhood association. The specific choice of the elements of

66

Spatial Conditional Overdispersed Models

the weight matrix is a very relevant issue, mainly because it should be the resultof both interpreting and understanding the spatial structure the variable understudy has. In the proposed models we assume that observational units correspondto each of the areas in the study, where we control for or model the variation
generated by the existing spatial correlation between experimental units, we use
the weights wij , which are the elements in the model reflecting the level of dependence between the spatial units in the study, indexed by i and j. These values are
the corresponding elements of the weight matrix W. One of the most commonlyused specifications for the weight matrix W assumes that wij = 1/ni , if the observational unit or region j belongs to the neighborhood of an observational unit
i, or if there is geographical first or second order contiguity between regions iand j, where ni is the number of first or second order adjacent regions for regioni; and wij = 0 otherwise. The generalized spatial conditional models proposed
here are structurally different from the Poisson or binomial CAR and SAR models. More specifically, the proposed spatial conditional models are formulated interms of generalized overdispersed models, which include spatial lag structuresin both the mean and dispersion regression structures. In addition, nonstructural
random effects are also included in both of the aforementioned regression structures, so that the models are able to capture the overdispersion not explained by
the spatial structures included in the proposed spatial models. Our proposed spatial conditional overdispersion regression model assumes that the spatial variable
under study, Yi , i = 1, . . . , n, conditioned on the values in all of the neighborhoods of the i-th region, but not including the i-th region itself (i.e., Y∼i ), has aoverdispersed conditional distribution denoted by f (yi |y∼i ), i = 1, . . . , n, where
the conditional mean and the conditional dispersion parameters follow given regression structures that, besides some covariates affecting the response variable,
also include spatial lags of the variable under study. These models assume thatthe conditional overdispersion density functions follow either a Poisson or a binomial distribution, leading to the proposal of the (generalized) spatial conditional
Poisson, negative binomial, normal Poisson, binomial, beta binomial and binomial normal regression models, respectively. As an illustrative example of thesemodel proposals, we consider the Poisson normal model for overdispersed countdata. In this model, the overdispersion is included in the model with the use of
a normally distributed random effect term in the mean model. In this way,g(λi ) = xT
i β + νi ,(1)

where g(.) is usually the logarithm function, xi is the q × 1 vector of explanatory
variables for the i-th observation, β is a q × 1 vector of unknown regression parameters, and νi ∼ N (0, τ ). In this model, (Yi |νi ), i = 1, . . . , n, follows a Poisson
distribution with mean λi = E(Yi |νi ). In the spatial conditional normal Poissonmodel, if Yi , i = 1, . . . , n, represent area count data from different regions or
areas, such as departments or states, a portion of the existing overdispersion canbe explained by the neighborhood spatial structured assumed by the researcher,which is given by the spatial conditional normal Poisson model where we assume
that (Yi |Y∼i , νi ) follows a Poisson distribution with mean λi = E(Yi |Y∼i , νi ), withg(λi ) = xT
i β + ρWi y + νi ,(2)and ρ is the parameter explaining the first order spatial association in the mean
model, Wi is the i-th row of the n × n weight matrix W, which follows theNúñez-Antón and Cepeda-Cuervo

67

assumed first order neighborhood structure (i.e., regions, units or departments
share a common border), and y is the n × 1 vector of the observed values of
the response variable under study. Finally, we propose the generalized spatialconditional normal Poisson model, which also allows for modelling spatial neighborhood structures. That is, we assume that (Yi |νi ) follows a Poisson distribution
with parameter µi , i = 1, . . . , n, where νi ∼ N (0, σi2 ). In addition, it is assumed
that conditional mean µi and the variance terms in the random effect distribution, σi2 ’s, are modelled as functions of some explanatory variables, so that its
regression structures are given by2Tlog(µi ) = xT
i β + ρWi y + νi and log(σi ) = zi γ + ηWi y,(3)

where zi is the qφ ×1 vector of explanatory variables for the i-th observation, γ is aqφ ×1 vector of unknown regression parameters, and η is the parameter explaining
the first order spatial association in the dispersion model. In these models, therandom factor is associated to the overdispersion generated, for example, fromthe heteroscedasticity or the possible spatial correlation of higher order than thatconsidered in the model. In the generalized spatial conditional normal Poissonmodel, if we have a fixed σi2 = σ 2 and keep σi2 unstructured (i.e., do not propose
any model for it), we have the spatial conditional normal Poisson regression modelwith τ = σ 2 . If, in addition, ρ = 0, we have the Poisson normal regression model.

3

Application

Data considered here correspond to the 32 departments (regions, geographicalunits or states) in Colombia. Variables available for each one of the geographical units are: the number of children under five who died in the five-year period
2000-2005 (i.e., variable ND), the percentage of women over 18 who had suffered any type of physical abuse from their current partners (i.e., variable Viol),the percentage of the population that had basic services not being satisfactorily
attended to for the year 2004 (i.e., variable NBI), the resources (in thousands)
provided for academic achievement or education and integral attention for young
children per household for families coming from the general participation system(i.e., the general plan for the allocation of resources from the central government
to the departmental or municipal governments) in the year 2005 (i.e., variableRec), the percentage of young people (i.e., between 18 and 23 years old) who hadaccess to a higher academic achievement level (i.e., to a higher educational level)in the year 2005 (i.e., variable HE), the percentage of children under one year ofage who had the third dose of the polio vaccine applied in the year 2004 (i.e.,the variable Vac), the number of mothers (in thousands) that had their last childborn between 1999 and 2005 and that underwent a postnatal screening period(i.e., the variable NScree), the number of mothers (in thousands) that had theirlast child after 1999 (i.e., variable NMother99), and the percentage of mothers
who had to pay for the total cost of the postnatal screening period (i.e., the variable Pay). For the weight matrix, W, we assume that the elements wij is equal to
one if region j belongs to the neighborhood of region i, and equal to zero, otherwise. As for the prior distributions, we assume independent normal distribution,
N (0, 105 ), for all of the regression parameters, including the spatial association
parameters, ρ and η. In the specific application considered here, the behavior

68

Spatial Conditional Overdispersed Modelsof the chains, all of them with small transient periods, seems to show that theconvergence was quickly achieved. Therefore, a burn in period of 2000 iterations
is assumed from the 10000 iterations developed in the estimation process. The
best fitting model for this data was the generalized spatial conditional normal
Poisson model with BIC and DIC values of 200.78 and 251.88, respectively, andmean and variance regression models given by:log(µi )=β0 + β1 Reci + νi , νi ∼ N (0, σi2 )(4)log(σi2 )=γ0 + ηWi y,(5)

with the corresponding estimates reported in Table 1.
TABLE 1. Parameter estimates, together with their standard deviations for themean and variance parameters for the infant mortality data and the generalized
spatial conditional normal Poisson model.

Estimate
Standard deviation

4

β0
3.283
0.271

β1
−4.564 × 10−04
2.091 × 10−04

γ0
0.522
0.352

η
−6.617 × 10−03
2.069 × 10−03

Conclusions

Results from a simulation study and from the analysis of the infant mortality
rates data, for the Poisson case, and to data including the proportion of mothers
who underwent a postnatal screening period in Colombia, for the binomial case,conclude that the proposed models fit better than both the previously proposedmodels (i.e., the ones not modelling the overdispersion in the data sets) or thewell known intrinsic conditional autoregressive (ICAR) models. More specifically,
for the infant mortality rates data, the best fitting model was the generalized
spatial conditional normal Poisson model, with spatial structures and random
effects included both in the mean and overdispersion regression models. As forthe proportion of mothers who underwent a postnatal screening period, they
were the spatial conditional binomial normal and beta binomial models, withspatial structures and random effects included only in the mean regression model.
Finally, a sensitivity analysis was performed to assess the effect or influence of
the assumed prior distributions for the different regression parameters for themean and overdispersion parameters have on the resulting parameter posterior
estimates.

Acknowledgments: This work was supported by Ministerio de Economı́ay Competitividad, FEDER, the Department of Education of the Basque Government (UPV/EHU Econometrics Research Group), and Universidad del Paı́s
Vasco UPV/EHU under research grants MTM2013-40941-P, MTM2016-74931-P,
IT-642-13, UFI11/03 and US15/11, and also by the Department of Statistics,
National University of Colombia.

Núñez-Antón and Cepeda-Cuervo

69

References
Cepeda-Cuervo, E., Córdoba, M. and Núñez-Antón, V. (2017). Conditional overdispersed models: application to count area data. Statistical Methods in Medical Research. In press.
Hinde, J. and Demétrio, C.G.B. (1998). Overdispersion: models and estimation.
Computational Statistics & Data Analysis, 27(2), 151 – 170.
Quintero-Sarmiento, A, Cepeda-Cuervo, E. and Núñez-Antón, V. (2012). Estimating infant mortality in Colombia: some overdispersion modelling approaches.
Journal of Applied Statistics, 39, 1011– 1036.
Wakefield, J. (2007). Disease mapping and spatial regression with count data.
Biostatistics, 8(2), 158 – 183.

Constructing wildebeest density distributions
by spatio-temporal smoothing of ordinal
categorical data using GAMs
Elaine A. Ferguson1 , Jason Matthiopoulos1 , Dirk Husmeier2
1

Institute of Biodiversity, Animal Health and Comparative Medicine, College ofMedical, Veterinary and Life Sciences, University of Glasgow, Glasgow, UK
2
School of Mathematics and Statistics, College of Science and Engineering, University of Glasgow, Glasgow, UK
E-mail for correspondence: e.ferguson.2@research.gla.ac.uk
Abstract: Spatio-temporal smoothing of large ecological datasets describing
species distributions can be made challenging by high computational costs anddeficiencies in the available data. We present an application of a GAM-basedsmoothing method to a large ordinal categorical dataset on the distribution of
wildebeest in the Serengeti ecosystem.
Keywords: GAMs; Ordinal categorical; Smoothing; Spatio-temporal; Wildebeest.

1

Introduction

Spatio-temporal smoothing of species distribution data has many potential uses
in ecology; for example, to provide a smooth density function that can be usedwith gradient matching approaches (Xun et al. 2013) to fit partial differential
equation (PDE) models of animal movement. A range of smoothing methods(kernel density estimation, splines, Gaussian processes, etc.) have been developedin the statistical literature. However, the practicalities and expense involved incollecting species distribution data over large areas in the field can mean thatthe data are not in a form that these methods can readily be applied to. Ordinal categorical data, for example, may be collected when it is infeasible toaccurately count all individuals in a population, so that the abundance at eachpoint in space and time is instead estimated as belonging to a broader abundance category. A relatively small number of approaches have been developed
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Ferguson et al.

71

for smoothing data of this type, where we need to recover the underlying true
density of individuals from the categories (Chu and Ghahramani 2005, Woodet al. 2016). Smoothing large datasets in multiple dimensions can also be madechallenging by high computational costs. Methods that allow smoothing of thesedatasets even when computational resources are limited would therefore be very
useful. Here we present an application of a method for applying spatio-temporalsmoothing to a large ordinal categorical dataset on the distribution of wildebeestin the Serengeti ecosystem of Tanzania and Kenya.

2

Methods

The wildebeest distribution data, which have been described and utilised ina number of previous studies (Norton-Griffiths 1973, Maddock 1979, Boone etal. 2006, Holdo et al. 2009), were obtained from monthly aerial surveys of the
Serengeti ecosystem during the period from August 1969 to August 1972. Eachcell in a grid of 25km2 cells was assigned to one of five wildebeest abundance
categories: 0, 1-25, 26-250, 251-2,500 and >2,500 individuals per 25km2 . There
were 2,576 cells making up the spatial grid, all of which were sampled on 33
occasions during the time period, resulting in a large dataset with a total 85,008
data points.
To smooth the data in time, t, and the two spatial dimensions (x, y) we fitted
GAMs (generalised additive models) with a tensor product (composed of cubic
regression spline smooths, where overfitting was prevented by penalisation of theintegral of the squared second derivatives) between these three variables using
the mgcv package (Wood 2011) in R (R Core Team 2015). We used the ordinal
categorical GAM method described in Wood et al. (2016), where the linear predictor gives the value of a latent variable, here representing the wildebeest density
underlying the ordinal categories. The cut-off points that demarcate the five ordinal categories were specified, and the probability that a point in space and time
belongs to a given category equals the probability that the latent variable lies
between the corresponding category cut-offs at that point.
In Wood et al. (2016), the latent function can range from −∞ to ∞, but weknow that wildebeest density has a minimum 0 and a finite maximum Wmax . We
can introduce these constraints by applying a sigmoidal transformation to thelatent function L after the GAM has been fitted, giving a preliminary wildebeest
density Ŵ as follows:Ŵ (x, y, t) =Wmax
1 + exp (−L (x, y, t))(1)

Note that this also required that an inverse sigmoid transform be applied to thecategory cut-offs c prior to the GAM fitting:c̄ = − log

Wmax
−1
c


(2)

Wmax was estimated by first assuming that the wildebeest densities in the gridcells assigned to the lower four ordinal categories, which had known upper andlower bounds, were equal to the mid-points of those categories. The sum of the

72

Smoothing ordinal categorical data using GAMs

densities in these lower category cells for each month was then subtracted from thetotal number of wildebeest WT known to be in the region from a population countin 1971 (Norton-Griffiths 1973). The remaining wildebeest for each month wereassumed to be divided evenly between the cells in the highest ordinal category(which was unbounded above) for that month. We took Wmax to be the largest
wildebeest density estimated for cells in the highest abundance category over all
months.
Even after applying sensible upper and lower bounds to the latent function, large
fluctuations in the area under Ŵ (which represents the total number of wildebeestin the region) can occur over time. This is undesirable, since we expect wildebeest
numbers to remain relatively stable at WT over the time period of interest. We
therefore consider the normalised wildebeest density W̄ , where the total numberof animals is maintained at WT by normalising Ŵ as follows:W̄ (x, y, t) = R

Ŵ (x, y, t) WTŴ (x, y, t)dxdy(3)

Due to computational time and memory constraints, a sufficiently flexible GAM
could not be fitted to the entire large dataset simultaneously. We therefore divided
the time series into three contiguous intervals and fitted a GAM in (x, y, t) toeach interval separately. Each GAM had 20 knots in the marginal smooth in eachspatial dimension, and a number of knots in the marginal smooth in time that was
equal to the number of time points present in the data subset to which the GAM
was fitted (11 or 12). This resulted in the effective degrees of freedom, which
are determined by the degree of penalization (selected during fitting) applied tothe integral of the squared second derivatives, being considerably lower than the
maximum number available, suggesting that the number of knots was sufficient(Wood 2006). The three GAMs were joined together by averaging at the linktimes li (i ∈ 1, 2), with smoothness being maintained by allowing the influence ofeach GAM on the others to decline smoothly, according to the parameter σ, asdistance from the point of joining increased. For a given point (x̄, ȳ, t̄), therefore,
we obtain a final estimate of wildebeest density W by

W (x̄, ȳ, t̄) = W̄GAMj (x̄, ȳ, t̄) +2Xai expi=1−(t̄ − li )22σ 2

mi (t̄)(4)

Here W̄GAMj is the normalised wildebeest density obtained from the GAM fitted
to time interval j, where

 1
2
j=

3

t̄ ≤ l1
l1 < t̄ ≤ l2t̄ > l2(5)W̄GAMi (x̄, ȳ, li ) − W̄GAMi+1 (x̄, ȳ, li )
2

(6)

if
if
if

The ai are given byai (x̄, ȳ, li ) =Ferguson et al.

73

FIGURE 1. Model fit in space at three different time points. A-C: The wildebeest
spatial distribution data for months 1, 18 and 35. D-F: The smooth wildebeest
density distribution estimated in space by the model for months 1, 18 and 35.
The two contours indicate the boundaries between abundance categories 0, 1 and
2. G-I: Estimated wildebeest abundance categories based on D-F.

and the mi , which ensure that the adjustments are made in the correct direction
on either side of each link point, aremi (t) =

−1
1

if
if

t̄ ≤ lit̄ > li(7)

If the influence of the adjoining GAMs declines too slowly with distance fromthe link points, relative to the rate at which changes occur in W̄GAMi (i.e. σ is
too large), unrealistic negative values of W can occur. We therefore tuned σ bystarting with a relatively large value and gradually decreasing it until no negativevalues of W occurred.

74

3

Smoothing ordinal categorical data using GAMs

Results and Conclusion

The method described was found to successfully produce a smooth function in
space that resembles the original data (Figure 1). The resulting function is also
observed to be smooth in time, with no evidence that the wildebeest densitychanges either more slowly or more rapidly around the GAM link times than it
does elsewhere in the time period (Figure 2). This suggests that our approach of
linking models that have been fitted to subsets of a larger dataset is a promisingmeans of reducing the high computational costs of smoothing large datasets in
multiple dimensions. Using this method, we have recovered realistically bounded
wildebeest abundance estimates from coarse ordinal categories; an ability that
could be useful in the field of ecology where such imperfect data are common. By
producing a smooth surface from which spatial and temporal gradients in density
can

FIGURE 2. Changes in the estimated wildebeest density in six grid cells (indicated by different colours/line types) over the time period of interest. The linktimes between the three GAMs are indicated by dashed vertical lines.
be calculated, our method also promises to enable statistical inference for PDE
models of animal movement using the gradient matching approach of Xun et al.
(2013), which we will investigate in future work.

Acknowledgments: Special thanks to Ricardo M. Holdo for providing accessto the wildebeest distribution data. E.A.F. is funded by a University of Glasgow
Lord Kelvin/Adam Smith PhD scholarship.

References
Boone, R.B., Thirgood, S.J. and Hopcraft J.G.C. (2006). Serengeti wildebeest migratory patterns modeled from rainfall and new vegetation growth. Ecology,
87, 1987 – 1994.

Ferguson et al.

75

Chu, W. and, Ghahramani, Z. (2005). Gaussian processes for ordinal regression.
Journal of Machine Learning Research, 6, 1019 – 1041.
Holdo, R. M., Holt, R. D. and Fryxell, J. M. (2009). Opposing rainfall and plant
nutritional gradients best explain the wildebeest migration in the Serengeti.
The American Naturalist, 173, 431 – 445.
Maddock, L. (1979). The migration and grazing succession. In: Serengeti: dynamics of an ecosystem, Sinclair, A. R. E., and Norton-Griffiths, M. (editors), Chicago: University of Chicago Press, 104 – 129.
Norton-Griffiths, M. (1973). Counting the Serengeti migratory wildebeest using
two-stage sampling. East African Wildlife Journal, 11, 135 – 149.
R Core Team (2015). R: A language and environment for statistical computing.
R Foundation for Statistical Computing, Vienna, Austria.
Wood, S.N. (2006). Generalized Additive Models: An Introduction with R. CRC
Press.
Wood, S.N. (2011). Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. Journal of
the Royal Statistical Society, Series B, 73, 3 – 36.
Wood, S.N., Pya, N., and Sfken, B. (2016). Smoothing parameter and model selection for general smooth models. arXiv:1511.03864v2.
Xun, X., Cao, J., Mallick, B., Maity, A., and Carroll, R.J. (2013). Parameter Estimation of Partial Differential Equation Models. Journal of the American
Statistical Association, 108, 1009 – 1020.

Spatio-temporal clustering of traffic networks
Ashwini Venkatasubramaniam123 , Ludger Evers2 ,Konstantinos Ampountolas13
1

Urban Big Data Centre, University of Glasgow, United Kingdom,
School of Mathematics and Statistics, University of Glasgow, United Kingdom,
3
School of Engineering, University of Glasgow, United Kingdom
2E-mail for correspondence: a.venkatasubramaniam.1@research.gla.ac.uk
Abstract: We present a novel Bayesian clustering method for spatio-temporal
data observed on a network and apply this model to cluster an urban traffic
network. This method employs a distance dependent Chinese restaurant process(DDCRP) to provide a cluster structure, by incorporating the observed data andgeographic constraints of the network. However, in order to fully capture thedependency structure of the data, a conditional auto-regressive model (CAR) is
used to model the spatial dependency within each cluster.
Keywords: Network; Spatial; Clustering; Bayesian1

Introduction

Heterogeneous urban traffic networks with regions of varying congestion levels
have unique fundamental properties and clustering aids in the division of a city
into homogeneous regions. We propose a novel Bayesian clustering technique forspatio-temporal network data which is based on an amalgamation of a distancedependent Chinese restaurant process (DDCRP) and a spatio-temporal conditional autoregressive model (CAR). We assume that we observe a time series ofmeasurements that represent congestion levels aggregated over each junction in
the network and the degree of similarity between adjacent junctions can be used
to define spatially contiguous clusters. Existing literature relevant to clustering
techniques for transportation networks account for spatial constraints but typically do not incorporate changes over time within a cluster. Traditional clustering
algorithms such as k-means and probability mixture models also require choices
to be made about the number of clusters.
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Venkatasubramaniam et al.

277

MethodIn our model, the road network forms an undirected graph with junctions acting
as nodes and road segments between junctions as edges. This graph can be used
to define the spatial component of the precision matrix. The spatial precision
Σ−1within a cluster is modelled as a CAR model and the temporal precisionS
Σ−1
T as an auto-regressive (AR1) model. Other models that incorporate temporal dependency such as the Matern covariance function are also possible. Due tothe presence of a grid network topology with a limited number of road segments
between junctions (typically not more than four per junction), the precision matrices exhibit sparsity and we utilize a CAR model proposed by Leroux (2000) to
define this spatial precision. We define an adjacency matrix W and the precision
matrix Q = Σ−1= ρ(diag(Wk++ ) − W) + (1 − ρ)InJ , where ρ controls how
Sstrongly correlated adjacent junctions are, diag(Wk++ ) is a diagonal matrix withelements equivalent to the row sums of W, and InJ is an nJ × nJ identity matrix(nJ = number of junctions). With the presence of a unique observation for every
space-time combination (nJ junctions and nT time points), a covariance matrix
Σ can be written as ΣS ⊗ ΣT . Clusters can be obtained by removing edges suchthat the graph can be partitioned into components not connected to each other.
In Figure 1, a network composed of eight nodes can be divided into two clusters
such that there are road segments but no links between adjacent junctions of two
differing clusters.

FIGURE 1. Graph showing two clusters formed in the network based on thepresence or absence of a road segment between junctions.
Individually removing edges not supported by the data would yield a sparsegraph, but would be unlikely to result in a graph with more than one component. Instead, we use a prior to enforce that edges are omitted in a way that leads
to a clear partitioning of the graph. We utilize a modified version of the DDCRP
first introduced by Blei (2011) that allows our model to incorporate geographic
constraints of the network, account for the shape, and determine the numberof clusters. The DDCRP makes assumptions of non-exchangeability to accountfor components of distance such as time, space, etc. In a traditional Chinese
restaurant process (CRP) (also a special case of the DDCRP), a restaurant canbe assumed to consist of an infinite number of tables. Customers i = 1 . . . n are
individual data points that enter and take a seat at a randomly chosen table k.
Tables are deemed to be clusters and after a finite number of customers nk havebeen seated, the seating plan represents a partition. In the usual representationof the CRP, customers choose tables. In a DDCRP, clusters are instead determined based on friendships between customers i and j, with a group of friends
then sitting at an assigned table zi , i.e., forming a cluster. Thus, z(c) are tableassignments that follow from customer assignments. In a non-sequential DDCRP,
clusters arise from some customers choosing to befriend themselves or someone
already connected to them, resulting in a redundant assignment. We also mod-

78

Spatio-temporal clustering of traffic networks

ify the DDCRP to allow customers to befriend more than one customer, which
controls for the number of singleton clusters. Let ci be the index of a customerthat is sitting with customer i and we describe the distribution of this customer
assignment as:
0, j 6= i and i  j
P (ci = j|α) ∝ 1, j 6= i and i ∼ jα, j = iIn our method, customers are junctions and friendships can only occur alongroad segments between junctions. Since this modified DDCRP suggests a priorover a combinatorial number of junction assignments, the posterior is intractable
and inference is carried out using a Metropolis within Gibbs sampler. We assume
that the measure of congestion levels Y follows a Gaussian distribution and thelikelihood at partition z(c) gives the product of probabilities calculated for setsof observations at each determined cluster. To account for the spatial dependency within the cluster, we define ΣS and ΣT to represent the precision matrix
described earlier. Accordingly, the likelihood can be defined as:nJ nTln(2π) − 0.5 ln |σ 2 I + τ 2 ΣS ⊗ ΣT |2
−0.5vec(Y)T [σ 2 I+τ 2 ΣS ⊗ ΣT ]−1 vec(Y)ln(P (Y|ΣS , ΣT , σ 2 , τ 2 )) = −We can rewrite terms in the likelihood, vec(Y)T [σ 2 I + τ 2 ΣS ⊗ ΣT ]−1 vec(Y) =
T
2
2
T
T
vec(Y)T (ΓS ⊗ΓT )(σ 2 I+τ 2 Λ)−1 (ΓT
S ⊗ΓT )vec(Y) = vec(ΓT YΓS ) (σ I+τ ΛS ⊗
2
2
−1
T
ΛT )
vec(ΓT YΓS ), where σ is variance of the noise, τ is a prior variance,
ΛT represents a diagonal matrix of the eigenvalues of ΣT , and ΓT represents
a matrix of the eigenvectors of ΣT . Here, we only need to compute the diagT
onal of [σ 2 I + τ 2 ΛS ⊗ ΛT ]−1 on the rotated data vec(ΓT
T YΓS ) . In addition,nT nJ
XX− ln(τ 2 ΛT [t, t] · ΛS [s, s] + σ 2 I).
ln |σ 2 I + τ 2 ΣS ⊗ ΣT | can be rewritten as
t=1 s=1

Together, these terms can be evaluated for an efficient solution in O(n2J nT +n2T nJ + n3J + n3T ) rather than O(n3J n3T ). Sampling from this posterior can happen in two phases where we first remove the customer and then consider how
the likelihood term can be changed when this customer is replaced. The sampler
thus has the potential to change multiple cluster assignments through a single
change in customer assignment and using these moves is able to explore the spaceof possible partitions to determine a partition structure conditional on observed
data.

3

Results

Occupancy is defined as the percentage of time that a location on the road is
occupied by vehicles. In our example, occupancy data was generated using the
AIMSUN microscopic simulator for a network in downtown San Francisco composed of 316 links and 158 junctions. This was recorded over a period of six hours
with a sampling frequency of 180 seconds. We cluster the simulated network suchthat each individual cluster represents a level of occupancy that is distinct fromother clusters, as shown in Figure 2.

Venkatasubramaniam et al.

79

121

36

22
4752

14
25

7

44

93

99

16

38

118

90

30

152

1357

135

126

84
63

3764
66 46

105 39

33 51

96
15542
142129151114 2076

29

113

80

2

82

158

100

50 5

21 98 27

95

43

111

60 74

124 13261
116 108104
77 106 85

73

92

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17

89
35

1

28

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125 149
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69 54 147
12855
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11512 53

120

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18 134 41

6

130

11

127

3
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123

26

24107

143 4 109

81

133

87

119

FIGURE 2. Traffic network with clusters that indicate different congestion levels.

This paper proposes a Bayesian clustering algorithm that accounts for spatialconstraints and is modelled in a computationally efficient manner on data withvarying temporal patterns. Further work seeks to identify clusters that changeover time. However, current Kronecker product tricks that enhance efficiency
cannot be utilized since spatial precision would change over time.

Acknowledgments: Research funded by the Lord Kelvin Adam Smith scholarship, University of Glasgow, 2014–2018

References
Blei, D. and Frazier, P. (2011). Distance dependent Chinese restaurant processes.
Journal of Machine Learning Research, 12(Aug):2461– 2488.
Leroux, B., Lei, X., and Breslow, N. (2000). Estimation of disease rates in small
areas: a new mixed model for spatial dependence. In Statistical models in
epidemiology, the environment, and clinical trials, pages 179 – 191. Springer.

Bayesian variable selection for identifying thesource of food-borne disease outbreaks
Rianne Jacobs1 , Emmanuel Lesaffre2 , Peter Teunis1 , Jan vande Kassteele11National Institute for Public Health and the Environment (RIVM), Bilthoven,
NL
2
L-Biostat, KU Leuven, Leuven, Belgium
E-mail for correspondence: rianne.jacobs@rivm.nl
Abstract: Early identification of contaminated food products is crucial in reducing social and economic burdens of food-borne disease outbreaks. Analytic
case-control studies are primarily used in this identification. In this paper, we develop a Bayesian variable selection method to account for misclassified responses
and missing covariates. The method, implemented in JAGS/R Software is used
to analyse the Salmonella Thompson 2012 outbreak data.
Keywords: Bayesian variable selection; Missing value imputation; Scaled logistic
regression; Spike-slab variable selection.

1

Introduction

With food chains becoming increasingly complex and food products being transported across the globe with increasing ease, contaminated food products canrapidly cause food-borne disease outbreaks. Such outbreaks have a large socialand economic burden on society; see for example the Salmonella Thompson 2012
outbreak in the Netherlands (Friesema et al. 2014). Early detection of such outbreaks and the subsequent identification of the contaminated food product(s) is
crucial in reducing these burdens.
Identification of contaminated food products is a long, cumbersome process involving several steps which are not clear cut - much like a criminal investigation
where information is incomplete, delayed, uncertain and continually updated.
Analytic case-control studies are the main epidemiological tool in this process ofidentification. Once an outbreak has been detected, patients fill out an extensive
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Jacobs et al.

81

food consumption questionnaire for some period prior to becoming ill. Simultaneously, controls are sampled matched according to municipality, age and genderand they also fill out the food consumption questionnaire. One can well imagine
the practical difficulties that subjects have trying to recall their dietary consumption and the resulting amount of missing values in such data. In addition,although controls are questioned on their symptoms, it is impossible to confirm
whether they are indeed true controls (i.e. not infected) or rather asymptomatic
infections (i.e. infected but not ill), resulting in misclassification of the response.
The analysis of the questionnaire data typically involves classical logistic regression. Due to the large number of different food products people may have consumed, one often has a variable selection problem, where one attempts to identify
relevant exposures. Moreover, in the beginning of an outbreak, the number of covariates (i.e. food products) may very well be close to or even greater than thenumber of observations. Classical variable selection procedures, i.e. a combination of univariable analysis and stepwise forward or backward selection based on
p-values, are most employed (Friesema et al. 2014), thereby ignoring the classicalproblems of multiple testing and bias they imply. When searching for the causeof an outbreak we, therefore, need a far more sophisticated variable selection
procedure.
We argue that the Bayesian approach offers tools to deal with variable selection
problems with missing covariates and misclassified responses. Indeed, Bayesian
methods allow us to use external information to aid the modelling when data
are scarce. This is crucial in the analysis of our case-control data especially inthe light of early identification when very few data are yet available. In addition,
Bayesian methods provide us with the flexibility to account for the additional
problems of missing covariates and misclassified responses which is hard to solvein the frequentist setting. In this paper, we, therefore, develop a Bayesian variable selection method which also accounts for misclassified responses and missing
covariates.

2

Data and method

The Salmonella Thompson 2012 outbreak data was obtained from a case-control
study performed by the National Institute for Public Health and the Environment (RIVM) in which smoked salmon was found to be the source of the outbreak but only after six weeks since the onset of the outbreak. The percentageof missing covariates is high - up to 67% for a given observation. Salmonella
infections are often asymptomatic, i.e. an infected person does not become ill,
implying a sensitivity of less than one, P (Y = 1|T = 1) < 1 (Y : observed response, T : true response). Moreover, as a case only entered the dataset if it was
twice laboratory-confirmed, we assume that no non-infected person entered thedataset as a case, i.e. P (Y = 1|T = 0) = 0, implying that the specificity is one,
P (Y = 0|T = 0) = 1.
To deal with the problem of misclassified responses, our Bayesian variable selection model extends the logistic regression model. Logistic regression models theprobability of an observed success, P (Y = 1), which is equal to a true success,
P (T = 1), for a correctly classified response. In our case, however, this equality
is contaminated by the sensitivity (Se), P (Y = 1) = P (T = 1)P (Y = 1|T = 1),

82

Bayesian variable selection in food-borne disease outbreaks

resulting in the scaled logistic regression model:
yi ∼ Bernoulli(µi )µi = πi × Se

(1)plogit(πi ) = β0 +Xxi,j × βj .

j=1

The scaled logistic model is, however, unidentifiable without extra information
from, e.g., validation data. In a Bayesian setting, one may, in addition also use
historical information to feed a prior distribution on the sensitivity.
To incorporate Bayesian variable selection, we apply the stochastic search variable selection (SSVS) method (George and McCulloch, 1993) in which a mixture
prior on the parameters, βj , with one spike and one slab Gaussian component isconstructed. The variances of the spike and slab are τ 2 and c2 τ 2 , respectively.
The spike and slab prior is given byβj |τ 2 , c2 ∼ γj N(0, τ 2 c2 ) + (1 − γj )N(0, τ 2 ),(2)γj |ωj ∼ Bernoulli(ωj )ωj ∼ Beta(aj,0 , bj,0 )where γj is the indicator variable for inclusion of βj into the model with inclusion
probability, ωj . Parameters aj,0 and bj,0 are chosen to reflect prior knowledge
about the probability that a covariate should be in the model.
The large percentage of missing covariates in the data implies that many of thexi,j ’s in Eq. 1 are missing. To impute these covariates, we construct a covariate
probability model asf (x1 , x2 , ..., xp ) = f (x1 )pXf (xj |x1 , ..., xj−1 )

(3)

j=2

with f (xj |x1 , ..., xj−1 ) a Bernoulli distribution with logit link function. In thecase of many covariates, as in our data, it is reasonable to assume that covariate
xj does not depend on all j − 1 covariates. Similarly to the variable selection of
the response model (Eq. 2), we perform variable selection in each of the regression
models of the covariate probability model in Eq. 3, applying the 2-level variableselection model of Mitra and Dunson (2010).

3

Data analysis

We applied our Bayesian variable selection model to the Salmonella Thompson
data. The model was implemented in R Software using JAGS for the MCMC
sampling, running 5 chains with a burn-in of 1000 iterations and then a further
4000 iterations per chain. Trace plots showed good mixing. In this analysis, priors
for the inclusion probabilities were chosen to favour slightly parsimonious models,
namely ωj ∼ Beta(1, 2), based on the idea that a food product is not guilty unlessproven so by the data. In order to make the scaled logistic model identifiable,
we need an informative prior for the sensitivity. In this analysis, we used Se ∼Beta(33, 4), assuming a median sensitivity of 0.9 and 5th percentile of 0.8. The

Jacobs et al.
ground_meat 10 Sep

0.53

ground_pork

0.48

smoked_fish

0.59

Aldi

0.33

raw_veg_readymade

0.49

raw_ham0.32

ground_meat

0.41

raisin_bun

0.28

saveloy

0.37

liver_sausage

0.32

shrimps

0.34

peanut0.3

sprout_veg

0.35

ice_cream

0.28

gammon_steak

0.31

tartare0.27

hard_cheese

0.26

grill_sausage

0.27

pork_steaks

−2

0

2

4

Covariates

ice_cream 17 Sep

ice_cream 12 Sep

83

6
0.82

0.95

0.28

−2

0

2

4

ice_cream 18 Sep

60.98

smoked_fish

0.79

spread_sausage0.79

raw_veg_readymade

0.67

smoked_fish

0.71

spread_sausage

0.55

tartare0.54

lettuce_butterhead

0.54

raw_veg_readymade

0.52

gammon_steak

0.46

gammon_steak

0.46

Aldi

0.39

shrimps0.4

shrimps

0.36

Aldi

0.35

tartare0.36

chick_turk_filet

0.35

saveloy

0.34

ground_pork

−2

0

2

4

smoked_fish 24 Sep

60.99

ice_cream

0.74

gammon_steakchick_turk_filet

0.35

−2

0

2

4

smoked_fish 09 Oct

61

ice_cream

0.71

0.6

nuts

0.65

0.5

raw_veg_readymade

0.61

tartare0.48

hamburger

0.63

chicken_wing

0.4

Aldi

0.46

ground_pork

0.39

gammon_steak

0.47

spread_sausage

0.36

hard_cheese

0.39

Deen

0.37

salad_pasta

0.4

chicken_thigh

0.28

ground_pork

0.37

−2

0

2

4

6

−2

0

2

4

6

Regression coefficient
Posteriorpercentiles

25% − 75%

10% − 90%

5% − 95%

2.5% − 97.5%

1% − 99%

medianFIGURE 1. Posterior percentiles of regression coefficients and corresponding
one-sided posterior inclusion probabilities, P (βj > 0.05), in the analysis of subsets of the Salmonella Thompson data simulating the available data at different
time points during the outbreak.

variance parameters, τ and c, of the spike and slab components, were chosensuch that the two distributions intersect at a practical significance level of 0.05
and that the slab distribution is relatively narrow avoiding excessively large β’s,
which are very unlikely in practice. The intercept terms of response and covariate
models were given a diffuse normal prior distribution. Looking at the posterior
inclusion probabilities of each covariate (Fig. 1), the covariate for smoked fish
clearly had the highest inclusion probability and, therefore, our model correctly

84

Bayesian variable selection in food-borne disease outbreaks

identified the contaminated food product.

4

Conclusion

In this paper, we developed a method that deals with the problems of variableselection, missing covariates and misclassified responses in the context of source
identification in food-borne disease outbreaks. We have shown how a Bayesiananalysis allows a relatively easy implementation of these concepts in the reanalysis of the Dutch Salmonella Thompson 2012 outbreak data. Moreover, theBayesian analysis performed better than the standard logistic backward variableselection model (not shown in this paper).
The method presented in this paper constitutes a first attempt at formalizing themethodology necessary for the analysis part of food-borne disease outbreak investigations. Current procedures are very much ad hoc in nature resulting in difficult
to interpret and misleading results. The method in this paper is methodologicallysound and the analysis results are intuitive and easy to interpret.

References
Friesema, I., de Jong, A., Hofhuis, A., ... and van Pelt, W. (2014). Large outbreakof Salmonella Thompson related to smoked salmon in the Netherlands,
August to December 2012. Euro Surveillance, 19, 20918.
George, E.L. and McCulloch, R. (1993). Variable selection via Gibbs sampling.
Journal of the American Statistical Association, 88, 881 – 889.
Mitra, R. and Dunson, D. (2010). Two-Level Stochastic Search Variable Selection in GLMs with Missing Predictors. The International Journal of
Biostatistics, 6, 33.

Using extended dgLARS to study Diabetes
Hassan Pazira1 and Ernst Wit2
1

Johann Bernoulli Institute (JBI), University of Groningen, The Netherlands

E-mail for correspondence: h.pazira@rug.nl
Abstract: Diabetes is one of the most common long-term health conditions.
Today, diabetes takes more lives than AIDS and breast cancer combined. It is aleading cause of blindness, kidney failure, amputations, heart failure and stroke.
The aim of this study is to use the differential geometric generalization of the
LARS algorithm for the double-parameter Gamma GLM with a canonical linkfunction to identify candidate factors that may be associated with diabetes. To
compute the solution path faster, we use the improved predictor-corrector algorithm, as proposed in Pazira et al. (2017).
Keywords: High-dimensional inference, Gamma GLMs, Least angle regression,
Improved predictor-corrector algorithm, Dispersion parameter.

1

Introduction

In the last decade, the cases of people living with diabetes jumped almost 50
percent. Worldwide, it afflicts more than 380 million people. And the World
Health Organization estimates that by 2030, that number of people living withdiabetes will more than double. Today, diabetes takes more lives than AIDS
and breast cancer combined. It is a leading cause of blindness, kidney failure,amputations, heart failure and stroke. Living with diabetes places an enormous
emotional, physical and financial burden on the entire family.
In this paper we consider the benchmark diabetes data used in Efron et al. (2004)
and Ishwaran et al. (2010), among others. The response y is a quantitative measure of disease progression for patients with diabetes one year later. The data
includes 10 baseline measurements for each patient, such as age, sex, bmi (bodymass index), map (mean arterial blood pressure), and six blood serum measurements: ldl (high-density lipoprotein), hdl (low-density lipoprotein), ltg (lamotrigine), glu (glucose), tc (triglyceride) and tch (total cholesterol), in addition to 45
interactions and 9 quadratic terms, for a total of 64 variables for each patient, soThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

86Using extended dgLARS for Diabetes

that this data has n = 442 observations on p = 64 variables. The aim of the study
is to identify which of the covariates are important factors in disease progression.
This diabetes data can be found in the new version of our dglars package.
In literature several methods have been proposed to identify variables that canaffect the disease. For this kind of problems the number of variables, say p, canbe much larger than the sample size n. In this case, it is often assumed that onlya small number of covariates (factors) contributes to the response, which leads toassume the sparsity of the model, many elements of the coefficients vector β areequal to zero. In recent statistical literature, many variable selection techniques
for sparse regression models are based on the penalized likelihood approach toestimate a solution curve embedded in the parameter space and then to findthe point that represents the best compromise between sparsity and predictive
behaviour of the model. Some important examples are the Least Absolute Shrinkage and Selection Operator (LASSO) estimator (Tibshirani, 1996), the Smoothly
Clipped Absolute Deviation (SCAD) method (Fan and Li, 2001), among others.
Differently from the methods cited above, Efron et al. (2004) introduced a new
method to select important variables in a linear regression model called least angle regression method (LARS). Augugliaro et al. (2013) proposed a new approachbased on the differential geometrical representation of a GLM. The method, which
does not require an explicit penalty function, has been called differential geometric LARS (dgLARS) method because it is defined generalizing the geometricalideas on which LARS is based. The later authors considered a class of the exponential family for a GLM, so that, they assumed that the dispersion parameter isknown, φ = 1. They also used the predictor-corrector (PC) algorithm to computethe solution curve.
In this paper we consider the dgLARS method for a larger class of the exponential
family, namely the exponential dispersion family, when the dispersion parameter
φ is unknown, and obtain the extended dgLARS estimator for Gamma GLM with
arbitrary link function. Aim of this paper is to identify a set of important factorsthat can affect the diabetes disease using the extended dgLARS.

2

Differential Geometric LARS for general GLM

Let Y = (Y1 , Y2 , · · · , Yn )> be a n-dimensional random vector with independent
components. In what follows we shall assume that Yi is a random variable with
p.d.f belonging to an exponential dispersion family (Jorgensen, 1987, 1997), i.e.,yi θi − b(θi )pYi (yi ; θi , φ) = exp+ c(yi , φ) ,yi ∈ Yi ⊆ R,
(1)a(φ)
where θi ∈ Θi ⊆ R is the canonical parameter, φ ∈ Φ ⊆ R+ is the dispersion parameter, and a(.), b(.) and c(., .) are given functions. In the following,
we assume that each Θi is an open set and a(φ) = φ. We consider φ as an
unknown parameter. The expected value of Y is related to the canonical pai)rameter by µ = {µ(θ1 ), · · · , µ(θn )}> , where µ(θi ) = ∂b(θis called mean value∂θimapping, and the variance of Y is related to its expected value by the identity
Var(Y) = φV(µ), where V(µ) is an n × n diagonal matrix with elements, called2b(θi )
the variance functions, V (µi ) = ∂ ∂θ
. Since µi is a reparameterization, model2i(1) can be also denoted as pYi (yi ; µi , φ).

Pazira and Wit

87

Following McCullagh and Nelder (1989), a Generalized Linear Model (GLM) is
defined by means of a known function g(·), called link function, relating the expected value of each Yi to the vector of covariates xi = (1, xi1 , . . . , xip )> byththe identity g{E(Yi )} = ηi = x>linear predici β where ηi is called the i>tor and β = (β0 , β1 , . . . , βp ) is the vector of regression coefficients. In order
to simplify our notation we let µ(β) = {µ1 (β), . . . , µn (β)}> where µi (β) =g −1 (x>i β). Therefore,
Q the joint probability density function can be written as
pY (y; µ(β), φ) = n
i=1 pYi (yi ; µi (β), φ). In the following of this paper we shalluse `(β, φ; y) = log pY (y; µ(β), φ) as notation for the log-likelihood function,∂m `(β, φ; y) = ∂`(β,φ;y)= φ−1 ∂m `(β; y) as notation for the mth score function,∂βmand Imn (β, φ) = E[∂m `(β, φ; y) · ∂n `(β, φ; y)] = φ−1 Imn (β) as notation for aelement of the Fisher Information matrix function.
m `(β,φ;y)= φ−1 rm (β), helps
The Rao’s score test statistic, given as rm (β, φ) = ∂√Im (β,φ)to define ρm (β), the angle between the mth basis function ∂m `(β, φ; Y) and theP∂`(β,φ;y)tangent residual vector r(β, φ, y; Y) = n, defined as follows
i=1 (yi − µi )∂µiρm (β, φ) = arccos
rm (β, φ),kr(β, φ, y; Y)kp{µ(β)}(2)

where k·kp{µ(β)} is the norm defined on the tangent space Tp{µ(β)} M, where
the set M is a p-dimensional submanifold of the differential manifold S. From(2), the Rao’s score test statistic contains the same information as the angleρm (β, φ). Thereby we can define the extended dgLARS method with respect to
the Rao’s score test statistic rather than the angle as respects the smallest angle
is equivalent to the largest Rao’s score test statistic.
The extended dgLARS solution curve, which is denoted by β̂A (γ) ⊂ Rk+1 , withγ ∈ [0, γ (1) ], whereby 0 6 γ (p) 6 · · · 6 γ (2) 6 γ (1) , is defined in the following
way: For any γ ∈ (γ (k+1) , γ (k) ], the dgLARS estimator is chosen in such a way
that
A(γ) = {a1 , a2 , · · · , ak },|rai (β̂(γ), φ)| = |raj (β̂(γ), φ)| = γ,
|r

ac
h

(β̂(γ), φ)| < |rai (β̂(γ), φ)| = γ,∀ai , aj ∈ A(γ),(3)∀ach ∈ Ac (γ) and ∀ai ∈ A(γ),

where A(γ) = {m : β̂m (γ) 6= 0} is called active set and Ac (γ) = {m : β̂m (γ) =0} is the complement of the active set. The new covariate is included in the active
set at γ = γ (k+1) where the following condition is satisfied:∃ach ∈ Ac (γ (k+1) ) :|rach (β̂(γ (k+1) ), φ)| = |rai (β̂(γ (k+1) ), φ)|, ∀ai ∈ A(γ (k+1) ).

(4)

From Pazira et al. (2017) we know that the value of the estimated dispersion
parameter does not change the order of the variables included in the active set
while it affects the value of `(β̂, φ̂; y), and as a result it affects the value of various
information criteria such as AIC or BIC.
For the high-dimensional setting (p ≥ n), we use the dispersion estimator φ̂P (γ)
at γ ∈ [0, γmax ] by the Pearson-like dispersion estimator, as proposed by Wood

88

Using extended dgLARS for Diabetes(2006);nφ̂P (γ) =2
X (yi − g −1 (x>1i β̂A (γ))),
n − |A(γ)| i=1 V (g −1 (x>i β̂A (γ)))(5)

where |A(γ)| = #{j : β̂j (γ) 6= 0} such that β̂j (γ) is the element of the extended
dgLARS estimator β̂A (γ).
To compute the solution curve, we use the improved predictor-corrector (IPC)
algorithm, described in Pazira et al. (2017), because this algorithm leads to an
decrease in the run times needed for computing the solution curve.

3

Application to Diabetes Dataset

In the recent literature, variable selection techniques, such as LARS and Spikeand Slab, were used in a linear regression model applied to the explained diabetes
data. While we spotted that, surprisingly, the response is markedly right-skewed
which can arise from a non-normal distribution, e.g. Gamma. Therefore, we fit a
Gamma regression model for this diabetes data and use the extended dgLARS
method by means of the IPC algorithm.
We first apply a number of variable selection methods such as LARS (Efron etal., 2004), LASSO (Tibshirani, 1996), Elastic Net (Zou and Hastie, 2005), andSpike and Slab (Ishwaran et al., 2010) by using the lars, glmnet and spikeslab
packages, and then compare the results to the results obtained from the proposed
dgLARS method implemented by our dglars package. Note that, for the dgLARS
method we use the Gamma family in our package, while this family is not available
in other packages, so that we fit the Gaussian family to the data to be able touse these packages.
When we compare the results of the dgLARS Gamma method to the results
obtained from other algorithms, we find out the remarkable results. From Table
1 we can see that, the variables selected by the LARS, LASSO and Elastic Net
methods are the same, and almost in all models the first 4 variables (3, 9, 4 and 7)
are the same. Moreover, importantly, all models (except the dgLARS) have thesame selected variables just in the different order. While all algorithms (except
the dgLARS) select the covariates 37, 12, 22, 27, 33 and 52 in the first 20 variables,
our proposed algorithm does not select them among the top 20 variables. Instead,the dgLARS algorithm by the Gamma model selects several new other variables(indicated in bold in Table 1) which none of the other algorithms do. For instance,
the variables 60, 18, 42, 35 and 40 are selected with the inverse link function.

TABLE 1. The sequences of the top 20 predictors selected by the LARS, LASSO,
Elastic Net, Spike and Slab and dgLARS algorithms obtained for diabetes data.
Algorithm
LARS
LASSO
Elastic Net
Spike and SlabdgLARS (inverse)

3
3
3
3
3

9
9
9
9
9

4
4
4
4
4

7
7
7
7
7

37
37
37
2
20

20
20
12
20
60

19
19
20
37
2

12
12
19
19
46

Selected Variables
22 28
2 10 27
22 28
2 10 27
10 22 28
2 27
12 27 52 11 10
18 10 42 28 11

11
11
30
22
19

30
30
11
63
30

46
46
52
30
35

33
33
46
24
29

52
52
33
58
40

24
24
24
43
24

29
29
29
5
63

Pazira and Wit

89

TABLE 2. A list of the top 20 selected variables and their parameter estimatesobtained using dgLARS Gamma method (with inverse canonical link, ηi = − µ1i )for diabetes data. |ACV | = 20, |AAIC | = 24 and |ABIC | = 10.
Step
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Variable
Name
Number
bmi
ltg
map
hdl
age : sex
hdl : ltg
sex
map : hdl
ltg^2glu
bmi : ltg
age : glu
age^2
glu^2
sex : map
sex : ltg
sex : bmi
bmi : hdl
age : ldl
tch : glu

3
9
4
7
20
60
2
46
18
10
42
28
11
19
30
35
29
40
24
63

Coefficient Estimate
CV
AIC
BIC
0.01820.02620.0129-0.01450.00670.0046-0.00900.0040-0.00530.0001-0.00260.00080.00210.00160.00120.00070.00060.0004-0.000200.01870.02780.0136-0.0159
0.00690.0053-0.01130.0052-0.0067-0.0001-0.00320.00100.00270.00120.00200.00150.00150.0015-0.00140.00130.01710.02050.0101-0.0105
0.00420.0032-0.00350.0011-0.0015
0
0
0
0
0
0
0
0
0
0
0

In Table 2, we also report the sequence of the top 20 variables and their parameter
estimates obtained using the dgLARS Gamma method with canonical link, ηi =− µ1i , based on variable selection methods: AIC, BIC and CV.
As a result, the extended dgLARS method based on a Gamma model, with theinverse canonical link function, finds out that the variables ”hdl : ltg”, ”ltg^2”,”bmi : ltg”, ”sex : ltg” and ”bmi : hdl” (namely: 60, 18, 42, 35 and 40) are more
important factors in disease progression than the variables ”bmi : map”, ”bmi^2”,”age : map”, ”age : ltg”, ”sex : hdl” and ”tc : tch” (namely: 37, 12, 22, 27, 33 and52).

References
Augugliaro, L., Mineo, A.M., and Wit, E.C. (2013). Differential Geometric Least
Angle Regression: A Differential Geometric Approach to Sparse Generalized Linear Models, J. R. Statist. Soc. B, 75, 471 – 498.
Efron, B., Hastie, T., Johnstone, I., and Tibshirani, R. (2004). Least Angle Regression. Ann. Statist., 32, 407 – 499.
Fan, J. and Li, R. (2001). Variable Selection Via Nonconcave Penalized Likelihood and Its Oracle Properties. J. Am. Statist. Ass., 96, 1348 – 1360.
Hastie T., Tibshirani, R., and Friedman, J. (2009). The Elements of Statistical
Learning: Data Mining, Inference, and Prediction. New York: Springer.
Ishwaran, H., Kogalur, U.B., and Rao, J.S. (2010). spikeslab: Prediction and variable selection using spike and slab regression. The R Journal. 2, 68 – 73.

90

Using extended dgLARS for Diabetes

McCullagh, P. and Nelder, J.A. (1989). Generalized Liner Models. London: Chapman and Hall.
Pazira, H., Augugliaro, L., and Wit, E.C. (2017). Extended differential geometric LARS for high-dimensional GLMs with general dispersion parameter.
Submitted.
Tibshirani, R. (1996). Regression Shrinkage and Selection Via the Lasso. J. R.
Statist. Soc. B, 58, 267 – 288.
Wood, S.N. (2006). Generalized Additive Models: An Introduction with R. Boca
Raton: Chapman & Hall/CRC.
Zou, H. and Hastie, T. (2005). Regularization and Variable Selection via the Elastic Net. J. R. Statist. Soc. B, 67, 301 – 320.

Stable Variable Selection with AdaSub
Christian Staerk1 , Maria Kateri1
1Institute of Statistics, RWTH Aachen University, Germany

E-mail for correspondence: christian.staerk@rwth-aachen.de
Abstract: The recently proposed Adaptive Subspace Method (AdaSub) aims
at finding the best model according to a certain variable selection criterion. We
illustrate the performance of AdaSub and its stability with respect to changesof its tuning parameters. Furthermore we demonstrate through simulations thatin case the model selected by AdaSub does not coincide with that proposed by
the criterion, AdaSub often reduces the number of falsely selected variables (false
positives) and provides a more stable model.
Keywords: Adaptive Subspace Method; Variable Selection; Linear Regression;
Bayesian Information Criterion.

1

Introduction

We consider the problem of variable selection in linear regression models. Classical variable selection criteria include the Akaike Information Criterion (AIC)
and the Bayesian Information Criterion (BIC), aiming at optimal predictions andidentification of the true generating model, respectively. The challenging problem
with these `0 -type selection criteria is that they lead to combinatorial and in general NP-hard optimization problems: If there are p possible explanatory variables,
then there are 2p possible models for which the criterion has to be evaluated in afull enumeration. To overcome this prohibitive computational approach, Staerk
et al. (2016) propose the Adaptive Subspace Method (AdaSub) which is based on
the idea of adaptively solving lower-dimensional sub-problems in order to provide
a solution to the original problem. It can be shown that, under certain conditions,
AdaSub identifies the best model according to the criterion used (Staerk et al.,2017).
This work focuses on two important issues related to the performance of the AdaSub algorithm. Since AdaSub is a stochastic algorithm in which certain tuning
parameters have to be specified, it is crucial to investigate the role of these parameters and their effect on the stability of the model selected by AdaSub. AfterThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

92Stable Variable Selection with AdaSuba brief presentation of the AdaSub algorithm in Section 2, this issue is addressedin Section 3.
The second issue refers to the interesting scenario in which AdaSub does not
identify the best model according to the criterion. It is well-known that the BIC
is variable selection consistent in the classical asymptotic setting, i.e. the best
BIC model coincides with the true generating model with probability tending to
one if the sample size n tends to infinity and the number of explanatory variables
p is bounded. However, this does not necessarily imply that the best BIC modelis always a good approximation to the “truth” for finite sample sizes n. In fact,
it will be demonstrated that in small sample size situations the best BIC modeltends to select many noise variables (false positives) and to overfit the data if
the true underlying model is relatively sparse. In Section 3 we show throughsimulations that AdaSub can mitigate the problem of selecting many “unstable”
variables in such situations.

2

Adaptive Subspace Method (AdaSub)

Let {Xj ; j ∈ P} be the set of explanatory variables with index set P = {1, . . . , p}
and let M = P(P) = {S ⊆ {1, . . . , p}} be the corresponding space of linear
models. Suppose that we observe data D = (X, Y ) with design matrix X andresponse Y . Let C : M → R be any model selection criterion for the given data
D (e.g. the BIC). We assume that C(S) 6= C(S 0 ) for all S 6= S 0 and that we wantto find the model S ∗ ∈ M that minimizes C, i.e. S ∗ := arg minS∈M C(S).
Define the map fC : M → M by fC (V ) := arg minS⊆V C(S) for V ∈ M. So
fC (V ) denotes the best model according to criterion C among all models includedin V . The steps of AdaSub are given by:(1) Initialize expected search size q ∈ (0, p), learning rate K > 0 and numberof iterations T ∈ N.
(0)(2) For j ∈ P initialize rj = pq .
(3) For t = 1, . . . , T :
(t−1)
(t)
) independently for j ∈ P.
(a) Draw bj ∼ Bernoulli(rj(t)(t)(b) Set V= {j ∈ P; bj = 1}.
(c) Compute S (t) = fC (V (t) ).
P(t)(d) For j ∈ P update rj

q+Kt

1 (i) (j),(i) (j)= p+K Pti=1 1 S
i=1where 1A denotes theVindicator function of a set A.
As the final subset selected by AdaSub one can either (i) choose the “best”
sampled model Ŝb for which C(Ŝb ) = min{C(S (1) ), . . . , C(S (T ) )}, or (ii) consider(T )Ŝρ = {j ∈ P; rj > ρ} with some threshold ρ ∈ (0, 1).

3

Choice of Tuning Parameters in AdaSub

In order to illustrate the performance of AdaSub in a high-dimensional setup and how it is effected by the choice of the tuning parameters K and q,
we consider an example with p = 1000 and n = 100. For this we generate
data D = (X, Y ) by simulating X = (Xij ) ∈ Rn×p with independent rows

Staerk and Kateri

93

Xi,∗ ∼ Np (0, Σ), where Σkl = 0.3 for k 6= l and Σkk = 1. Furthermore let
β 0 = (1, −1, 1, 2, −2, 2, 0, . . . , 0)T ∈ Rp be the true vector of coefficients withactive set S0 = {1, . . . , 6}. The response Y = (Y1 , . . . , Yn )T is simulated via
ind.

Yi ∼ N (Xi,∗ β 0 , 1), i = 1, . . . , n. The criterion C we adopt is the Extended BIC(EBIC) with parameter γ = 0.5, which is especially suited for high-dimensionalsituations (Chen and Chen, 2008). The R-package “leaps” (Lumley and Miller,2009) is used to compute fC (V ) for V ∈ M.

FIGURE 1. AdaSub for high-dimensional example: Plot of the evolution of
EBIC(S (t) ) along the iterations (t) for different values of K (q = 5 fixed).

(t)

FIGURE 2. AdaSub for high-dimensional example: Plot of the evolution of rj
for j ∈ {1, 3, 12} along the iterations (t) for different values of K (q = 5 fixed).

We apply AdaSub with T = 10, 000 iterations on a dataset simulated as above
and fix q = 5 as the initial expected search size. Figure 1 and Figure 2 show(t)the evolution of EBIC(S (t) ) and rj along the iterations t for different values ofthe learning rate K ∈ {10, 100, 1000}. In all three cases the algorithm identifies
the correct model S0 = {1, . . . , 6} with EBIC(S0 ) ≈ 371 based on Ŝb and Ŝρ forρ ∈ [0.2, 0.9], i.e. S0 = Ŝb = Ŝρ .
However, there is a trade-off in choosing K > 0: If K is small (K = 10), then
AdaSub adapts slowly to the information learned about the variables and hence
a very diverse range of models is considered. If instead K is large (K = 1000),then the algorithm might actually “converge” too fast. Suppose for example thata variable Xj is not chosen when it is first considered in the model search (i.e.
(t)
qj ∈ V (t) but j ∈/ S (t) ), then rj = p+K≈ 0 for K very large, so variable Xj
will probably not be considered in the model search for a long time. In our case,the choice K = 100 = n seems favourable, for which AdaSub only needed 1123

94

Stable Variable Selection with AdaSub

iterations to find S0 . Nevertheless, an important observation is that the selectedmodel by AdaSub is stable with respect to changes of K, as long as the numberof iterations is large enough.

FIGURE 3. AdaSub for high-dimensional example: Plot of the sizes of the sampled sets V (t) (grey dots) and the sizes of the “best” subsets S (t) (red crosses)
along the iterations (t) for different values of q (K = 100 fixed).

Expected Size of V^(t)

15

q=2
q=5
q = 10

10

5

0
0

2000

4000

6000

8000

10000

FIGURE 4. AdaSub for high-dimensional example: Plot of the evolution of the
expected search size along the iterations (t) for different values of q (K = 100
fixed).
We now apply AdaSub with T = 10, 000 iterations on the same dataset as above
for different values of q ∈ {2, 5, 10}, while K = 100 is fixed. Figure 3 shows thesizes of the sampled sets V (t) and the sizes of the “best” subsets S (t) = fC (V (t) )along the iterations t, while Figure 4 depicts the evolutionof the expected sizesP(t−1)
of the sets V (t) which are given by E[|V (t) |] =r
for t = 1, . . . , T .
j∈P j
We can see that the AdaSub algorithm automatically and quickly adjusts the
expected search sizes E[|V (t) |] and that the algorithm “converges” against thetrue underlying model S0 with six variables, no matter which initial expected
search size q is used. Ideally, the tuning parameter q should be chosen in a way
such that it reflects the subjectively expected size of the best model S ∗ = fC (P)according to criterion C. However, a general observation is that the choice of thetuning parameter q seems not to be as crucial as the proper choice of the learning
rate K. A more detailed and refined analysis of theses issues is an interesting topic
for future research.

Staerk and Kateri

495

Improving Stability of BIC by AdaSub

In order to investigate the stability of BIC, we consider a low-dimensional scenario
with p = 30, so that the determination of the best BIC model S ∗ is computationally feasible. The sample size n is increased from 40 to 200 in steps of size 20 andfor each value of n we simulate 100 different datasets according to the simulation
setup described above, with the following modification: For each dataset, we select s0 ∈ {0, . . . , 10} and S0 ⊂ P of size |S0 | = s0 randomly; then for j ∈ S0 we
independently simulate βj0 ∼ U [−2, 2] from the uniform distribution, while we set
βj0 = 0 for j ∈/ S0 . Based on these simulated data, we compare the performanceof the thresholded model Ŝρ with threshold ρ = 0.9 from AdaSub with the best
BIC model S ∗ . In AdaSub we set T = 2000, q = 5 and K = n.
Mean false positives

Rel. frequency true model selected

6

1.0

5

0.8

4

MSE on average
2.0

0.6

1.5

0.4

1.0

1

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0

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3
2

40

60

80

100 120 140 160 180 200

40

60

80

100 120 140 160 180 200

0.0

40

60

80

100 120 140 160 180 200

n

n

n

Mean false negatives

Rel. frequency AdaSub yields best BIC

MSPE on average

6

1.0

5

0.8

2.5

4

0.6

2.0

3
0.4

2

1.5
0.2

1
0

0.0
40

60

80

100 120 140 160 180 200

n

40

60

80

100 120 140 160 180 200

n

1.0

40

60

80

100 120 140 160 180 200n

FIGURE 5. Comparison of Ŝ0.9 from AdaSub (solid lines with crosses) withbest BIC model S ∗ (dashed lines with dots) in terms of mean number of false
positives/ false negatives, relative frequency of selecting the true model S0 , rel.
frequency of agreement between Ŝ0.9 and S ∗ , Mean Squared Error (MSE) andMean Squared Prediction Error (MSPE) on independent test set with sample
size 100.
Figure 5 shows that the best BIC model S ∗ tends to select many false positives
if the sample size is small. In contrast, the thresholded model Ŝ0.9 selected by
AdaSub is often sparser and yields less false positives in situations where the
BIC is too liberal (at the prize of a slightly increased mean of false negatives).
When the sample size increases, the best BIC model S ∗ becomes more “stable”
and the relative frequency that Ŝ0.9 and S ∗ agree tends to 1. However, selecting
the thresholded model from AdaSub is beneficial for small sample sizes yielding
higher relative frequencies of selecting the true model S0 , smaller Mean Squared
Errors (MSE) and smaller Mean Squared Prediction Errors (MSPE).
A reason for the undesirable behaviour of the best BIC model is that the discrete
nature of the `0 -penalty can lead to “overfitting” of the criterion, since the optimization is carried out among all possible 230 ≈ 109 models. This problem has
been addressed both by the Statistics community (e.g. Breiman, 1996) and theMachine Learning community (e.g. Loughrey and Cunningham, 2005). However,96

Stable Variable Selection with AdaSub

the simulation results show that AdaSub can mitigate the “overfitting problem”
of BIC in the given situation of a sparse underlying true model.
The tendency that AdaSub selects a sparser model in unstable situations is also
observed in additional simulations with different correlation structures of X anddifferent selection criteria C. In ongoing research we want to investigate this phenomenon further and aim to provide theoretical explanations for the promising
performance of AdaSub.

References
Breiman, L. (1996). Heuristics of instability and stabilization in model selection,
The Annals of Statistics, 24(6), 2350 – 2383.
Chen, J. and Chen, Z. (2008). Extended Bayesian Information Criteria for Model
Selection with Large Model Spaces, Biometrika, 95(3), 759 – 771.
Loughrey, J. and Cunningham, P. (2005). Overfitting in wrapper-based feature
subset selection: The harder you try the worse it gets, In Research andDevelopment in Intelligent Systems XXI, Springer London , 33 – 43.
Lumley, T. and Miller, A. (2009). leaps: Regression Subset Selection. R package
version 2.9, http://CRAN.R-project.org/package=leaps.
Staerk, C., Kateri, M. and Ntzoufras, I. (2016). An Adaptive Subspace Me-thodfor High-Dimensional Variable Selection. In: Proc. of the 31st International
Workshop on Statistical Modelling, Rennes, 295 – 300.
Staerk, C., Kateri, M. and Ntzoufras, I. (2017). High-Dimensional Variable Selection via Low-Dimensional Adaptive Learning. (submitted)

Boosting distributional regression models for
multivariate responses
Andreas Mayr12 , Janek Thomas2 , Matthias Schmid3 , Florian
Faschingbauer1 , Nadja Klein4
1

Friedrich-Alexander-University (FAU) Erlangen-Nürnberg, Germany
Ludwig-Maximilians-University Munich, Germany
3
Rheinische Friedrich-Wilhelms-University Bonn, Germany,
4
University of Melbourne, Melbourne Business School, Australia
2E-mail for correspondence: andreas.mayr@fau.de
Abstract: We introduce a boosting algorithm for multivariate distributional
regression that is able to estimate these complex models while simultaneously
selecting the most informative variables in potentially high-dimensional settings.
Our proposed method is evaluated empirically and applied in a recent birth cohort
study. The aim is to model the growth of children via the combined distributionof height and weight during early childhood to identify possible predictors.
Keywords: correlated responses; gradient boosting; multivariate GAMLSS.

1

Introduction

One of the most popular semi-parametric statistical modelling approaches beyond
the classical regression of the mean are generalized additive models for location,
scale and shape (GAMLSS, Rigby and Stasinopoulos, 2005). The main idea of
GAMLSS is that each parameter of the conditional distribution – not only the
expected value – is modelled by its own additive predictor. This flexible framework can be further extended towards multivariate outcomes, in order to model
the joined distribution of two or more responses (Klein et al., 2014). We combine this approach with an extended statistical boosting algorithm (Mayr et al.,2012) which allows to estimate statistical models while simultaneously selecting
the most influential variables. Besides evaluating its performance in simulations,
we use this approach for the joint distribution of height and weight of a Germanbirth cohort study in order to select predictors for the growth of children. Particularly, we are looking for predictors in the clinical information from mother andThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

98Boosting multivariate distributional regression

child at birth and in a questionnaire on the socio-demographic characteristics ofthe parents (e.g., education).

2

Multivariate GAMLSSIn case of a univariate Gaussian response Y ∼ N (µ, σ 2 ) in GAMLSS we modelboth the location µ = ηµ (x) and the scale σ = exp(ηµ (x)) of the conditional
distribution via additive predictors. In case of a bivariate Gaussian distribution,e.g., for Y = (weight, height)> ∈ R2 ,   2
weightµ1
σ1∼N,heightµ2
ρσ1 σ2

ρσ1 σ2σ22

we follow the same principle,leading to five different additive predictors, including
one for the correlation parameter ρ.
ηµ1 (x) = µ1 ,
ηµ2 (x) = µ3 ,3ησ1 (x) = log(σ1 ),ησ2 (x) = log(σ2 ),pηρ (x) = ρ/ (1 − ρ2 )Boosting multivariate GAMLSS

While variable selection is already a complicated issue for GAMLSS, the complexity further increases for multivariate distributions. In order to deal with this issue,
we use a further extended version of the gradient boosting algorithm introducedin Mayr et al. (2012). The new algorithm fits in every step m = 1, ..., mstop all partial derivatives of the joint likelihood one-by-one to the base-learners h1 (x1 ),...,
hp (xp ), selecting the best-performing one j ∗ for each dimension. Afterwards, thebest overall update is selected based on the potential increase in the likelihood(Thomas et al., 2017):∂
selectl(y1 , y2 , µˆ1 [m] ,σˆ1 [m] ,µˆ2 [m] ,σˆ2 [m] ,ρ̂[m] ) −→∂ηµ1∂
selectl(y1 , y2 , µˆ1 [m] ,σˆ1 [m] ,µˆ2 [m] ,σˆ2 [m] ,ρ̂[m] ) −→∂ησ1∂
selectl(y1 , y2 , µˆ1 [m] ,σˆ1 [m] ,µˆ2 [m] ,σˆ2 [m] ,ρ̂[m] ) −→∂ηµ2∂
selectl(y1 , y2 , µˆ1 [m] ,σˆ1 [m] ,µˆ2 [m] ,σˆ2 [m] ,ρ̂[m] ) −→∂ησ2∂
selectl(y1 , y2 , µˆ1 [m] ,σˆ1 [m] ,µˆ2 [m] ,σˆ2 [m] ,ρ̂[m] ) −→
∂ηρ

jµ∗1
jσ∗1
jµ∗2 best

update [m+1]−→ η̂µ2jσ∗2jρ∗Only this best overall-update is finally carried out, leading to data driven variable selection and mode-choice. Note, that the boosting algorithm is fitting theMayr et al.
1.0

µ1

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X12

FIGURE 1. Resulting coefficients for 12 partly overlapping informative variablesfrom 100 simulations with n = 500 and p = 1000. Each plot refers to one additive
predictor (ηµ1 , ησ1 , ηµ2 , ησ2 , ηρ ). The horizontal lines are the corresponding true
values: Generally, the algorithm selects the correct variables – the amount ofshrinkage differs among the predictors.

negative gradient of the joint likelihood (with respect to the different additive
predictors) and not the actual observations.
An implementation of this extended algorithm is provided via a new option in
the R add-on package gamboostLSS (Hofner et al., 2016).

4

Simulation

We first carried out a simulation study with n = 500 bivariate Gaussian distributed outcomes and p = 1000 explanatory variables. Only 12 variables havean actual effect on any of the distribution parameters (see Figure 1). Our results
suggest that the algorithm converges to the correct solution and is able to identify a small subset of informative variables in potentially high-dimensional data
situations . There is small tendency to falsely include informative variables inboth scale parameters; e.g. X10 , which actually has only an effect on σ2 and ρ isoften also included for σ1 .

100

Boosting multivariate distributional regressionµweightµheight
15

2

10

partial effectpartial effect

5
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age [years]

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3
age [years]

0.95
0.90

partial effect

0.85
0.80
0.75
0.70
0.65

1

2

3

4

5

FIGURE 2. Partial effect of the age on the parameters of the joint distributionof weight and height in early childhood.

5Birth cohort study

We apply our new approach analysing the joint distribution of weight and heightin early childhood. The underlying data set is a birth cohort study including
453 children born at the University Hospital Erlangen, with measurements atfive different time points. As possible explanatory variables we have the clinicalinformation of the mother and the baby at birth, the weight and the height of theparents and results of a questionnaire on their socio-demographic background. For
continuous variables we use P-splines as base-learners while all other variables
enter via simple-linear models. To account for the longitudinal structure of thedata we include subject-specific random intercepts.
Picking the most influential variables, the results of our models suggest, that theweight in early childhood is influenced to a much greater extend by the sociodemographic background of the parents (e.g., education level) than the height.
The latter (µheight ) only depends on the parents’ height at birth. Generally, thealgorithm selected more variables for σweight and σheight than for the mean parameters (see Table 1) . The correlation between height and weight follows au-type shape, while the variance increases for both outcomes with the children
getting older (Figure 2).

Mayr et al.

101

TABLE 1. Selected variables for the different additive predictors and the typeof their effect (linear positive, linear negative, smooth) on the growth of children.
The results are based on the analysis of a recent birth cohort study with thebivariate outcome Y = (weight, height)> .

6

Variableµweightµheight

σweight

σheightρ

age childbirth-weightweight father
weight motherheight father
height motherfemale childeducation motherage motherbreast-fed
cesarean-section
stress-level (birth)gestational age
alcohol (pregnancy)smoothsmooth
pos.
pos.
neg.
neg.
neg.
neg.

smoothsmoothsmooth
pos.
pos.
neg.
neg.

smoothsmoothsmooth

pos.
pos.

neg
pos.

neg.

pos.
pos.

neg.

neg.
neg.
neg.
neg.
neg.
neg.

neg.
neg.

Discussion

We propose a boosting algorithm to estimate and select GAMLSS distributional
regression models (Rigby and Stasinopoulos, 2005) for multivariate outcomes.
GAMLSS had been already extended towards multivariate distributions by Klein
et al. (2014) in a Bayesian setting.
The proposed boosting algorithm, due to being based on a machine-learning
approach, is applicable to high-dimensional data with more candidate variables
than observations (p > n) – as illustrated in the simulation. It takes advantage
of a recent extension (Thomas et al., 2017) of the original algorithm for boosting
GAMLSS (Mayr et al., 2012). Following this new approach, tuning of the algorithm (via the number of boosting iterations) boils down to a one-dimensional
problem, making the computationally burdensome grid search (Hofner et al.,2016) for the optimal combination of stopping iterations unnecessary.
In the application to the birth cohort study we illustrated how the algorithm
simultaneously selects only the most influential variables for the different predictors: although being a relatively low-dimensional data set, this task would havebeen infeasible for most other variable selection approaches for distributional
regression models.

Acknowledgments: The work on this article was supported by the Deutsche
Forschungsgemeinschaft (http://www.dfg.de, grant number SCHM 2966/1-2) andthe Interdisciplinary Center for Clinical Research of the Friedrich-Alexander Uni-

102

Boosting multivariate distributional regression

versity Erlangen-Nürnberg (project J49).

References
Klein, N., Kneib, T., Klasen, S. and Lang, S. (2014). Bayesian Structured Additive Distributional Regression for Multivariate Responses. Applied Statistics, 64, 569-591.
Hofner, B., Mayr, A., Schmid, M. (2016). gamboostLSS: An R Package for Model
Building and Variable Selection in the GAMLSS Framework. Journal of
Statistical Software , 74(1), 131.
Mayr, A., Fenske, N., Hofner, B., Kneib, T. and Schmid, M. (2012). Generalized
additive models for location, scale and shape for high-dimensional data –a flexible approach based on boosting. Applied Statistics, 61(3), 403-427.
Mayr, A., Binder, H, Gefeller, O. and Schmid, M. (2014). The Evolution of Boosting Algorithms - From Machine Learning to Statistical Modelling. Meth Inf
Med, 53(6), 419-427.
Thomas, J., Mayr, A., Bischl, B., Schmid, M., Smith, A., Hofner, B. (2017). Stability selection for component-wise gradient boosting in multiple dimensions. arXiv : 1611.10171.
Rigby, R. A. and Stasinopoulos, D. M. (2005). Generalized additive models forlocation, scale and shape. Applied Statistics, 54, 507 – 554.

LASSO-type penalization in the frameworkof Generalized Additive Models for Location,
Scale and ShapeAndreas Groll1 , Thomas Kneib1 , Nikolaus Umlauf2
1
2

Department of Statistics, University of Göttingen, Germany
Department of Statistics, University of Innsbruck, Austria

E-mail for correspondence: agroll@uni-goettingen.de
Abstract: We propose a regularization approach for high dimensional data setups in the generalized additive model for location, scale and shape (GAMLSS)
framework. It is designed for linear covariate effects and is based on L1 -type
penalties. The following three penalization options are provided: the conventional
least absolute shrinkage and selection operator (LASSO) for metric covariates,and both group and fused LASSO for categorical predictors.
Keywords: GAMLSS; Lasso; Penalization; Variable selection; Variable fusion.

1

Introduction

A model class that has gained increasing attention in recent years is the classof the GAMLSS, introduced by Rigby and Stasinopoulos (2005). In contrast toconventional regression approaches, where usually the expected mean is regressed,in the GAMLSS all distribution parameters (as, for example, the location, scaleand shape) can be simultaneously modeled in terms of covariates. However, inhigh dimensional data set-ups classical fitting procedures for the GAMLSS often
become very unstable and methods for variable selection are desirable.
The first ones who addressed the issue of variable selection, i.e. the selection of areasonably small subset of informative covariates to be included in a particular
GAMLSS, were Mayr et al. (2012). They extended boosting techniques, which
originated in the machine learning field, to the framework of the GAMLSS. The
approach is called gamboostLSS and is based on classical gradient boosting,which they successfully adapted to the GAMLSS characteristics. Both variableselection and model choice are naturally available within their regularized regresThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

104

L1 -type penalization for GAMLSSsion framework. For an implementation into the statistical software R, see Hofneret al. (2016).
The present work describes a different regularization approach for the high dimensional GAMLSS framework, which is designed for linear covariate effects
only and is based on L1 -type penalties. Using adequate penalties, the cases of
conventional LASSO for metric covariates, and both the group and fused LASSO
for categorical predictors are covered. The implementation of the methods is incorporated into the unified modeling architecture for distributional generalized
additive models (GAMs) established in Umlauf et al. (2017a), which exploits
the general structure of GAMs and encompasses many different response distributions, estimation techniques, model terms etc. The corresponding R-package
bamlss (Umlauf et al., 2017b) embeds many different approaches suggested inliterature and software and serves as a unified conceptional “Lego toolbox” forcomplex regression models. Furthermore, within its framework both the implementation of algorithms for complex regression problems and the integration of
already existing software are substantially facilitated.
For illustration purposes, the proposed methods are applied to Munich rent standard data, which are used as a reference for the average rent of a flat depending
on its characteristics and spatial features.

2

Model specification

Along the lines of Rigby and Stasinopoulos (2005), who regard the GAMLSS
as a semiparametric regression-type model with both linear and smooth covariate effects, in the following we focus on the fully parametric model with solely
linear effects. Let y = (y1 , . . . , yn )T be the response vector with single observations yi , i = 1, . . . , n, being conditionally independent given a set of covariates.
The corresponding conditional density f (yi |θi ) usually depends on several distribution parameters θi = (θi1 , . . . , θid )T that commonly represent distribution
characteristics like location, scale, shape and/or kurtosis, but generally may be
any of the distribution’s parameters. After all, the key feature of a GAMLSS isthat each distribution parameter θk can be modeled by its own predictor ηθk for
k = 1, . . . , d, which, in our case, depends linearly on a set of pk covariates together
with an intercept β0k . Following Mayr et al. (2012), we denote by gk (·) known
monotonic link functions, corresponding to the linear predictor of the submodelof parameter θk . Then, a generalized linear model for location, scale and shapeis given by the following set of equations
gk (θk ) = β0k +

pk
X

xT
jk βjk = ηθk .

j=1

As the covariates can be metric and/or categorical, we use the general notationxT
jk βjk for a single predictor term, which either collects all covariate dummies
and regression coefficients corresponding to the jk-th group of variables, if thecovariate is categorical, and which reduces to a product of scalar values, if thecovariate is metric, i.e. xjk βjk . Estimation of regression parameters can be basedon maximizing the model’s log-likelihoodl(β) =nX
i=1

log (f (yi |θi )) ,(1)

Groll and Umlauf

105

with vector β collecting the effects of all linear predictors ηθk , k = 1, . . . , d.
Note that the log-likelihood (1) depends on the parameters βjk through the relations θik = g −1 (ηθik ). Suitable fitting schemes are implemented in the R-package
gamlss (Stasinopoulos and Rigby, 2007) and base on the following principle: ineach iteration, backfitting steps are successively applied to all distribution parameters, using the submodel fits of previous iterations as offset values for thoseparameters that are not involved in the current step. However, in high dimensional situations these fitting procedures often become very unstable and methods
for variable selection are needed.

3

L1 -type penalization

In the following, three L1 -type penalties are introduced, which are designed forlinear covariate effects: the conventional LASSO for metric covariates, and thegroup and fused LASSO if categorical covariates are present. Altogether, a termλJ(β) is subtracted from the log-likelihood (1), where J(β) is a combinationof (parts of) the three penalty terms from below and λ a tuning parameter,
controlling the strength of the penalty.
Classical LASSO: For a metric covariate xjk , following Tibshirani (1996), theabsolute value of its (scalar) regression coefficient is penalized, i.e.
Jm (βjk ) = |βjk | .
Group LASSO: For a (dummy-encoded) categorical covariate xjk , the L2 -normof vector βjk , which collects all corresponding coefficients, is penalized (compare,e.g., Meier et al., 2008), i.e.
Jg (βjk ) = ||βjk ||2 .
Fused LASSO: Alternatively, for categorical covariates often clustering of categories with implicit factor selection is desirable. Depending on the nominal (firstexpression) or ordinal scale level (second expression) of the covariate, one of thefollowing two penalties can be used (compare Gertheiss and Tutz, 2010):
Jf (βjk ) =

X
l>m

(jk)

wlm |βjkl − βjkm |,orJf (βjk ) =

cjk
X

(jk)

wl

|βjkl − βjk,l−1 | ,

l=1
(jk)

(jk)

where cjk is the number of levels of categorical predictor xjk and wlm , wl
denote suitable weights. Choosing l = 0 as the reference, βjk0 = 0 is fixed.

All proposed penalties have the attractive property to be able to set the coefficients of single (groups of) covariates to zero and, hence, to perform variableselection. Within the estimation procedure, i.e. the corresponding back-fitting algorithm implemented in bamlss, local quadratic approximations of all presented
penalty terms are used (see Oelker and Tutz, 2015). Further, note that bamlss
generally also allows to assign to each linear predictor of the d submodels (corresponding to parameters θk , k = 1, . . . , d) its own penalty term, i.e. a term λk J(β).
This way, the estimated models become even more flexible, with the drawbackthat then the grid search for the optimal tuning parameters λk has to be carriedout on d dimensions and, hence, becomes computationally more demanding.

106

L1 -type penalization for GAMLSSFIGURE 1. Marginal BIC curves for parameters µ and σ, holding the othertuning parameter fix at the respective minimum of the BIC.

4

Application on Munich rental guide data

We now apply the proposed penalization approach to the Munich rent data, whichcome from 3015 households interviewed for the Munich rent standard 2007. The
response is the monthly rent per square meter in Euro, and from a large setof covariates we incorporate a selection of nine factors, both ordered as well asnominal and binary, similar to Gertheiss and Tutz (2010). We fit a Gaussian
GAMLSS and use for both distribution parameters, i.e. µ and σ, a combinationof the two different fused LASSO penalties introduced above. In order to obtain aflexible fit, the penalty terms of both corresponding linear predictors are assignedwith separate tuning parameters λµ and λσ , respectively.
The optimal tuning parameters are selected by BIC on a 2-dimensional grid. Figure 2 shows the corresponding marginal BIC curves for both µ and σ, holding theother tuning parameter fix at the respective minimum of the BIC. Figure 2 and3 show the paths of the dummy coefficients of both the ordinal covariate year of
construction and the nominal district, which are penalized by the two different
fused LASSO penalties from above. It is seen that with increasing tuning parameters λµ and λσ , respectively, categories are successively fused, i.e. the coefficientsare set equal. In addition, it can be seen that for the ordinal covariate year of
construction in Figure 2 only neighboring coefficients are fused, while for thenominal factor district in Figure 3 any groups of coefficients can be aggregated.

References
Gertheiss, J. and Tutz, G. (2010). Sparse modeling of categorial explanatory variables. The Annals of Applied Statistics, 4(4), 2150 – 2180.
Hofner, B., Mayr, T., and Schmid, M. (2016). gamboostLSS: An R package formodel building and variable selection in the GAMLSS framework. Journal

Groll and Umlauf

107

FIGURE 2. Ordinal fused coefficient paths for the year of construction for parameters µ (left) and σ (right); vertical dashed lines: optimal tuning parameters.

FIGURE 3. Nominal fused coefficient paths for the district effect for parametersµ (left) and σ (right); vertical dashed lines: optimal tuning parameters.

of Statistical Software, 74(1), 1 – 31.
Mayr, A., Fenske, N., Hofner, B., Kneib, T., and Schmid, M. (2012). Generalized
additive models for location, scale and shape for high-dimensional data - a
flexible approach based on boosting. Journal of the Royal Statistical Society, Series C, 61(3), 403 – 427.
Meier, L., Van De Geer, S., and Bühlmann, P. (2008). The group lasso for logistic regression. Journal of the Royal Statistical Society B, 70(1), 53 – 71.
Oelker, M.-R. and Tutz, G. (2015). A uniform framework for the combination ofpenalties in generalized structured models. Advances in Data Analysis and108

L1 -type penalization for GAMLSSClassification, 1 – 24.

Rigby, R.A. and Stasinopoulos, D.M. (2005). Generalized additive models for location, scale and shape (with discussion). Journal of the Royal StatisticalSociety, Series C, 54, 507 – 554.
Stasinopoulos, D.M. and Rigby, R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software,23(7).
Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journalof the Royal Statistical Society B, 58, 267 – 288.
Umlauf, N., Klein, N., and Zeileis, A. (2017a). BAMLSS: Bayesian additive models for location, scale and shape (and beyond). Working Paper, Faculty of
Economics and Statistics, University of Innsbruck.
Umlauf, N., Klein, N., Zeileis, A. and Köhler, M. (2017b). bamlss: Bayesian additive models for location, scale and shape (and beyond). R package version
0.1-2, URL: http://cran.r-project.org/package=bamlss.

Boosting Generalized Additive Models for
Location, Scale and Shape for Functional
Data
Almond Stöcker1 , Sarah Brockhaus1 , Sophia Schaffer2 ,Benedikt von Bronk2 , Madeleine Opitz2 , Sonja Greven1
1
2

Department of Statistics, LMU Munich, Germany
Soft Condensed Matter Group, LMU Munich, Germany

E-mail for correspondence: sonja.greven@stat.uni-muenchen.de
Abstract: We extend Generalized Additive Models for Location, Scale, andShape (GAMLSS) to regression with functional response. GAMLSS are a flexible
model class allowing for modeling multiple distributional parameters at once. The
model is fitted via gradient boosting, which provides inherent model selection.
We apply the functional GAMLSS to analyze bacterial interaction in E. coli andshow how the consideration of variance structure fruitfully extends usual growthmodels.
Keywords: Functional data; GAMLSS; Gradient Boosting.

1

Introduction

We propose a flexible approach to regression with functional response allowing
for simultaneously estimating multiple distributional characteristics of response
curves. It therefore generalizes usual functional mean regression models.
In functional data analysis (Ramsay and Silverman; 2005), functional response
regression aims at estimating covariate effects on response curves. Depending on
the application, covariates might effect the response curves in quite different ways.
This makes it particularly important to have a general framework for functional
regression to draw on.
Rigby and Stasinopolous (2005) introduced GAMLSS extending usual Generalized Additive Models (GAM) to multiple distributional parameters: covariate
effects on all parameters can be analyzed simultaneously, while doubtful assumpThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

110Boosting GAMLSS for Functional Data

tions of homoscedasticity can be overcome. At the same time the range of applicable distributions of GAM is extended to non-exponential family distributions.
We develop and implement functional GAMLSS combining a GAMLSS framework developed by Mayr et al. (2012) and a general functional regression framework (Brockhaus et al.; 2016a). For scalar-on-function regression the combination
with GAMLSS was discussed by Brockhaus et al. (2016b) providing a selection ofdifferent functional covariate effect types. Thus, the framework now provides flexible functional GAMLSS for scalar-on-function, function-on-scalar and functionon-function regression.
We apply the model to analyzing bacterial interaction of two competing Escherichia coli bacteria strains: a toxin producing ‘C-strain’ and a toxin sensitive‘S-strain’. Both are exposed to different external stress levels. Our aim is to model
the S-strain growing behavior in dependence of C-strain and stress level. As thegrowth process can be influenced in various ways, the covariates might effect both
location and scale of growth curves.

2

Model formulation

Consider a data scenario with N observations of a functional response Y andrespective covariates X. Y is a stochastic process, such that its realized trajectories yi : T → R, t 7→ yi (t) for i = 1, ..., N present the response curves over an
index set T . Let xi = (xi,1 , ..., xi,p )T denote the collection of the i-th observed
covariates. Each single covariate might be categorical, scalar or functional.
We assume that for all t ∈ T the conditional distribution of the response FY (t)|X
is known up to the unknown parameters ϑ(t) = (ϑ(1) (t), ..., ϑ(Q) (t)). For eachparameter an additive regression model is assumed specifying
g

(q)

(q)

(ϑ

)=h

(q)

(x) =

(q)
J
X

(q)

hj (x),

q = 1, ..., Q,

j=0

where g (q) is a link function for the q-th parameter of interest. The model structure corresponds to the GAMLSS introduced by Rigby and Stasinopoulos (2005).
However, covariates and response may now be functions. Correspondingly, alsoϑ(1) , ..., ϑ(Q) are modeled as functions.
(q)

The effect functions hj (x, t) are constructed modularly using tensor productbases, which allows for flexible specification of a variety of effect types of scalarand functional covariates (see Brockhaus et al.; 2016a).
The model coefficients are estimated using gradient boosting. It generalizes former model based boosting approaches (Brockhaus et al.; 2016b, Thomas et al.;2016) to GAMLSS for functional response regression. Gradient boosting is a gradient descend method for model fitting, which aims at minimizing a loss function.
This is performed in a component-wise and stepwise procedure and thereby yields
automatic model selection, prevents over-fitting and allows for fitting models withmore parameters than observations.

Stöcker et al.

3111

Analysis of bacterial interaction in Escherichia coli

Equilibria in biodiversity stand and fall with bacterial interaction. A comprehensive analysis of underlying processes involves investigating both expectation andvariability of growth curves.
Von Bronk et al. (2016) establish an experimental setup with two cohabiting
Escherichia coli bacteria strains: a colicin producing ’C-strain’ and a colicin sensitive ’S-strain’. Single bacteria of the C-strain can liberate the colicin, which kills
numerous S-strain bacteria on contact. However, the S-strain might still outgrow
the C-strain. The arising population dynamics are influenced by external stressinduced with the antibiotic agent Mitomycin C (MitC). At N = 334 observationsites, bacteria under consideration are exposed to four different MitC concentrations. Bacterial growth curves Si (t) of the S-strain and Ci (s) of the C-strain,
i = 1, ..., 334, are observed over 48 hours.
We model the i-th S-strain curve Si (t) in dependence on the C-strain growth.
To this end, we assume Si (t) to be gamma distributed with mean and standard
deviation ϑi (t) = (µi (t), σi (t))T . Both mean and standard deviation are modeled
analogously as
log ϑi (t) = f0 (t) + fM itCi (t) + h0 (Ci , t) + h1 (Ci0 , t),
(q)

(q)

(q)

(q)

with historical effects hj (Ci , t) =

(q)

(q)

Rt
0

q = 1, 2,(q)

Ci (s)βj (s, t) ds, j = 0, 1.

For both mean and standard deviation, the model includes functional intercepts(q)(q)
f0 (t) and group specific deviations fM itCi (t) per MitC level. Ci (s) and its
0
derivative Ci (s) reflect C-strain spread and growth. In the historical effect, thecomplete history of the C-strain up to the current time point t may influence theS-strain growth (see (Brockhaus et al.; 2016a)). Smooth functional components
are modeled using tensor-product B-Spline basis.
We observe MitC effects on both mean and standard deviation curves of the Sstrain. Especially, the standard deviation substantially increases with time andwith the MitC concentration.
Regarding the effects of the C-strain, we observe the effect of the derivative
C 0 to be dominant. Moreover, the coefficient function shows temporal regimes.
Such regimes were already noticed by von Bronk et al. (2016) and may now bequantitatively analyzed (Figure 1).

4Discussion

Functional GAMLSS present an extremely flexible model class, which combines
two conceptual extensions of GAM. They are of particular interest, if not only
the mean curve but rather the full dynamic of the response process is underconsideration. In this sense, it provides a natural extension to the usual analysis
of bacterial growth.

References
Brockhaus, S., Fuest, A., Mayr, A. and Greven, S (2016b). Signal regression models for location, scale and shape with an application to stock returns.
arXiv:1605.04281.

112

Boosting GAMLSS for Functional Data
σ

µ

(q)

sgn(βj50(s, t))

1s in [h]

40

−1

(q )

30

βj

(s, t)

20
0.05
10
0.00
0
0

10

20

30

40

50

0

10

20

30

40

50

−0.05

t in [h]FIGURE 1. Coefficient functions β (q) (s, t) for the historical effects of Ci0 (s) onmean and standard deviation of S-strain growth curves. The t-axis presents thetime line for the response curve, the s-axis the one for the C-strain. For a fixeds = s0 , β (q) (s0 , t) describes the influence of the covariate curve at that time over
the whole S-strain growth curve.

Brockhaus, S., Melcher, M., Leisch, F. and Greven, S (2016a). Boosting flexible
functional regression models with a high number of functional historicaleffects. Statistics and Computing, accepted.
Mayr, A., Fenske N., Hofner B., Kneib T. and Schmid, M. (2012). Generalized additive models for location, scale and shape for high dimensional data: a flexible approach based on boosting. Journal of the Royal Statistical Society,
Series C, 61, 403 – 427.
Ramsay, J. and Silverman, B.W. (2005). Functional Data Analysis. New York:
Springer-Verlag.
Rigby, R.A. and Stasinopoulos D.M. (2005). Generalized additive models for location, scale and shape. Journal of the Royal Statistical Society, Series C,
54, 507,– 554.
Thomas, J., Mayr, A., Bischl, B., Schmid, M., Smith, A. and Hofner, B. (2016).
Stability selection for component-wise gradient boosting in multiple dimensions. Manuscript submitted for publication.
von Bronk, B., Schaffer, S., Götz, A. and Opitz, M. (2016). Heterogeneous toxin
production promotes success in two-strain bacterial competition. Manuscript
submitted for publication.

Markov-Switching GAMLSS with an
Application to Operational Losses
Julien Hambuckers1 , Thomas Kneib1 , Roland Langrock2 ,Alexander Sohn1
1
2

Chair of Statistics, Georg-August-Universität Göttingen, Germany
Chair of Statistics, University of Bielefeld, Germany

E-mail for correspondence: tkneib@uni-goettingen.de
Abstract: This paper investigates modeling the dynamics of random sums representing the total operational losses where randomness is both in the loss frequencies and the loss sizes such that a compound Poisson process model is employed.
To this end, we formulate a Markov-switching generalized additive model for
location, scale and shape that allows all parameters of the compound loss distribution to depend on economic covariates in a flexible way while simultaneously
allowing this dependence to vary over time according to a hidden state process.
Relying on this approach, we analyze a novel dataset of 817 losses resulting fromfrauds in the Italian bank UniCredit.
Keywords: distributional regression; extreme events; hidden Markov model;
generalized Pareto distribution.

1

Markov-Switching Compound Poisson Process
Models for Total Losses

We are interested in modeling the temporal dynamics of the distribution of random sums
Nt
X
Lt =
Yi,t ,
i=1

for t = 1, 2, . . ., where Nt is the number of events occurring in period t, and Yi,t isthe severity of the ith event occurring during period t. In the actuarial literature,
this model is used for modeling the total claim of an insurance company overtime whereas in the banking literature it is used to model the total operationalloss suffered by a bank. Operational losses are defined by the Basel Committee
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

114Markov-Switching GAMLSS
7

4

60

x 10

3.5

50

Number of events

3

Total loss

2.5

2

1.5

40

30

20

1

10
0.5

0
3/20059/20096/20140
3/20059/20096/2014TimeTime

FIGURE 1. Total losses (left) and number of losses per quarter (right).

for Banking Supervision (BCBS) as direct or indirect losses resulting from inadequate or failed internal processes, people and systems or from external events.
In the financial literature, the distribution of Nt is referred to as the frequency
distribution whereas the distribution of Yi,t is termed the severity distribution.
The distribution of Lt is determined indirectly by making assumptions about thefrequency distribution and the severity distribution as well as their dependence.
For our application, we will assume
Nt
Yi,t

iid

∼

iid

∼

Poisson(λ),
GPD(γ, σ),where realisations of Nt and Yi,t , ∀{t, i} are assumed to be independent, λ > 0is the frequency parameter of the Poisson distribution for the counts and theseverities follow ageneralized Pareto distribution GPD(γ, σ) with shape parameter γ ≥ 0, scale parameter σ > 0 and density(y −1/γ
1− 1+γ, γ 6= 0f (y; γ, σ) =σ
γ=01 − exp − σy ,In combination, this leads to a compound Poisson process model for the total
losses Lt .
We are then interested in• relating the distributional parameters λ, γ and σ to economic covariates byembedding the compound Poisson model within the framework of generalized additive models for location, scale and shape (GAMLSS, Rigby andStasinopoulos, 2005), and• allowing for temporal variation in the regression effects via a Markov
switching structure implied by a latent state variable St .
We therefore propose Markov-switching GAMLSS that rely on combining a Markov
chain structure for the latent state St with a GAMLSS specification for the total loss distribution extending the Markov-switching generalized additive models
framework by Langrock et al. (2016). For each distributional parameter θ ∈Hambuckers et al.

115{λ, γσ}, we assume an additive predictorg(θ) = ηθ = x0 βθ t +
(S )

J
X

(S )

hθ,jt (zj ),j=1

where g(·) is a monotonic link function, x is a vector of covariates with (state(S )dependent) linear effects βθ t , and hθ,j (zj )(St ) are the (state-dependent) nonlinear effects of the continuous covariates zj modelled via penalised splines. All
model parameters (relating to the Markov chain for the latent state and to theconditional compound Poisson process given state membership) can then be determined based on penalised likelihood inference with smoothing parameters determined by cross-validation.

2

Application to Operational Losses

We apply the proposed Markov-switching GAMLSS to a novel data set of 817
operational losses at UniCredit, one of the largest European banks. We focus
on the particular class of operational losses resulting from external frauds, see
Figure 1 for some descriptives on the data set. The collection period of the losses
ranges between January 2005 and June 2014. Losses have been scaled by an
unknown factor for anonymity reasons. The minimal amount considered is Euro
25,000, so that we have a sample of extreme losses, where the GPD is a reasonable
hypothesis for the severity distribution. We have access to the exact date of eachloss, meaning that we can assign each loss to a specific year and a specific quarter,and compute the total loss for each quarter. From a regulatory point of view,banks are expected to define the capital reserve for a horizon of one year but wework on a quarterly basis, as we wouldn’t have enough time periods to estimate
correctly the transition probabilities of the model otherwise.
In our analyses, we rely on the following covariates (see Figure 2):• To model the frequency parameter λ, we use the percentage of the total
revenue coming from fees (PRF) which can be interpreted as a measure of
the economic well-being of the bank (with higher ratios implying smaller
dependence on market interest rates). The PRF also measures the level ofactivity of the bank on behalf of clients.
• To model the scale parameter σ, we use the Italian unemployment rate,serving as a proxy for the overall economic performance of Italy, where
UniCredit has its main activities.
• To model the shape parameter γ, we consider the values of the VIX index
which is a measure of the market volatility, based on put and call optionsof the S&P500. It is also considered as a barometer of market sentiments.
The estimated covariate effects for the different latent states are shown in Figure 3
while estimated state probabilities are shown in Figure 4 (left).
For operational losses, the capital reserve is derived from the estimated 99.9%
quantile of Lt such that this is a quantity of major interest for banking regulators. Figure 4 (right) shows estimates for this quantile derived from our model.
One interesting feature is that the Markov-switching GAMLSS, unlike simpler
alternatives, never entails quantile breeches on the observed data.

116

Markov-Switching GAMLSS

0.45

0.14

0.13

0.12

Unemployment rate

0.4

PRF

0.35

0.3

0.25

0.11

0.1

0.09

0.08

0.07

0.06

0.05

0.2
3/20059/20090.04
3/20056/20149/20096/2014

Time

Time
45

40

VIX index

35

30

25

20

15

10
3/20059/20096/2014Time

FIGURE 2. Values of the explanatory variables over the considered period. Fromleft to right: lagged values (one quarter) of the PRF, the Italian unemployment
rate and the VIX index.

Acknowledgments: Financial support by the German Research Foundation(DFG), Research Training Group 1644 “Scaling Problems in Statistics” andproject KN 922/4-2 is gratefully acknowledged. The authors acknowledge Dr.
F. Piacenza for providing us with the data.

References
Hambuckers, J., Kneib, T., Langrock, R. and Sohn, A. (2016). A Markov-switching
Generalized Additive Model for Compound Poisson Processes, with Applications to Operational Losses Models. Technical Report.
Langrock, R., Kneib, T., Glennie, R. and Michelot, T. (2016). Markov-switching
generalized additive models. Statistics and Computing, to appear.
Rigby, R. and Stasinopoulos, D. (2005). Generalized additive models for location, scale and shape. Journal of the Royal Statistical Society: Series C(Applied Statistics), 54, 507–554.

Hambuckers et al.

117

0.2

0.15

h(PRF)

0.1

0.05

0

−0.05

−0.1
0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

PRF
0.6

h(unemployment rate)

0.5

0.4

0.3

0.2

0.1

0

−0.1
0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.35

0.4

0.13

Unemployment rate
0.7
0.6
0.5
0.4

h(VIX)

0.3
0.2
0.1
0
−0.1
−0.2
−0.3
0.1

0.15

0.2

0.25

0.3

VIX index (10−2)

FIGURE 3. Estimated nonparametric effects of the covariates for state 1 (solidline) and state 2 (dotted line).

118

Markov-Switching GAMLSS

7

1

8

x 10

0.9
7

0.8
6

5

0.6

Euro

State probability

0.7

0.5

4

0.4
3

0.3
2

0.2
0.1

1

0
3/20056/2007 3/2008

9/2009Time

9/201012/20116/20140
3/20056/2007 3/2008

9/20099/201012/20116/2014Time

FIGURE 4. State probabilities for the different regimes (left; state 1 as solid line,
state 2 as dotted line) and Q̂0.999 (Lt ; Xt , St ) over time (right; state 1 as dashedline, state 2 as dotted line; total losses superimposed as grey bars).

Aggregate Claims Modelling via Dynamic
Score Driven Models
Mariana Melo12 , Cristiano Fernandes1 , Eduardo Melo23
1

Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Brazil
SUSEP - Brazilian Insurance Supervisor, Brazil
3
CPES - Insurance Economics Research Center, ESNS, Brazil
2

E-mail for correspondence: mariarozo@gmail.com
Abstract: In the insurance industry, the measurement of obligations and theevaluation of actuarial risks depend fundamentally on the aggregate claims distribution. From a mathematical point of view, the aggregated claims variable isa random sum of random variables. However, obtaining the analytical expression
for this probability distribution is a hard task. In this paper, a new approach is
proposed for the modelling of the aggregated claims predictive distribution. We
combine the newly proposed Generalized Autoregressive Score (GAS) models, tospecify a non-Gaussian distribution for both the number of claims and for theclaims severity. By use of the Fast Fourier Transform (FFT), we are then able tonumerically obtain the aggregated claims distribution. The proposed method isapplied to real data, provided by a leading Brazilian motor insurer.
Keywords: GAS models; Aggregate Claims; Collective risk model; Random sum.

1

Introduction

In the insurance industry, the measurement of obligations and the evaluation of
actuarial risks depend fundamentally on the aggregate claims distribution. New
solvency supervision principles have required more sophisticated approaches toassess the claims expenses of an insurer. In addition to obtaining the expectedvalue of these expenses, it is essential to insurers to assess the inherent risk ofthese cash flows, whose distribution of probability becomes the main objective.
From a mathematical point of view, the aggregated claims variable is a random
sum of random variables. Therefore, randomness is present on both the valuesof the individual claims and on the number of incurred claims. However, theaggregate claims distribution, except in rare circumstances, does not present a
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

120Aggregate Claims Modelling via Dynamic Score Driven Models

closed form.
In this paper, a new class of observation driven models proposed by Creal etal. (2013) and Harvey and Chakravarty (2008), named the Generalized Autoregressive Score (GAS) models, is proposed for the modelling of the aggregatedclaims predictive distribution. By construction GAS models allow parameters of
the probability distribution to evolve on time according to an updating equation which use the score as a driving mechanism. We combine the GAS models,applied for the number of claims and also for the claims severity distribution,with the use of Fast Fourier Transform (FFT) to compute the aggregated claims
distribution.
Differently from the traditional approach for the number and severity of claims
distribution, parameter time varying models allows a more precise forecast oncethe model follows the parameter dynamic on time. Aggregate claims distribution,
except in rare circumstances, does not present a closed form. FFT was chosen
to obtain this distribution since it is an efficient numerical method with low
computational effort. The proposed method is applied to real data, provided
by a leading Brazilian motor insurer. The number of claims was both modeled
trough a Poisson distribution and a negative binomial distribution. Claim severity
was assumed both gamma or lognormal distributed. In all models appropriate
parameters were made time varying according to the score driven mechanism.

2

Models

The aggregate claims represents the total amount paid for all claims incurred ina certain period of time in an insurance portfolio. In the collective risk model, theaggregate losses is represented by a sum of a random number (claim frequency) ofindividual payments (claim severity). In this work, the claim frequency and claim
severity data are both consideredas times series, so the aggregate claim at time
P t
t is represented by St∗ = N
i=1 Xit , t = 1, ..., T , where Nt is the random variable
that represents the claim frequency at time t and Xit , i=1,...,Nt , is the randomvariable that represents the severity of the claim i at time t. The assumptions arethat X1t , X2t , . . . are independent and identically distributed (iid) at each time tand independent of Nt .

2.1

Score driven models

GAS models provide a general framework for modelling distributions with timevarying parameters. The time variation of the parameters is introduced by letting
parameters be functions of lagged dependent variables, using the scaled score.
It is also possible to introduce exogenous variables on the parameter updating
equation. In this framework, the parameters are perfectly predictable given thepast information.
For a GAS(p,q) model, the predictive model density and the associatedupdatingPp
equation
are
given
by
y
A
i st−i+1 +t ∼ p(yt |ft , θ; Ft−1 ) and ft+1 = w +i=1Pq
j=1 Bj ft−j+1 , where Ft = {y1 , . . . , yt }, ft is the time-varying parameter vector,
θ is the static parameter vector, θ ∈ Θ ⊆ Rdθ , w is a vector of constants, Ai and
Bj are coefficient matrices and st is the scaled score (st = St ∇t ).

Melo et al.

121

t |ft ,θ)The score is defined as ∇t = ∂ln p(yand the scaling matrix St is, usually,
∂ft
h
i
0
−1/2
−1
chosen between It|t−1 , It|t−1 ou I, where It|t−1 = Et−1 ∇t ∇t is the information
matrix and I is the identity matrix. Fixed parameters estimation is done bymaximum likelihood (ML).

2.2

Claim frequency and severity models

As in most cases in the insurance field, we have used the Poisson and negative
binomial distributions for claim frequency (Nt ), considering the mean λt andparameter pt time-varying, respectively.
For the claim severity models, following the results of Creal et al. (2014), it isassumed that all severity claims (Xi,t ) at time t are identical distributed andcross-sectionally independent conditional on ft and on information set Ft−1 . In
this case, the log-likelihood, the score
and the information matrix
P vector
P att time(2)tt take a simple
form: lt = n; Ft−1 ), ∇t = n
i=1 ln p(xi,t |ft , θ
i=1 ∇t,i
Pnadditivetand It|t−1 = i=1
It|t−1,i , where nt is the claim frequency at time t. Gamma andlognormal are popular selections for distributions of motor insurance loss given
their appropriate characteristics, as non-negative support, positive skewness andvariance proportional to the mean-squared (constant coefficient of variation). In
this article, these two distributions are used to model the claims severity. Models
descriptions are given in Table 1. For the Gamma model, α is the shape parameterand µαt is the scale parameter.
TABLE 1. Details for the claim frequency and claim severity models.

ModelsClaim frequency (Nt ):Poisson(λt )NegBin(r, pt )Claim severity (Xt,i ):Gamma α, µαt
Lognormal(µt , σ 2 )ft∇t

It|t−1ln(λ t) 
pt
ln 1−ptnt − λ t

λt

ln(µt )µtr(1 − pt ) − nt pt
 Pnt
i=1 xt,i
α
−
n
t
µt
Pnt
2
i=1 ln xi,t − nt µt σr(1 − pt )
nt α
nt
σ2

Our goal is to obtain the aggregate claims distribution. In this sense, since this
distribution can not be obtained analytically for the vast majority of cases, themodels for series of claims numbers (Nt ) and for severity (Xit ) described previously were used with the FFT technique to obtain the target distribution. The
FFT is an efficient method to calculate compound distributions by inverting characteristic functions. Klugman et al.(2012) and Wang (1998) give an explanationof the method in application to aggregate claims distribution.

122

3

Aggregate Claims Modelling via Dynamic Score Driven Models

Empirical application

The data base used in this paper consist of motor insurance individual claimsof Brazilian leading insurer, occurred between 01/2006 and 02/2014, totalizing
30.959 claims in 98 months. The observations of the first 86 months were used
to fit the models, leaving the remaining 12 months for out-of-sample evaluation.
Figure 1 presents the data time series (claim frequency, claim severity mean
and total aggregate loss). The claim severity values are deflated by a consumer
inflation index. All models were run using R Software. We adopted models with−1
p = q = 1 on the updating equation of ft and St = It|t−1. The models were also
−1/2

fitted with St = It|t−1 , however the maximum likelihood optimization proved tobe more stable with the first option.
Claim severity mean

Total aggregate loss

12000

15

260

6000

20

280

25

8000

300

30

10000

320

35

340

40

360

45

14000

Claim frequency

2006

2008

2010

2012

2014

2006

2008

2010

2012

2014

2006

2008

2010

2012

2014

FIGURE 1. Data time series (claim frequency, claim severity mean and total
aggregate loss) from 01/2006 to 02/2014.
Quantile residuals were used to evaluate the models. The results for serial autocorrelation tests (Ljung-Box test up to the lag 30), conditional heteroscedasticity
tests (Ljung-Box test up to lag 30 on the squared residuals) and normality tests(Jarque-Bera) tests were satisfactory, except for the lognormal model. In order
to compare models, the Akaike Information Criteria (AIC), Bayesian Schwarz
Criteria (BIC), the log-likelihood value, as well as the MAPE (Mean Absolute
Percentage Error) were used. The results are shown on Table 2. Our results,
including residuals analyses, indicate that the gamma distribution is more suitable to the severity data. The Poisson and negative binomial models for claim
frequency presented similar results but the Poisson model produces a slightly
better out-of-sample performance. Also, in comparison with a static benchmark
model, where all parameters are fixed in time, our model obtained better results.
Figure 2 presents information about the aggregate claims distributions for 12months ahead, obtained using FFT and the results from Poisson and gamma
models for claim frequency and claim severity respectively.

4

Concluding Remarks

Following the methodology presented, we obtained the complete density for theaggregate claims random variable conditioned on time and not only means andupper moments. We used a new framework of time-series models, driven by score

Melo et al.

ModelClaim frequency
Poisson GAS(1,1)
Neg. Binomial GAS(1,1)
Claim severity
Gamma GAS(1,1)
Lognormal GAS(1,1)

123

log-Lik

AIC

BIC

MAPE

-381.967-380.412

771.935770.825

781.752783.097

4.872
4.870

-108790.3-109255.4

217590.6
218520.8217602.9218533.1

7.042
5.007

12000

TABLE 2. In sample results, using data from 01/2006 to 02/2013, for claim
frequency models (Poisson GAS(1,1) and Negative Binomial GAS(1,1)) and for
claim severity models (Gamma GAS(1,1) and Lognormal GAS(1,1)).

6000

7000

8000

9000

10000

11000

Observed
Q 1%
Q 5%
Mean
Q 95%
Q 99%

2013.42013.62013.82014.02014.2

FIGURE 2. Quantiles and means of the aggregate claims distributions obtained
from the Poisson GAS(1,1) and Gamma GAS(1,1) models - 03/2013 to 02/2014.

with time-varying parameters, to model the predictive distribution of the claim
frequency and the claim severity. In comparison to static fitted distributions, thepredictive distributions are more appropriate as they take into account the time
dynamics and the parameters uncertainty. To the best of our knowledge, there arefew studies on obtaining the predictive distribution of the aggregate claims andmost of them use the Bayesian approach. In addition to the application proposedin this article, the presented method can be applied to other lines of business orother problems involving sum of random variables and calculation of compound
distributions, as well as developed with different probability distributions thatbest fit the data.
From the aggregate claims distribution, it is possible to calculate the probability that the insurer has a portfolio loss above a given value, perform premium
calculations, company ruin probability, and obtain subsidies for technical provisions and risk-based capital calculations. The applications of our method meetthe demands of the current solvency principles and the measurement of insurance
contracts by making it possible to obtain predictive distributions, conditioned intime, from the loss of insurance portfolios. Assessment uncertainty in relation

124

Aggregate Claims Modelling via Dynamic Score Driven Models

to this stochastic process allows for the practice of risk management practices,
either through the acquisition of measures such as VaR, TVaR or others, as wellas to calculate risk margins.

Acknowledgments: The first author thanks FUNENSEG for the support.
References
Creal, D., Koopman, S.J., and Lucas, A. (2013). Generalized autoregressive scoremodels with applications. Journal of Applied Econometrics, 28(5), 777 –795.
Creal, D., Schwaab, B., Koopman, S.J., and Lucas, A. (2014). Observation-driven
mixed-measurement dynamic factor models with an application to credit
risk. Review of Economics and Statistics, 96(5), 898 – 915.
Harvey, A. and Chakravarty, T. (2008). Beta-t-(e) garch. University of Cambridge, Faculty of Economics.
Klugman, S.A., Panjer, H.H., and Willmot, G.E. (2012). Loss models: from data
to decisions. John Wiley & Sons.
Wang, S. (1998). Aggregation of correlated risk portfolios: models and algorithms. In Proceedings of the Casualty Actuarial Society, 85, 848 – 939.

Enhancing predictive performance ofvulnerability to poverty estimates
Maike Hohberg1 , Katja Landau1 , Thomas Kneib1 , Stephan
Klasen1 , Walter Zucchini11

University of Göttingen, Germany

E-mail for correspondence: mhohber@uni-goettingen.de
Abstract: We analyze two modifications to a simple measure of vulnerability topoverty, which defines vulnerability as expected income poverty. First, to model
income, we use distributional regression and relate each parameter of the conditional income distribution to a set of covariates. Second, instead of defining
a household as vulnerable if its probability of being poor is larger than 0.5, weconstruct a vulnerability cutoff employing the receiver operating characteristic(ROC). Using panel data from Germany, we find that our new cutoff method
considerably increases predictive performance while placing the income regression model into the distributional regression framework improves predictions forsome years.
Keywords: vulnerability to poverty, gamlss, ROC

1

Introduction

Although not living in poverty at the moment, many households are extremely
vulnerable to events such as job loss, unexpected expenditures, economic downturns, and weather phenomena, and can easily fall below the poverty line. Policymakers are therefore often interested in knowing which people are at risk ofpoverty in order to adequately design anti-poverty programs. Even though a few
empirical applications of vulnerability to poverty measures evaluated predictive
performance of their estimates, very little attention has been paid to their improvement.
One popular approach considers vulnerability as expected poverty (Chaudhuri
et al., 2002) and defines the vulnerability of a household h as the estimatedprobability of falling below a poverty threshold z of some normally distributed
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

126

Enhancing vulnerability to poverty estimatesmeasure of welfare yh (e.g. log of income) given a covariate vector x0 h . That is


0
ln
z
−
x
β̂
0
c yh < ln z|xh ) = Φ  q h 
Pr(ln
(1)
x0h θ̂
where β̂ is a vector of coefficients from a regression specification with log income
as the dependent variable. Implying a relationship between higher volatility inincome and poverty risk, the variance is allowed to vary with the covariates acrosshouseholds yielding coefficients θ̂. Both coefficient vectors are estimated via athree-step feasible generalized least squares (FGLS) procedure as in Amemiya(1977). The household is then classified as vulnerable if the probability in (1) isequal to or greater than 0.5.
The need for an enhanced method results from three major drawbacks of the standard approach: First, the welfare measure is always assumed to be (log)normally
distributed but in many applications income or expenditures do not behave(log)normally. Second, once we depart from the normality assumption, parameters other than mean and variance could be modeled to capture the full effecton the whole conditional income distribution. Third, setting the vulnerability
cutoff at 0.5, neglects the variability a household faces. If the expected income
equals the poverty line, the normal density function in equation (1) yields 0.5 independent of the standard deviation (McCarthy et al., 2016). Additionally, thisclassification does not always perform well in terms of prediction (Bergolo et al.,2012; Celidoni, 2013).
We tackle all of these drawbacks by two modifications: Distributional regression
is aiming at the first and second point. Within this flexible framework, different
distributions can be selected to model income, and all parameters of this distribution are related to a structured additive predictor which can incorporate
non-linear effects. This is equivalent to generalized additive models for location
scale and shape (GAMLSS, Rigby & Stasinopoulos, 2005). However, we prefer
the term “(structured additive) distributional regression” as some distributions
neither have location nor scale parameters but potentially only shape parameters(Klein et al., 2015). Our approach to use the ROC to determine the vulnerability
cutoff focuses on the third drawback of the traditional method. We thus directly
address recent criticism of the traditional vulnerability threshold raised by e.g.
Bergolo et al.(2012) and McCarthy et al. (2016) and propose a endogenous cutoff
which improves predictive precision.

2

Estimation strategy

2.1

Data

To demonstrate the benefits of our modifications we use data of 15 years, namelyfrom 1993 to 2008 of the German Socio-Economic Panel (SOEP), which offers
comprehensive coverage of household characteristics and income enabling us toretrospectively observe whether a household did become poor or not.

Hohberg et al.

2.2

127

Model

Placing the income regression within the framework of structured additive distributional regression, the conditional income distribution is given by a density
conditioned on parameters ϑhk , k = 1, . . . , K. Each of the K parameters is itself
dependent on the covariates including variables such as past income, number ofchildren and elderly in the household h, education and gender of the household’shead. We thus write
gk (ϑhk ) = ηh k = x0h βh k +
ϑ

ϑ

Jk
X

ϑ

fj k (νh )

(2)

j=1
ϑ

where gk is a link function, ηh k the predictor for the k-th parameter, the vectorϑx0h contains covariates which are assumed to have a linear effect, βh k are the corϑk
responding coefficients for these covariates, and fj (νh ) are smooth functions of
Jk continuous covariates νh which have non-linear effects. More precisely, for thecovariate past income we relax the restrictive assumption of a linear effect and use
P(enalised)-splines (Eilers & Marx, 1996) to flexibly model its relationship to thedependent variable. As conditional distributions, we use in addition to the lognormal, the Burr distribution. The parameters are simultaneously estimated via aback-fitting algorithm that maximizes the penalized likelihood avoiding the stepwise FGLS procedure of the traditional approach. The models are implemented
in the software R (R Core Team, 2016) using the package gamlss (Stasinopoulos& Rigby, 2007).

2.3

Constructing a vulnerability cutoff using the ROC

After fitting the models, a cutoff (the vulnerability line) is specified such thathouseholds whose predicted income falls below the cutoff are classified as vulnerable. To put it more precisely, we first calculate predicted incomes from themodels above. We then determine which households are actually poor in the next
year by using the observed incomes and a predefined poverty line. After that,
we use each of the predicted incomes as a hypothetical vulnerability line. Households with an income below the hypothetical vulnerability line will be declared
vulnerable. For each possible cutoff, that is each predicted income, we construct
the ROC by comparing the actual poor and non-poor with vulnerable and nonvulnerable households. A higher cutoff naturally leads to a higher true positive
rate (TPR) as more households are declared vulnerable but also increases thefalse positive rate (FPR). Depending on the costs of ”false alarms” or ”missing ahousehold”, a prerequisite can now be arbitrarily established. We decided to define the predicted income which leads to a TPR of 80 percent as the vulnerability
line.
With the method described so far, we determine a cutoff ex post. To estimate
vulnerability as a forward looking perspective on poverty, we rely on the pastvulnerability lines to make a prediction for the following year. Thus, the predictions will not exactly meet a TPR of 80 percent but will be close to it.

128

3

Enhancing vulnerability to poverty estimates

Main results and concluding remarks

We compare how well the traditional and our proposed models predict poverty
status using the TPR, FPR, and scoring rules. The models differ with respect
to the distribution of the response (and hence parameters that are related tocovariates), nonlinearities in the covariate effects, and the vulnerability cutoff.
We find that for some years using distributional regression does improve predictive performance mainly due to incorporating non-linear effects of past income.
We suppose that effects of modeling shape parameters are relatively small dueto the use of data from an industrialized country where social safety nets allow
households to cope with idiosyncratic shocks. Since modeling scale and shape parameters should account for idiosyncratic shocks, it is possible that for Germany
those idiosyncratic shocks are either little prevalent in the data or households areable to cope reasonably well with them. Nonetheless, distributional regression is
an attractive alternative to estimate vulnerability to poverty as it estimates mean
and variance simultaneously and is able to incorporate non-linear effects. These
effects seem important in our dataset especially when looking at incomes at thelower end of the distribution.
Regarding our new cutoff method, we find that it largely outperforms the traditional approach. Figure 1 shows the improvement in TPR and FPR when both of
our proposed modifications are applied but with the new cutoff being the main
driver. The plane’s division is based on our arbitrary prerequisites or, to put
it bluntly, which TPR and FPR are acceptable for us for being a good prediction. The result for the predictions of each year is represented by a point. Bestpredictions lie in the 4th quadrant, worst predictions in the 2nd quadrant.

3

1

2

3

1

2

TPR

0.40
0.00

0.20

0.40
0.00

0.20

TPR

4

0.60

4

0.80

(b)

0.60

0.80

(a)

0.00

0.10

0.20
FPR

0.30

0.40

0.00

0.10

0.20

0.30

0.40

FPR

FIGURE 1. Plots of accuracy for different models. Plot (a) shows results for the
traditional approach, i.e. 0.5 poverty probability cutoff, log normal distribution,no splines. Plot (b) represents the modified model, i.e. new cutoff, 3-parameter
Burr distribution, spline for past income.
The low TPRs for the traditional approach are somewhat surprising and weput further effort in investigating why we obtain these striking differences in
predictive performance. Looking at the effects of past income on current income,Hohberg et al.

129

we find that for the most part the relationship is roughly linear whereas nonlinearities occur especially at the bottom end of the income distribution where theeffects are much lower and can even be negative. These effects are not accounted
for in models assuming a linear relationship. On the other hand, models that doincorporate non-linear effects also have difficulties in predicting very low incomesas our dataset only comprises about 10-14 percent poor households. Hence, thepoor observations only have a small impact on the overall models’ prediction
ability, and thus the models predict overly optimistic incomes resulting in too
few households being classified as vulnerable.
Our new cutoff method is able to mitigate these weaknesses of the models since
the cutoff is determined endogenously and constructed in a way to fulfill a prescribed TPR. In contrast, the traditional method relies heavily on the model
specification and its prediction abilities can be weak, especially at the lower
extremes of the income distribution. As the 0.5 probability was advocated fordeveloping countries, where the share of the poor is generally higher than in Germany, it is possible that the traditional cutoff performs better in other settings.
One might argue that the 0.5 cutoff simply needs to be set lower in our case. We
tested several other probability cutoffs but the results where still not satisfactory
yielding very high FPRs.
Since our new cutoff method is constructed to improve predictive performance,
can mitigate weaknesses in the income generating model specification, and is easy
to implement, it is a useful tool if researchers or policymakers are particularlyinterested in correctly identifying vulnerable households rather than in measuring
overall vulnerability.

130

Enhancing vulnerability to poverty estimates

References
Amemiya, T. (1977). The Maximum Likelihood and the Nonlinear Three-Stage
Least Squares Estimator in the General Nonlinear Simultaneous Equation
Model Econometrica, 45, 955 – 968.
Bergolo, M., Cruces, G., and Ham, A. (2012). Assessing the Predictive Power ofVulnerability Measures: Evidence from Panel Data for Argentina and Chile
Journal of Income Distribution, 21, 28 – 64.
Celidoni, M. (2013). Vulnerability to Poverty: An Empirical Comparison of Alternative Measures. Applied Economics, 45, 1493 – 1506.
Chaudhuri, S., Jalan, J., and Suryahadi, A. (2002). Assessing Household Vulnerability to Poverty from Cross-sectional Data: A Methodology and Estimates from Indonesia Discussion Paper Series 0102-52, Department of
Economics, Columbia University.
Eilers, P. H. C. and Marx, B.D. (1996). Flexible smoothing with B-splines andpenalties Statistical Science, 11, 89 – 121.
Klein, N., Kneib, T., Lang, S., and Sohn, A. (2015). Bayesian Structured Additive Distributional Regression with an Application to Regional Income Inequality in Germany Annals of Applied Statistics, 9, 1024 – 1052.
McCarthy, N., Brubaker, J., and de La Fuente, A. (2016). Vulnerability to Poverty
in rural Malawi Policy Research Working Paper WPS7769, The WorldBank.
R Core Team (2016). R: A language and environment for statistical computing.
R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/.
Rigby, R.A. and Stasinopoulos, D.M. (2005). Generalized Additive Models for
Location, Scale and Shape Journal of the Royal Statistical Society: Series
C (Applied Statistics), 54, 507 – 554.
Stasinopoulos, D.M. and Rigby, R.A. (2007). Generalized Additive Models for
Location Scale and Shape (GAMLSS) in R Journal of Statistical Software,23.

A heteroscedastic probabilistic temperature
forecasting model incorporating spread-error
correlation and high-resolution forecasts
Annette Möller 1 , Jürgen Groß 2
1Institute of Applied Stochastics and Operations Research, Clausthal Universityof Technology, Germany,
2
Institute for Mathematics and Applied Informatics, University of Hildesheim,
Germany
E-mail for correspondence: annette.moeller@tu-clausthal.de
Abstract: Ensembles of deterministic numerical weather predictions require correction with respect to forecast bias and dispersion properties. A prominent approach is the use of statistical postprocessing models, yielding calibrated andsharp predictive probability distributions. A recently developed postprocessing
method based on an AR-model for the forecast errors is now extended to incorporate a heteroscedastic variance parameter, accounting for the well-known spreaderror correlation of forecast ensembles. Making use of a high-resolution forecast
provides the model with additional information leading to improved propertiesof the predictive distribution.
Keywords: Statistical postprocessing model; predictive probability distribution;
autoregressive process; high-resolution forecast; spread-error correlation.

1

Introduction

The method of choice to quantify and assess sources of uncertainty in deterministic numerical weather prediction (NWP) models is the employment of forecast
ensembles (Leutbecher and Palmer, 2008). However, ensembles of NWP forecasts
typically exhibit biases and dispersion errors, thus require statistical postprocessing to improve reliability and forecast skill (Gneiting and Katzfuss, 2014).
Recently, Möller and Groß (2016) developed an ensemble postprocessing modelfor temperature, based on autoregressive information present in the forecast errors of the raw ensemble members.
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

132

Probabilistic temperature forecasting with a heteroscedastic model

We present an extended version of the model in Möller and Groß (2016), wherethe variance parameter is defined to depend on the ensemble spread, thus leading
to a heteroscedastic model accounting for the well-known spread-error correlation(Gneiting et al., 2005), stating that there exists a positive association between
ensemble spread and ensemble mean forecast error. Further, an additional highresolution forecast is utilized for the postprocessing model. This single forecast
was obtained by running the respective NWP model with higher resolution thanused for the other ensemble members. Thus, the high-resolution forecast provides
more detailed information on small-scale processes not captured by the lowerresolution forecast ensemble.

2

A heteroscedastic postprocessing model

Let {x1 (t), . . . , xm (t)} denote an ensemble of NWP forecasts for a univariate(Gaussian distributed) weather variable Y (t) at a fixed location. Further, letxhres (t) = xm+1 (t) denote an additional high-resolution forecast of Y (t). The respective forecast error series are given by ei (t) = Y (t) − xi (t), i = 1, . . . , m + 1.
Möller and Groß (2016) assume that the individual series {ei (t)}, i = 1, . . . , mfollow a weakly stationary AR(p) process (Shumway and Stoffer, 2006). Fitting
autoregressive models to each individual error series yields an AR-adjusted ensemble {ex1 (t), . . . , xem (t)}. We now follow the idea of Möller and Groß (2016)
and Gneiting et al. (2005) and assume the predictive distribution to be Gaussian. However, in the extended model, the high-resolution forecast is considered
as additional covariate, that is
Y (t)|x1 (t), . . . , xm (t), xhres (t) ∼ N (µ(t), σ 2 (t)) .

(1)

To account for the fact that the members x1 (t), . . . , xm (t) form an exchangeablegroup, while the high-resolution forecast has somewhat different properties, theparameters of the Gaussian distribution are defined as (weighted) sums of the respective group-wise parameters. That is, µ(t) = 21 (µens (t) + µhres (t)), and σ(t) =Pm
11(σens (t)+σhres (t)). Here, µens (t) is estimated by the mean xe(t) = mei (t)
i=1 x
2of the AR-adjusted forecast ensemble {ex1 (t), . . . , xem (t)}. To explicitly incorporate
the spread-error correlation of ensemble forecasts, our model for σ 2 (t) combines
the estimated dispersion of the error process retrieved solely from the past ofeach member xi (t) individually with the spread of the AR-adjusted members.
Specifically, the standard deviation σens (t) is parameterized as
v
u
m
p
u1 X
σens (t) = w t
γ 2 (t) + (1 − w) S 2 (t).
(2)
m i=1 iHere, S 2 (t) is the empirical variance of the AR-adjusted forecast ensemble {ex1 (t), . . . , xem (t)}and γi2 (t) = Var(ei (t)) is the variance of the AR(p) process assumed for the forecast error ei (t) = Y (t)−xi (t) of ensemble member xi (t). The parameters γi (t) can
directly obtained from the fitted AR(p) model, while the weights w are estimatedby the so-called minimum-CRPS approach (Gneiting et al, 2005).
The high-resolution parameters µhres (t) and σhres (t) are estimated in the sameway. However, the above formulas are reduced to the (m+1)-st summand, thus areMöller and Groß133

TABLE 1. Verification statistics averaged over 76 stations and 4341 test dates.

EMOS(ENS)
EMOS(ENS,HRES)
H-AR-EMOS(ENS)
H-AR-EMOS(ENS,HRES)

CRPS

DSS

RMV

Var(PIT)

0.906
0.876
0.903
0.873

2.152
2.153
2.013
1.936

1.491
1.435
1.532
1.502

0.095
0.093
0.088
0.085

solely based on the AR-adjusted high-resolution forecast x
em+1 . Although the partof the predictive variance corresponding to the ensemble spread is zero for a single
ensemble member, the variance γi2 (t) of the AR(p) error series corresponding toan individual member xi (t) can still be computed - on the basis of past values.
This fact highlights an additional advantage of the autoregressive postprocessing
model introduced by Möller and Groß (2016), as standard postprocessing models
are not able to estimate the variance parameter based only on a single ensemble
member.

1.5
1.0
0.5
0.0

0.0

0.5

1.0

1.5

2.0

EMOS(ENS,HRES)

2.0

EMOS(ENS)

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

PIT

H−AR−EMOS(ENS)

H−AR−EMOS(ENS,HRES)

1.5
1.0
0.5
0.0

0.0

0.5

1.0

1.5

2.0

PIT

2.0

0.0

0.0

0.2

0.4

0.6

PIT

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

PIT

FIGURE 1. PIT histograms aggregated over 76 stations and 4341 test dates.

134

Probabilistic temperature forecasting with a heteroscedastic model

As described in Möller and Groß (2016), the proposed model can be combinedwith other predictive distributions (e.g. based on EMOS) in a spread-adjusted linear pool (Gneiting and Ranjan, 2013), to further improve predictive performance.
The basic approach of combining the group-wise mean and variance parametersof the regular members and the high-resolution forecast by a (weighted) sum constitutes a simple 2-group model. The assumption of equal weights in both groups
can be relaxed leading to weights estimated from training data, which provides amore data driven version of the 2-group model. The SLP combination proposed inMöller and Groß (2016) allows a further generalization of this approach. For eachof k ensemble member groups a separate predictive distribution (e.g. obtained by
the heteroscedastic AR-EMOS model, however, any postprocessing model could
be used) is fitted and the distributions of the k groups are then SLP-combined.

0.0

0.2

0.4

0.6

0.8

1.0

p−values

FIGURE 2. p-values of the 76 station-wise Diebold-Mariano tests for CRPS
values of H-AR-EMOS vs. EMOS.

3

Case study with ECMWF temperature forecasts

We employ 24-h ahead forecasts from the European Centre for Medium-Range
Weather Forecasts (ECMWF, see e.g. Buizza et al., 2007) for 2-m surface temperature over Germany. A data set over a period of 12 years is considered, ranging
from 2002-01-01 to 2014-03-20 and containing forecasts and observations at 76
stations in Germany resulting in a total of 329916 forecast cases.
The ECMWF ensemble consists of 50 exchangeable members, a control forecast
and a high-resolution forecast. In the original analysis of Möller and Groß (2016),
only the 50 regular members where considered, while the two additional forecasts
have not been utilized. For the analysis presented here, we make explicit use of
the high-resolution forecast.

Möller and Groß135

Table 1 presents results for the continuous ranked probability score and theDawid-Sebastiani score (CRPS and DSS; see, e.g. Wilks, 2011; Gneiting and
Katzfuss, 2014) for the EMOS model, and the heteroscedastic AR-EMOS model(named H-AR-EMOS here). It is clearly visible that incorporating the highresolution forecast into any of the models improves the predictive performance
substantially. Further, the H-AR-EMOS model substantially improves on the thestandard EMOS approach in terms of the DSS, while only a small improvement
is visible in the CRPS.
The last two columns of Table 1 present the root mean variance (RMV) and thevariance of the probability integral transform (PIT) values. The RMV is used
as sharpness measure for the predictive distribution, while the variance of the
PIT values assesses the dispersion properties of the distribution, and thus its
calibration. Neutral dispersion, corresponding to a uniform PIT histogram, is1= 0.0833, the variance of the uniform
indicated by a PIT variance equal to 12
distribution on [0, 1] (see Gneiting and Ranjan, 2013). In case the predictive distribution matches the distribution the observations are drawn from, the respective
PIT values would (approximately) follow a uniform distribution. In line with theprinciple “maximizing sharpness of the predictive distribution subject to calibration” (Gneinting and Katzfuss, 2014), both aspects of the distribution should
be investigated in conjunction. That is, although EMOS has a slightly sharper
predictive distribution this is at the expense of calibration, thus the better calibrated H-AR-EMOS model is preferable. Table 1 shows that the heteroscedastic
AR-EMOS exhibits the best dispersion properties, with the model incorporating
the high-resolution forecast closest to the variance indicating neutral dispersion.
On the contrary, the EMOS models are slightly sharper than AR-EMOS, however, at the expense of calibration.
Figure 1 presents the corresponding PIT histograms of the considered models.
Their appearance is in line with the values obtained for the PIT variance. While
the EMOS models exhibit strong underdispersion (indicated by the strongly pronounced U-shape), the H-AR-EMOS models are able to correct the dispersion
properties further. In addition to this, it can be noted that incorporating the highresolution forecast improves the dispersion properties of both models comparedto the version not using the high-resolution forecast.
To asses the significance of the difference in CRPS values a (one-sided) DieboldMariano test (Diebold and Mariano, 1995) is performed for the comparision HAR-EMOS(ENS,HRES) vs. EMOS(ENS,HRES). Figure 2 shows the resulting 76
p-values. When looking at the Figure, it can be seen that for a large portion of
the stations the test is highly significant, indicating that H-AR-EMOS indeed
yields an improvement over EMOS.

Acknowledgments: We are grateful to the European Centre for MediumRange Weather Forecasts (ECMWF) and the German Weather Service (DWD)
for providing forecast and observation data, respectively.

136

Probabilistic temperature forecasting with a heteroscedastic model

References
Buizza, R., Bidlot, J.R., Wedi, N., Fuentes, M., Hamrud, M., Holt, G. and Vitart, F.
(2007). The new ECMWF VAREPS (variable resolution ensemble prediction system). Quarterly Journal of the Royal Meteorological Society, 133,
681 – 695.
Diebold, F.X. and Mariano, R.S. (1995). Comparing predictive accuracy. Journal of Business & Economic Statistics, 13, 253 – 263.
Gneiting, T. and Katzfuss, M. (2014). Probabilistic forecasting. Annual Reviewof Statistics and Its Application, 1, 125 – 151.
Gneiting, T., Raftery, A.E., Westveld III, A.H. and Goldman, T. (2005) Calibrated
probabilistic forecasting using ensemble model output statistics and minimum CRPS estimation. Monthly Weather Review, 133, 1098 – 1118.
Gneiting, T. and Ranjan, R. (2013). Combining predictive distributions. Electronic Journal of Statistics, 7, 1747 – 1782.
Leutbecher, M. and Palmer, T.N. (2008). Ensemble forecasting. Journal of Computational Physics, 227, 3515 – 3539.
Möller, A. and Groß, J. (2016). Probabilistic temperature forecasting based onan ensemble AR modification. Quarterly Journal of the Royal Meteorological Society, 142, 1385 – 1394.
Shumway R.H., Stoffer D.S. (2006). Time Series Analysis and Its Applications.
With R Examples. Second Edition. Springer.
Wilks, D.S. (2011). Statistical Methods in the Atmospheric Sciences. Academic
Press.

Probabilistic temperature post-processing
using a skewed response distribution
Manuel Gebetsberger1 , Georg J. Mayr1 , Reto Stauffer2 , Achim
Zeileis2
1
2

Institute of Atmospheric and Cryospheric Sciences, Univ. of Innsbruck, Austria
Department of Statistics, Univ. of Innsbruck, Austria

E-mail for correspondence: Manuel.Gebetsberger@uibk.ac.at
Abstract: Weather forecasts are typically based on numerical weather prediction models, where possible forecast errors can be corrected by statistical postprocessing methods. A common quantity to develop, test, and demonstrate new
post-processing methods is the surface air temperature which is frequently assumed to follow a Gaussian distribution. Nevertheless, the classical Gaussian
distributional regression models which use only few covariates are not able toaccount for all local features leading to strongly skewed residuals. The authors
demonstrate two approaches to overcome this problem: assuming a skewed response distribution to directly account for skewness, and extending the classical
Gaussian distributional regression model by including all available informationfrom the weather prediction model in combination with a boosting variable selection method.
The preliminary findings show satisfying results especially for an Alpine stationby either using a generalized logistic type I distribution with few covariates,
or using all available information plus an appropriate variable selection procedure. Both approaches are able to improve the predictions with respect to overall
performance and calibration. Furthermore, similar results can be achieved for
non-Alpine sites although with smaller improvements.
Keywords: Temperature; Ensemble; Forecast; Skewed; Distributional regression

1

Introduction

Weather forecasts are typically generated by numerical weather prediction (NWP)
models. Nowadays, ensemble prediction systems (EPSs) are widely used where
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

138

Skewed Probabilistic Temperature Post-Processing

multiple NWP runs with slightly perturbed initial conditions and parameterizations try to capture the forecast uncertainty. Nevertheless, it was found thatsuch models often show systematic errors due to simplified physical equations,insufficient resolution, and unresolved processes (Bauer et al. 2015).
One possibility to correct for these errors are statistical post-processing methods. These methods have been tested extensively for various forecast quantities,
especially surface air temperature. Distributional regression models (Klein et al.
2015) are one common post-processing strategy. Corrected probabilistic forecasts
are obtained by modeling the parameters of a response distribution using linear
predictors including covariates provided by an EPS.
Statistical models using the Gaussian assumption are generally able to improve
the EPS output and to produce well-calibrated probabilistic temperature forecasts (Gneiting et al. 2015). However, this is not true for all geographical locations. Atmospheric processes associated with unresolved topographical features
such as wintry cold pools are per definition not present in the EPS model andoften lead to a skewed distribution. If these effects (e.g. wintry cold pools, strong
valley heating during summer) cannot be depicted by the regression model, theresiduals might be (strongly) skewed leading to inappropriately calibrated probabilistic forecasts.
To overcome possible skewness-related errors, this study assesses two different
strategies for an Alpine and two non-Alpine stations. The benefit of a skewed
response distribution will be evaluated using different distributional regression
models. In addition, it is tested whether additional covariates from the EPS
remove the need of a skewed response distribution. Focus will be on the Alpine
station where pronounced skewness is visible in observed surface air temperaturerecords.

2

Regression Framework

Distributional regression models can be expressed in a general form:y ∼ D h1 (θ1 ) = η1 , . . . , hK (θK ) = ηK )(1)

where D represents the parametric distribution for the response y with θk , k =1, . . . , K parameters. The parameters are linked to the additive predictors ηkusing a monotone link function hk (e.g., identity, logit, log). Each parameterhk (θk ) can be expressed by an additive predictor of the form:
ηk = ηk (x, βk ) = f1k (x, β1k ) + · · · + fP k (x, βP k )(2)including various (possibly non-linear) functions fpk , p = 1, . . . , P .

3

Data and Model Setup for Comparison

Results for three different sites and +12h–+96h forecasts in 6-hourly intervals areshown: Innsbruck located in the European Alps, Munich in the Alpine foreland,and Hamburg in a plain environment. These sites are selected to ensure spatial

Gebetsberger et al.

139

independence of observation-forecast data pairs and to investigate the influence ofdifferent topographical environments. The covariates are based on forecasts fromthe European Centre for Medium-Range Weather Forecasts. The EPS provides(discrete) probabilistic forecasts consisting of 50 exchangeable ensemble members for a wide range of atmospheric variables. Two types of regression models
are estimated: “simple” models which only use 2m temperature forecasts, and“complex” models using all available information (110 covariates) plus a smooth
cyclic spline effect representing the seasonality as a function of the day of theyear (season). Table 1 shows a summary of the models and their specification.
We make use of the logistic and generalized logistic distribution in order to test
distributions with slightly heavier tails compared to the classical Gaussian distribution. Additionally, the generalized logistic distribution type I (Johnson et al.
1995) is able to account for possible skewness and has the distribution function
F (x) = 1/(1 + exp(−(x − µ)/σ))ζ ,(3)

where ζ defines the additional shape parameter that has to be estimated. Dueto the large number of covariates in the “complex” models, a likelihood gradientboosting approach (R package bamlss, Umlauf et al. 2017, R package version
0.1-1, https://r-forge.r-project.org/projects/bayesr/) is used for variable selection and coefficient estimation. A 10-fold block-wise cross-validation is further
performed to ensure full out-of-sample results.
The overall performance of the models is evaluated on the ignorance score (Ign),
and PIT histograms to visually assess calibration (Gneiting et al. 2015). To quantify thePinformation provided by individual PIT histograms the reliability index
RI = Ii=1 |κi − I1 | is chosen (Feldmann et al. 2015). It accounts for absolutedeviations of each PIT-bin from perfect uniform calibration, where I defines thenumber of bins and κi the relative fraction in each bin. Ideally, RI values areclose to zero. Additionally, the sharpness of the predictive distributions is verified in terms of average width of predictions intervals which should be as small
as possible.

4

Results and Discussion

The response distribution for surface temperature is often assumed to be Gaussian in statistical ensemble post-processing studies. This is obviously not appropriate for the investigated Alpine station used in this study. The simple Gaussian

TABLE 1. Covariates used for distribution parameters µ, σ, ζ for all response distributions. “mean”/“sd”: ensemble means and standard deviations, and “season”:
smooth seasonal effect. Covariates in brackets only for “complex” models.

Response Distribution

location µscale σshape ζGaussian
Logistic
Generalized Logistic (I)mean (+season)mean (+season)mean (+season)

sd
sd
sd

—
—
season (+mean)

Skewed Probabilistic Temperature Post-Processing(c) Skewed Logistic (Ign. 2.31)

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140

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FIGURE 1. PIT histograms for the Alpine site. Left to right: Gaussian, logistic, and skewed logistic models. Top column shows the “simple” models, bottom
column the “complex” models. Ignorance shown in brackets. Horizontal line indicates perfect calibration.

regression model shows uncalibrated and strongly skewed predictions (Fig. 1a).
The skewed logistic distribution could clearly improve the predictive performance,
indicated by a smaller ignorance and an almost uniformly distributed PIT histogram (Fig. 1c). This improvement is also visible for all individual lead times interms of ignorance (Fig. 2a) and calibration (Fig. 2d).
A better overall performance in ignorance can be achieved using more complex
models which can select from a large number of EPS covariates. Although additional covariates remove the need of a skewed response distribution, the complex
Gaussian model still shows large errors in the tails (Fig. 1d), which might result from considerably smaller prediction intervals compared to simple Gaussianmodel (Fig. 2g).
Overall best ignorance for the Alpine site was achieved using all available covariates and a logistic response distribution (Fig. 1e), however, there is a distinct
trade-off between perfect calibration, lowest possible ignorance, and also computational time. While simple models suggest the use of a skewed distribution,complex models perform very similar on all three distributions and imply that
the Gaussian assumption might be sufficient. Nevertheless, these complex models
using a cross-validated boosting procedure are roughly ten times more expensive
in terms of computational time.
Similar results could also be found for the Alpine foreland site (Fig. 2b,e,h) andthe topographically plain site (Fig. 2c,f,i), if only with smaller magnitude of
improvements due to the two strategies and more visible at shorter lead times.
At these two sites, the raw EPS has generally a better forecast performance due
to less unresolved distinct local features in the EPS forecasts. Since better EPS
forecasts lead to more informative covariates, the overall predictive performanceof the statistical models is best at the plain site (Fig. 2c) and worst in the Alpine
environment (Fig. 2a).

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FIGURE 2. Verification measures for an Alpine, a foreland, and a plain site (leftto right) using “simple” models (black lines) and “complex” models (grey lines).
From top to bottom the ignorance score (Ign), reliability index (RI), and mean
width of the 80% prediction interval (PIW) are shown, evaluated at forecasting
lead times +12h–+96h. The 80% PIW defines the width between the predicted
0.1 and 0.9 quantile. Note that the Alpine site has a different range on the y-axis.

142

5

Skewed Probabilistic Temperature Post-Processing

Conclusion

The results shown highlight the importance of an appropriate distributional assumption for post-processed surface air temperature forecasts, especially when
only the corresponding temperature covariate is used in the regression model. A
skewed logistic assumption improves calibration and forecast performance for all
tested sites, but more pronounced in the Alpine environment.
Compared to this skewed regression model, complex models cannot further improve overall calibration but clearly obtain smaller prediction intervals which also
lead to better forecast performance. Generally, the difference in forecast performance between the chosen distributional assumptions for the complex models
is rather small. This indicates that the Gaussian assumption is already suitablegiven the large number of covariates. Nevertheless, these complex models arecomputationally roughly 10 times more expensive, and hence can be difficult toimplement in an operational system.
The authors are currently extending this study by including additional sites toanalyze the most influential covariates. A small set of such covariates might already lead to an improvement of similar magnitude, while considerably cutting
down the computational costs. While the Gaussian assumption looks suitable forthe complex models considered, it has to be investigated whether this is still true
if the models only contain the most important covariates, especially for stations
located in a complex environment.

Acknowledgments: Austrian Science Fund (FWF; TRP 290-N26), and Autonome Provinz Bozen, Abteilung für Bildungsförderung, Universität & Forschung(ORBZ110725). Results partly achieved using the HPC infrastructure LEO of the
University of Innsbruck.

References
Bauer, P., Thorpe, A., and Brunet, G. (2015). The quiet revolution of numerical
weather prediction. Nature, 525, 47 – 55.
Feldmann, K., Scheuerer, M., and Thorarinsdottir, T. L. (2015). Spatial postprocessing of ensemble forecasts for temperature using nonhomogeneous Gaussian regression. Monthly Weather Review, 143, 955 – 971.
Gneiting, T., Balabdaoui, F., and Raftery, A.E. (2015). Probabilistic forecasts,
calibration and sharpness. Journal of the Royal Statistical Society, 69(2),243 – 268.
Johnson, N. L., Kotz, S., and Balakrishnan, N. (1995). Continuous univariate distributions., Volume 2. 2nd edition. New York: Wiley & Sons.
Klein, N., Kneib, T., Lang, S., and Sohn, A. (2015). Bayesian structured additive distributional regression with an application to regional income inequality in Germany. The Annals of Applied Statistics, 9(2), 1024 – 1052.

Umlauf, N., Klein, N., and Zeileis, A. (2017). BAMLSS: Bayesian additive models for location, scale and shape (and beyond). Working Papers, Faculty of
Economics and Statistics, University of Innsbruck. URL: http://EconPapers.repec.org/RePEc:inn
05.

Boosting multivariate Gaussian models forprobabilistic temperature forecasts
Thorsten Simon1 , Nikolaus Umlauf2 , Georg J. Mayr1 , Achim
Zeileis2
1

Institute of Atmospheric and Cryospheric Sciences, University of Innsbruck,
Austria2
Department of Statistics, University of Innsbruck, Austria
E-mail for correspondence: thorsten.simon@uibk.ac.at
Abstract: Many weather prediction tasks are multivariate problems, e.g., predicting several quantities (such as temperature and precipitation) for a particulartime or predicting a single quantity over time. In the latter case, a state-of-theart method is to fit several marginal prediction models and then combine themusing ensemble copula coupling (ECC). As an alternative approach, we propose
to fit a single multivariate Gaussian model where all parameters (means, variances, and correlations) can be expressed by additive models. For estimation of
the resulting large number of parameters a gradient boosting algorithm is employed. Results for a case study show equal performance with respect to marginal
predictive distributions and better performance with respect to the full multivariate distribution in comparison to nonhomogeneous Gaussian regressions (NGRs)
combined with ECC.
Keywords: boosting; additive models; multivariate Gaussian; weather prediction.

1

Introduction

To obtain calibrated weather forecasts for several lead times, e.g., predicting temperature 12 h, 36 h, 60 h, . . . in advance, the output of numerical weather prediction (NWP) ensemble systems is often postprocessed using statistical models.
One popular choice for temperature forecasts is nonhomogeneous Gaussian regression combined with ensemble copula coupling (NGR-ECC, see Schefzik et al.,2013). In a first step (NGR) linear models are employed for the location and scaleparameter of a Gaussian distribution for several lead times separately. In a second
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

144

Boosting multivariate Gaussian models

step (ECC) the predicted quantiles are reordered according to the raw ensemble
output, in order to preserve the covariance structure of the NWP ensemble.
This study aims at extending NGR in the following way: Estimating predictive
distributions for several lead times and their correlations simultaneously. The
location, scale and correlation parameters of a multivariate Gaussian (MVN)
will be expressed by GAM-type additive predictors η. Estimating multivariate
distributions with these specifications is a complex task for dimensions higher
than 2 (Klein et al., 2015). Gradient boosting can offer an attractive solution to
fit a MVN with additive predictors for all parameters as only first derivativesof the log-likelihood with respect the predictors are required. Another benefit ofboosting is that selection and shrinkage of coefficients can be obtained.

2

Methods

The log-likelihood of the multivariate Gaussian for a k-dimensional observationy = (y1 , y2 , . . . , yk )T can be parameterized as follows,l(µ, Σ|y) = −k1
1
log(2π) − log(|Σ|) − (y − µ)T Σ−1 (y − µ),222

where µ = (µ1 , µ2 , . . . , µk )T denotes the vector of the mean parameters and Σ
denotes the covariance matrix. The latter can be decomposed into Σ = DΩD,
where D is a diagonal matrix with the standard deviations σ1 , σ2 , . . . , σk on the
diagonal, and Ω is the correlation matrix with the elements ρij .
The parameters µi , σi and ρij are linked to their predictors ηµi , ησi and ηρij by
the identity, the log and the rhogit function, respectively. The partial derivatives
with respect to the predictors of µi and σi are,kX∂l=ςij (yj − µj )
∂ηµi
j=1

and

k
X
∂l
ωij ỹj ,= −1 + ỹi
∂ησi
j=1

where ςij and ωij denote the elements of the inverse covariance matrix Σ−1 andinverse correlation matrix Ω−1 , respectively. Additionally, y˜i = (yi − µi )/σi . The
partial derivative with respect to the predictor of ρij is
!
!#
"
k
k
X
3
∂l
1
1 X
2 − 2
ωim ỹmωjm ỹm× 1 + ηρij
= − ωij +
.
∂ηρij
2
2 m=1
m=1
In order to fit the model a gradient boosting algorithm is applied as implemented
by Umlauf et al. (2017). The algorithm is an iterative procedure. The numberof iterations mmax has to be defined in advance. In each step the coefficientsof the term which would contribute most to maximizing the log-likelihood are
updated by the proportion ν of the local estimate of the coefficients. Thus, theboosting algorithm results in mmax distinct sets of coefficients. The optimal setof coefficients is selected by out-of-sample validation. A generic description ofgradient boosting is given by Mayr et al. (2012).

Simon et al.

3145Application in weather prediction

A case study is presented for predicting temperature in Innsbruck, Austria (47.260◦ N,
11.357◦ E). Six lead times are considered, 12 h, 36 h, . . . , 132 h in advance. Data
is on hand from January 2011 to December 2016 leading to a sample size ofroughly 2150. The ensemble predictions of the European Centre for MediumRange Weather Forecasts (ECMWF) serve as NWP input. The additive predictors are declared as follows,ηµi = αi,0 + αi,1 ∗ mean(ensi ) + fi,cc (yearday),ησi = βi,0 + βi,1 ∗ log(sd(ensi )) + gi,cc (yearday),ηρij = γij,0 + γij,1 ∗ cor(ensi , ensj ),where indices i, j ∈ {1, 2, . . . , 6} refer to the lead times 12 h, 36 h, . . . , 132 h,
respectively. ensi denotes the raw ECMWF ensemble temperature forecast, andfi,cc (yearday) and gi,cc (yearday) are cyclic smooth functions modeled by splines
to account for annual cycles. The model is trained on the period 2011–2015.
Validation on the data of year 2016 leads to an optimal set of coefficients after
mopt = 5000 iterations where ν = 0.05.
Figure 1 displays the fitted nonlinear functions fi,cc (·) and gi,cc (·), which contribute to the additive predictors ηµi and ησi , respectively.
The coefficients describing fi,cc (·) were not selected within the first 5000 iterations. Thus, fi,cc (·) remains flat. This suggests that the bias between the model
temperature and observations is constant throughout the year.
gi,cc (·) (Fig. 1, right) contributes to the predictor of σi on the log-scale. gi,cc (·)
reveals an annual cycle with two peaks. One occurs in January and one in
June/July. The fitted effects for all lead times vary only slightly among eachother.
However, the main focus of this study is to model the correlation structure oftemperature between the lead times, 12 h, 36 h, . . . , 132 h. Figure 2 summarizes
the distribution of the fitted correlations. All parameters on the first diagonal
next to the main diagonal of Ω vary around 0.74. The parameters on the second
diagonal vary around 0.46. Thus, the fitted correlation matrixes exhibit a structure similar to a symmetric Toeplitz matrix or or even close to the correlation
matrix of a AR-process.
f(yearday)

g(yearday)

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FIGURE 1. Nonlinear effects for all lead times. Left: fi,cc (yearday) contributing
to ηµi . Right: gi,cc (yearday) contributing to ησi .

146

Boosting multivariate Gaussian models

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FIGURE 2. Fitted correlation coefficients ρij for all days in 2016. Eachbox-and-whisker plot indicates the distribution of one correlation parameter overall sample cases.

The range indicated by the box-and-whisker plots (Fig. 2) suggests that the valuesof the intercepts γij,0 are more important for determining the structure of thecorrelation matrix than the coefficients of the linear terms, γij,1 . Thus, only asmall part of the correlation structure modeled by the numerical ensemble canbe retained by the statistical model.
mopt
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FIGURE 3. Boosting paths for the coefficients of the predictors of the correlation
parameters on the rhogit scale.
Figure 3 illustrates how the values of the coefficients γij,0 develop during theiterative boosting procedure. The intercepts γij,0 are selected before the γij,1 areSimon et al.

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selected. However, after 15000–25000 iterations the values of the intercepts start
dropping, which might be caused by an overfitting of the location parameter µi .

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NGR

MVN

NGR−ECC MVNFIGURE 4. Out-of-sample scoring. Left: continuous rank probability score(CRPS) for univariate NGR models and marginal predictive distributions of theboosted MVN model. Right: energy score (ES) of the NGR-ECC and the boosted
MVN.
Figure 4 compares the performance of the proposed method to the performanceobtained by the state-of-the-art method NGR-ECC. The NGR models are also
fitted via boosting. The marginal predictive distributions of the boosted MVN
model are compared to NGR models fitted for every single lead time separately
via the continuous ranked probability score (CRPS, Gneiting and Raftery, 2007).
The left panel of Figure 4 reveals that the two models perform equally wellwith respect to their marginal distributions. The multivariate performance of themodels is assessed in terms of the energy score (ES, Gneiting and Raftery, 2007).
The boosted MVN outperforms the NGR-ECC in our case (Figure 4, right).

4

Conclusions

This study suggests to fit multivariate Gaussian distributions via gradient boosting, where additive predictors can be defined for all location, scale and correlation
parameters. A case study in the field of weather forecasting shows promising results.
Further investigations are needed to fully understand the potential of boosting
MVN with additive predictors. There are alternative ways to parameterize thecorrelation matrix, i.e., modeling the parameters of its inverse or its Cholesky
decomposition (Pourahmadi, 2011).
In the present case one could assume an AR-process among the response variable
as they are temporally ordered. This kind of parameterization is implicated by thefindings in this study (cf. Fig. 2). However, more research is needed to examine
whether a parsimonious or flexible parameterization is superior in this kind of
application.
Depending on the problem changing the parameterization can have an effect on
the required iterations until convergence of the boosting algorithm, and could
yield different results when a shrunken version of the model is selected.

148

Boosting multivariate Gaussian models

Acknowledgments: We acknowledge the funding by the Austrian Research
Promotion Agency (FFG) project LightningPredict (Grant No. 846620).

References
Gneiting, T. and Raftery, A.E. (2007). Strictly proper scoring rules, predictionand estimation. Journal of the American Statistical Association, 102, no. 477,
359 – 378.
Klein, N., Kneib, T., Klasen, S. and Lang, S. (2015). Bayesian structured additive distributional regression for multivariate responses. Journal of theRoyal Statistical Society, Series C, 64, Part 4, 569 – 591.
Mayr, A., Fenske, N., Hofner, B., Kneib, T. and Schmid, M. (2012). Generalized
additive models for location, scale and shape for high dimensional data—a
flexible approach based on boosting. Journal of the Royal Statistical Society, Series C, 61, Part 3, 403 – 427.
Pourahmadi, M. (2011). Covariance estimation: The GLM and regularization
perspectives. Statistical Science, 26, no. 3, 369 – 387.
Schefzik, R., Thorarinsdottir, T.L., and Gneiting, T. (2013). Uncertainty quantification in complex simulation models using ensemble copula coupling.
Statistical Science, 28, no. 4, 616 – 640.
Umlauf, N., Klein, N., and Zeileis, A. (2017). BAMLSS: Bayesian additive models for location, scale and shape (and beyond). Working Papers, Faculty of
Economics and Statistics, University of Innsbruck.

Generalization of the Whittle likelihood for
nonparametric spectral density estimation
Claudia Kirch1 , Matthew Edwards2 , Alexander Meier1 , Renate
Meyer21
2

Otto-von-Guericke University, Magdeburg, Germany
University of Auckland, New ZealandE-mail for correspondence: meyer@stat.auckland.ac.nz
Abstract: Most nonparametric Bayesian approaches use Whittle’s likelihood toestimate the spectral density as the main nonparametric characteristic of stationary time series, as e.g. Choudhuri et al. (2004) and Rosen et al (2012). But
as shown in Contreras-Cristan et al. (2006), the loss of efficiency of the nonparametric approach using Whittle’s likelihood can be substantial. We show that
the Whittle likelihood can be regarded as a special case of a nonparametricallycorrected parametric likelihood which gives rise to a robust and more efficient
Bayesian nonparametric spectral density estimate based on a generalized Whittle likelihood. Its frequentist properties are investigated in a simulation study.
Applications to LIGO gravitational wave data and the El Niño Southern Oscillation phenomenon will be described.
Keywords: Bayesian nonparametrics; stationary time series; spectral density
estimation; Bernstein polynomial prior; gravitational waves.

1

Introduction

Most Bayesian nonparametric approaches to time series analysis are based on
Whittle’s likelihood approximation (Whittle, 1957), as e.g. Choudhuri et al.
(2004) and Rosen et al. (2012). We will show that the Whittle likelihood can be
regarded as the likelihood of a parametric working model, namely iid Gaussian,which has been nonparametrically corrected in the frequency domain. Borrowing
an idea from a periodogram bootstrap for time series in Kreiss and Paparoditis(2003), we propose a generalization of the Whittle likelihood that uses a more
realistic parametric working model, e.g. an AR(p) model, again nonparametrically corrected in the frequency domain and suggest a Bayesian semi-parametricThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

150Generalization of the Whittle likelihood

approach to spectral density estimation.

2

Generalized Whittle Likelihood

Let Zn = (Z1 , . . . , Zn ) be a real stationary time series with mean zero and absolutely summable autocovariance function γ(h). Then the spectral density exists,Pis given by the Fourier transform of the autocovariance function f (λ) =∞−ikλ
1
for 0 ≤ λ ≤ 2π, is bounded and periodic, i.e. f (2π − λ) =k=−∞ γ(k)e2π
f (λ). Nonparametric estimation of the spectral density is based on the periPn−itλ 2odogram In (λ) = 2π1 n. It is well known that the periodograms
t=1 Zt e
are asymptotically independent
evaluated at the Fourier frequencies λj = 2πjnand Exponential(f (λj )) distributed for j = 0, . . . , N = b(n − 1)/2c. This gives
rise to Whittle’s likelihood approximation (Whittle, 1957)
( N 
)
X
In (λj )W
log f (λj ) +.
p (Zn |f ) ∝ exp −f (λj )
j=0This approximation is exact only in the case of Gaussian white noise in whichcase f (λj ) = σ 2 /2π but yields a good approximation also in the case of nonGaussian time series, see e.g. Shao and Wu (2007). Now consider that we startwith a N (0, σ 2 ) working model. We then Fourier transform the time series, cor1/2
rect in the frequency domain by multiplication with a correction matrix Cn =f (λi )
1/2
diag(. . . , σ2 /2π , . . .) , and inverse Fourier transform into the time domain. Then
1/2

the density of the transformed time series FnT Cn Fn Zn induced by the Gaussian iid working model is the Whittle likelihood. This gives rise to a generalization: instead of a Gaussian iid working model, we take a more realistic
parametric working model, e.g. an AR(p) model, and define a correction maf (λi ), . . .) where fparam denotes thetrix Cn = Cn (f, fparam ) = diag(. . . , fparam(λi )spectral density of the parametric working model. Then we call the density of
1/2
the transformed time series FnT Cn Fn Zn the generalized Whittle likelihood:−1/2
pC
pparam (FnT Cn−1/2 Fn Zn ).
param (Zn |f ) ∝ det(Cn )It can be shown that the classical Whittle likelihood is a special case, the nonparametrically corrected likelihood of a Gaussian AR(0) model. Furthermore, if
the model is correctly specified, then pCparam = pparam but even in the misspecified case, the periodogram associated with the generalized Whittle likelihood is
asymptotically unbiased for the true spectral density. For a proof, we refer to thepreprint by Kirch et al. (2017).

3

Bayesian Spectral Density Estimation Using theGeneralized Whittle Likelihood

In the following, we propose a semi-parametric Bayesian approach to spectral
density estimation using the generalized Whittle likelihood and specify the priordistribution. We assume a Gaussian AR(p) working model and use a nonparametric Bernstein-Dirichlet prior for the spectral density as in Choudhuri et al.

Kirch et al.

151(2004), see also Petrone (1999). However, we put the prior not on the spectral
density f itself but on the ratio cη (λ) = f (λ)/fparam (λ; a)η where a denotes theparameters of the AR(p) working model and η ∈ [0, 1] captures the confidence
in the parametric model. Assuming a stationary mean zero Gaussian time series
with absolutely summable autocovariance function and a spectral density that isbounded away from zero, one can show that for fixed Gaussian AR(p) working
model and a fixed confidence parameter η the generalized Whittle likelihood, theclassical Whittle likelihood, and the true likelihood are all mutually contiguous.
Furthermore, under some technical assumptions on the prior, the posterior distribution is consistent, i.e. the posterior distribution computed using the generalized
Whittle likelihood will concentrate in a neighbourhood of the true spectral density for increasing length of the time series. For a detailed proof, see Kirch et al.
(2017).
In practical applications, we will also estimate the parameters a of the AR(p)working model and the confidence parameter η jointly with the nonparametric
spectral density. Therefore, prior specification is completed by putting a Uniform(0,1) prior on η and Uniform(-1,1) priors on the partial autocorrelations
corresponding to the autoregressive parameters a of the AR(p) working model toensure stationarity and causality. The degree p of the AR(p) model is fixed anddetermined in a preceding parametric AR(p) model selection procedure by thevalue of p minimizing the deviance information criterion (DIC), see Spiegelhalter
et al. (2002) and Meyer (2016).
We sample from the joint posterior distribution using an adaptive version of a
MH-within-Gibbs sampler, see also Roberts and Rosenthal (2009). Details regarding posterior computation can be found in Kirch et al. (2017) and software
is available in the R package beyondWhittle on CRAN, see Meier et al. (2017).

4

Simulation Study

We briefly summarize the major findings of a comprehensive simulation study in
Kirch et al. (2017). We generated data from various Gaussian ARMA(p, q) timeseries and compared the performance w.r.t. the average integrated absolute error(IAE) and the coverage probability of a uniform 95% credible interval (UCI) of
the Bayesian spectral density estimate based on a parametric AR(p) likelihood,the classical Whittle likelihood, and the generalized Whittle likelihood, i.e. a
nonparametrically corrected likelihood with an AR(p) working model where theorder p was chosen by minimization of DIC. In the correctly specified AR(p)scenario, of course the parametric estimator performed best. But the estimator
based on the generalized Whittle likelihood benefits from the correctly specified
AR(p) model and performed only marginally inferior to the parametric model
whereas the estimator based on the Whittle likelihood had much higher IAE andlower UCI coverage. In the misspecified MA(q) scenario, the parametric estimator
had highest IAE and lowest coverage as expected, whereas the nonparametricestimator based on the Whittle likelihood now had lowest IAE. However, thegeneralized Whittle estimator was only marginally inferior w.r.t. IAE but had
even higher UCI coverage.

152

5

Generalization of the Whittle likelihood

Applications

0.0
−1.0

−0.5

SOI

0.5

1.0

The Southern Oscillation Index (SOI), the monthly standardized anomaly of themean sea-level pressure difference between Tahiti and Darwin (available form
the Australian Bureau of Meteorology) is one of the key atmospheric indicesfor gauging the strength of El Niño events and their potential impacts on theAustralian region. We use the mean-centered SOI time series, shown in Figure 1,
to illustrate the effect of the choice of the AR(p) working model on the spectral
density estimate.

1950

1960

1970

1980

Time

FIGURE 1. Plot of the SOI index from 1950 to 1987.

70
40

50

60

NLL

80

90

100

Figure 2 shows the negative maximum log likelihood (NLL) value for different
AR(p) models.

0

5

10

15

20

25

30

p

FIGURE 2. Negative maximum log likelihood for various AR(p) models of SOI
data.
Figure 3 shows the log-periodogram overlaid by the posterior median spectral
density estimates of the parametric AR(p) model and generalized Whittle likelihood for the autoregressive orders p = 0, 5, 11 and 15 that correspond to “elbows”

Kirch et al.

153

0.0

0.0

−2.5

−2.5

log PSDlog PSDin the scree-like plot of the negative maximum log likelihood values in Figure 2.
For p = 0, the Bernstein polynomials of the nonparametric correction cannot yet

−5.0
−7.5
−10.0

−5.0
−7.5
−10.0

0

1

2

3

0

1

2

3

Frequency

0.0

0.0

−2.5

−2.5

log PSDlog PSD

Frequency

−5.0
−7.5
−10.0

−5.0
−7.5
−10.0

0

1

2

Frequency

3

0

1

2

3

Frequency

FIGURE 3. Posterior median spectral density estimates of the generalized Whittle likelihood (solid black) and AR (dashed black) for the SOI data on a logarithmic scale, for different autoregressive orders p = 0, 5, 11, 15.
capture the main peak whereas for p = 5 they can. With p = 11, minor peaks
are also well estimated with the generalized Whittle but not with the classicallikelihood. The estimators of a parametric AR(15) and the nonparametrically corrected model are very similar but all peaks of the AR(15) estimator are sharpenedeven further via the nonparametric correction.
Because of space constraints, we refer to Kirch et al. (2017) for an application to
LIGO gravitational wave data that we aim to discuss in detail at the conference.

Acknowledgments: This work was supported by DFG grant KI 1443/3-1.
We also thank the New Zealand eScience Infrastructure (NeSI) and the Universitätsrechenzentrum (URZ) Magdeburg for their high performance computing
facilities, and the Centre for eResearch at the University of Auckland and Jörg
Schulenburg for their technical support.
References
Choudhuri, N., Ghosal, S. and Roy, A. (2004). Bayesian estimation of the spectral density of a time series. JASA, 99, 1050 – 1059.
Contreras-Cristán, A., Gutiérrez-Peña, E. and Walker, S.G. (2006). A note on Whittle’s likelihood. Communications in Statistics – Simulation and Computation, 35, 857 – 875.
Kirch, C., Edwards, M., Meier, A., and Meyer, R. (2017). Beyond Whittle: Nonparametric correction of a parametric likelihood with a focus on Bayesiantime series analysis. https://arxiv.org/abs/1701.04846.

154

Generalization of the Whittle likelihood

Kreiss, J.-P., and Paparoditis, E. (2003). Autoregressive-aided periodogram bootstrap for time series. Annals of Statistics, 31, 1923 – 1955.
Meier, A., Kirch, C., Edwards, M., and Meyer, R. (2017). beyondWhittle: BayesianSpectral Inference for Stationary Time Series. R package.
Meyer, R. (2016). Deviance Information Criterion (DIC). N. Balakrishnan, P.
Brandimarte, B. Everitt, G. Molenberghs, W. Piegorsch, and F. Ruggeri(Eds.). Wiley StatsRef: Statistics Reference Online, Wiley.
Petrone, S. (1999). Random Bernstein Polynomials. Scandinavian Journal of Statistics 26, 373 – 393.
Roberts, G.O. and Rosenthal, J.S. (2009). Examples of adaptive MCMC. Journal of Computational and Graphical Statistics, 18, 349 – 367.
Rosen, O., Wood, S., and Stoffer, D.S. (2012). Adaptive Spectral Estimation for
Nonstationary Time Series. JASA, 107, 1575 – 1589.
Shao, X. and Wu, B.W. (2007). Asymptotic spectral theory for nonlinaer timeseries. Annals of Statistics, 35, 1773 – 1801.
Spiegelhalter, D.J., Best, N.G., Carlin, B.P., and Van der Linde, A. (2002). Bayesianmeasures of model complexity and fit. JRSSB, 64, 583 – 639.
Whittle, P. (1957). Curve and periodogram smoothing. JRSSB, 19, 38 – 63.

Model-based clustering for populations of
networks
Mirko Signorelli1,2 , Ernst C. Wit1
1
2

Johann Bernoulli Institute, University of Groningen, Netherlands
Department of Statistical Sciences, University of Padova, Italy

E-mail for correspondence: signorelli@stat.unipd.it
Abstract: We propose a clustering method for sequences of graphs. The method
exploits mixtures of generalized linear models and allows to detect subpopulationsof graphs in a model-based fashion. We provide an Expectation-Maximization
algorithm with Simulating Annealing (EMSAGC) for model estimation.
Keywords: EMSAGC; network; graph; mixtures of generalized linear models;model-based clustering.

1

Introduction

The last decades have witnessed a growing interest in network science. Althoughstatistical network models traditionally aimed at modelling relations encoded ina single network, the increasing availability of relational data has encouraged thecollection of several instances of the same network. Examples include longitudinal sequences of networks, which represent the dynamic evolution of a complex
system, and populations of networks, where each network represents the state ofa system for a given statistical unit.
Since it is reasonable to expect networks within such longitudinal or cross-sectionalsequences of networks to be similar to a certain degree, it is clear that modellingeach network separately would be a dispersive and ineffective strategy. Instead,by jointly modelling them we can borrow information between similar graphs andachieve a much more parsimonious answer.
For these reasons, we propose a novel method capable to characterize the joint
distribution of sequences of networks and to cluster networks, which relies on
mixtures of generalized linear models (Grün and Leisch, 2008). This approachcombines the flexibility of mixture models with the fact that many popular network models can be estimated by specifying a generalized linear model. We alsoThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

156Model-based clustering for populations of networks

propose EMSAGC, an estimation algorithm based on the integration of the EM
with simulated annealing.

2

Model-based clustering of networks

We consider a sequence of K graphs S = {G1 , G2 , ..., GK }, where each graph
Gk = (V, Ek ), k ∈ {1, ..., K}, defines a specific set of edges Ek between the same
set of v vertices V , and it is represented by the adjacency matrix Yk . We representkS with an array Y of size v × v × K. An entry yij
in Y refers to the presence(and intensity) or absence of edge eij ∈ Ek .
In principle, we could imagine that each graph Gk is drawn from a different
distribution f (Y |θk ), k ∈ {1, ..., K} with parameter vector θk - to wit, Yk ∼f (Y |θk ) . In the presence of many networks, however, this would result in a
cumbersome modelling exercise, leading to K different models.
Instead, we postulate the existence of M ≤ K subpopulations of graphs S1 , ..., SM
within S, each with density f (Y |θm ) , m ∈ {1, ..., M }. We denote by Zk ∈{1, ..., M } the latent identifying label of graph Gk , such that Zk = m if Gk ∈ Sm .
Therefore, we view each graph Gk ∈ S as a random draw from a mixture model
whose components are the densities f (Y |θm )
Yk ∼

M
X

πm f (Y |θm ) ,

(1)

m=1

with mixing proportions πm = P r(Zk = m), m ∈ {1, ..., M } denoting the priorprobabilities
Pthat a graph belongs to the mth subpopulation Sm . Clearly, πm ≥0 ∀m and m πm = 1. The likelihood of S is thenL(Y, Z|Θ) =KY

P r(Yk |Zk , Θ)P r(Zk |Θ) =

k=1

K
Y

πZk f (Yk |θZk ) ,

(2)

k=1

where Θ = (θ1 , ..., θM ) and Z = (Z1 , ..., ZK ).
Many popular network models, such as the p1 and p2 models (Holland and Leinhardt, 1981; van Duijn et al., 2004) as well as stochastic blockmodels and theirextensions (Wang and Wong, 1987), assume edges (or, for directed graphs, dyads)
to be statistically independent and, thus, they can be estimated with a generalized linear model (or, in the case of the p2 model, a generalized linear mixed
model). If one of such models is employed, each of the densities f (Y |θm ) in Equations (1) and (2) can thus be estimated by specifying a generalized linear model.
This can be done by considering densities f from exponential dispersion families,
and modelling the conditional expectation of each edge yij as
ηij = g [E (yij |xij , θm )] = xij β,(3)where xij indicates a set of nodal or edge-specific attributes and g is the linkfunction. Clearly, the density f (Y |θm ) can then be obtained asYf (Y |θm ) =f (yij |θm ) .
(4)i<jSignorelli and Wit

157

With these specifications, model (1) defines a mixture of generalized linear models(Grün and Leisch, 2008).
Within this framework, it is possible to cluster graphs according to features suchas node degree, reciprocation, membership of groups and nodal or edge-specificcovariates. For example, if the interest is simply on nodal degree, a mixture of p1
models can be considered; if a partition of nodes into groups is known, a mixtureof stochastic blockmodels can be specified as well (we provide an example of thisin Section 4). More generally, if one would simply like to cluster graphs without
assuming a specific network model, they can specify a model with one parameterkmfor each pair of nodes, i.e., ηij= γij. Note, however, that this approach does notallow to cluster graphs according to their tendency towards transitivity, since
Exponential Random Graph Models cannot be specified as generalized linear
models.

3

Model estimation

3.1

EMSAGC: Expectation-Maximization algorithm with
Simulated Annealing for Graph Clustering

In principle, maximization of (2) could be performed with a simple implementation of the EM algorithm. However, in our simulations this often resulted in low
accuracy of the clustering method, in particular for the cases of binary graphs andof sparse edge-valued graphs (Signorelli, 2017). Therefore, we propose EMSAGC,
an estimation algorithm that integrates the EM with Simulating Annealing andis capable to improve the clustering accuracy. Simulated Annealing (Eglese, 1990)
is a strategy that avoids the risk to get trapped in a local optimum of an objective function f by proposing a move from the current local maximum x̂ to a
proposal x̃, and by allowing a positive probability to accept the move even whenf (x̃) < f (x̂). The algorithm consists of the following iterative steps:
1. for k ∈ {1, ..., K} and m ∈ {1, ..., M }, define the initial probabilities
p1km = P r(zkm = 1). Denote by P 1 the K × M matrix which collects
these probabilities;
2. for t = 1, 2, ...:
 M step with Simulated Annealing.
M1. Given P t , estimate M network models(specified as GLMs) with
weights given by pt1m , ..., ptKm for the m-th component andderive Θ̂t .
M2. If t ≥ 2 and L(Y, Z|Θ̂t ) ≤ L(Y, Z|Θ̂t−1 ), consider the alternative state P̃ t and estimate Θ̃t :? if L(Y, Z|Θ̃t ) ≥ L(Y, Z|Θ̂t ), set Θ̂t = Θ̃t and P t = P̃ t .
? if L(Y, Z|Θ̃t ) < L(Y, Z|Θ̂t ), set Θ̂t = Θ̃t and P t = P̃ t withprobability equal to!1/T (t)log L(Y, Z|Θ̃t ),(5)
log L(Y, Z|Θ̂t )where T (t) = log1 t .

158Model-based clustering for populations of networks
 E step. Given Θ̂t , derive P t+1 ast
P r(Gk |θ̂m)
pt+1.
km = PMt
j=1 P r(Gk |θ̂j )(6)

3. Choose the best solution within the sequence {Θ̂1 , Θ̂2 , ..., }, i.e.
Θ̂EM SAGC = argmaxt=1,2,... L(Y, Z|Θ̂t ).

3.2

(7)

A comparison of the performance of the EM and EMSAGC
algorithms

We have assessed the performance of the EM and EMSAGC algorithms on simulated sequences of undirected networks with two mixture components. We haveconsidered both binary graphs and edge-valued graphs, employing mixtures oflogistic and of Poisson generalized linear models to estimate the clusters. When asequence of edge-valued graphs whose edge values follow a Poisson distribution isconsidered, the EM algorithm already results in a very good clustering accuracy.
Instead, as Table 1 shows, when we consider sequences of binary graphs and ofgraphs whose edge values are overdispersed (i.e., they follow a negative binomial
distribution) the accuracy of the EM algorithm is often low, especially wheneither K or v are very small. In these cases, application of EMSAGC results in ahigher accuracy of the retrieved clusters, yielding a substantial improvement overthe performance of the EM algorithm. Full details on the simulation settings canbe found in Signorelli (2017).

4

Example application

We consider 10 daily interaction networks between employees of the French Institute for Public Health Surveillance, collected by Génois et al. (2015). In partic-

TABLE 1. Comparison of the accuracy obtained with the EM and EMSAGC
algorithms, for binary graphs and edge-valued graphs with overdispersed degree
distribution. Each cell displays the percentage of correctly clustered graphs.

Binary graphs

Sparse edge-valued graphs

K

v

EM

EMSAGCEMEMSAGC

10
10
10
20
20
20
50
50
50

10
50
100
10
50
100
10
50
100

72%
68%
65%
92%
85%
81%
100%
100%
100%

74%
99%
98%
100%
100%
100%
100%
100%
100%

67%
65%
66%
81%
91%
84%
96.4%
100%
100%

86%
100%
100%
92%
100%
100%
98%
100%
100%

Signorelli and Wit

159

ular, we focus on those employees who have registered interactions for at least 6
days; each of them belongs to one of four departments (DISQ, DMCT, DSE and
SRH). It is reasonable to assume that interactions are affected both by department affiliation and by individual features. Therefore, we consider a mixture ofdegree-corrected stochastic blockmodels with two components, where the numberof daily interactions between node i from department r and node j from department s follows a Poisson distribution whose mean depends, on the log-scale, bothon nodal effects αi , αj and on block-interaction parameters φrs .
The application of the EMSAGC algorithm allows to detect a variation in thepattern of interactions among departments from the first 7 days (cluster 1) tothe remaining 3 days (cluster 2) considered in the study. The estimate of theblock-interaction parameters in the two clusters are shown in Table 2. Overall,
we find two changes in the pattern of interaction across departments. On the one
hand, members of DISQ and DSE are more active within their department in
the first 7 days considered, but then they interact more with each other in theremaining 3 days. On the other hand, employees in DMCT and SRH seem tofollow the opposite pattern: in the last 3 days, they reduce interactions between
their departments and are more active within their own department.

References
Eglese, R. (1990). Simulated annealing: a tool for operational research. European
journal of operational research, 46(3), 271–281.
Genois, M. et al. (2015). Data on face-to-face contacts in an office building suggests a low-cost vaccination strategy based on community linkers. Network
Science, 3(03), 326–347.

TABLE 2. Comparison of block-interactions between clusters. Employees in departments DSE and DISQ interact more within their department in the first 7days, and more between each other in the last 3 days (corresponding parameters
are emphasised in bold). Conversely, employees in DMCT and SRH interact morewith each other in the first 7 days, and within their own departments in the last
3 days (corresponding parameters are underlined).

Block-interaction
parameter
DMCT
DSE
DISQ
SRH
DMCT-DSE
DMCT-DISQDMCT-SRH
DSE-DISQ
DSE-SRH
DISQ-SRH

Estimates
Cluster 1 Cluster 2
0.71
0.92
1.43
1.83
-0.15
-0.22
-0.34
-0.24
-0.53
-0.96

0.86
0.58
1.10
2.01
-0.12
-0.18
-0.56
0.03
-0.49
-0.96

160

Model-based clustering for populations of networks

Grün, B. and Leisch, F. (2008). Finite mixtures of generalized linear regression
models. In: Recent advances in linear models and related areas, 205-230.
Springer.
Holland, P. W. and Leinhardt, S. (1981). An exponential family of probability
distributions for directed graphs. Journal of the American Statistical Association, 76(373), 33–50.
Signorelli, M. (2017). Inferring community-driven structure in complex networks.
Doctoral dissertation, University of Groningen. Retrieved from http://hdl.handle.net/11370/5e39c
168a-49f9-95e2-98860de508a9.
van Duijn, M. A. J., Snijders, T. A. B., Zijlstra, B. H. J. (2004). p2: a random
effects model with covariates for directed graphs. Statistica Neerlandica,
58(2), 234–254.
Wang, Y. J. and Wong, G. Y. (1987). Stochastic blockmodels for directed graphs.
Journal of the American Statistical Association, 82(397), 8–19.

Generalized Method of Moments for
Estimating the Parameters of Stochastic
Actor-oriented Models
Tom A. B. Snijders12 , Viviana Amati3 , Felix Schönenberger3
1

Department of Sociology, University of Groningen, The Netherlands
Nuffield College and Departement of Statistics, University of Oxford, UK
3
Department of Computer and Information Science, University of Konstanz,
Germany
2E-mail for correspondence: tom.snijders@nuffield.ox.ac.uk
Abstract: Stochastic actor-oriented models are models for dynamic network
data, collected by observing a network and a behavior over time in a panel design.
We present an estimator based on the generalized method of moments, using
more statistics than parameters. For some examples we study the resulting gainin statistical efficiency. We discuss algorithmic issues that have to be solved toobtain a stable procedure.
Keywords: Network panel data; Stochastic actor-oriented models; Generalized
method of moments; Selection and Influence processes.

1

Stochastic actor-oriented models

Stochastic actor-oriented models (SAOMs) (Snijders, 2001; Steglich et al., 2010)
are applied to analyze network panel data (x, z)t1 , . . . , (x, z)tM , defined as observations of a network x and a behavior z observed at M > 1 time points. The
network x is a digraph; the behavior z is an ordinal discrete variable with integer
values, defined at the level of actors.
In SAOMs the network and the behavior are jointly the two dependent variables.
The dependence of network dynamics and that of behavioral dynamics on the
network-behavioral configuration are referred to as “selection” process and “influence” process, respectively. SAOMs allow to distinguish the role of selection
and influence mechanisms in explaining the network and behavioral changes overtime.
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

162

GMoM for estimating the parameters of SAOMs

Following the approach originally proposed by Holland and Leinhardt (1977),
SAOMs assume that the time series (x, z)t1 , . . . , (x, z)tM is the outcome of acontinuous-time Markov chain. The observed panel data are assumed to be theresult of a sequence of unobserved changes happening between two consecutive
observation time points tm−1 and tm . For simplicity, we treat only the case M = 2.
The series of changes is modeled in the following way. At a random moment, one
actor is selected and has the opportunity to make either a network or a behavioral change. Due to the Markov assumption, the time between two consecutive
opportunities for a change must be exponentially distributed. The correspond[X][Z]
ing rate parameters are called λm and λm for network and behavioral changes,
respectively.
Probabilities of changes are determined by so-called evaluation functions, defined
separately for the two variables. For variable V = X or Z, the evaluation function
is given by[V ]fi (β [V ] , x, z) =

[V ]
K
X

[V ]

[V ]

βk sik (x, z)

(1)

k=1
[V ]

where sik (x, z) for V = X, Z are suitable statistics of which a description canbe found, e.g., in Steglich et al. (2010).
When an actor gets the opportunity for a network change, he can create one
new outgoing tie, terminate one existing outgoing tie, or do nothing. When an
actor gets the opportunity for a behavioral change, he can increase or decrease
his behavior by one unit, or do nothing. Changes are modeled according to a[X]
multinomial logit model. Given the current state (x, z), the probability p(x0 ,z)[Z]

that, for a network change, the next state is (x0 , z), and the probability p(x,z0 )that, for a behavioral change, the next state is (x, z 0 ), is given by


[X]
[Z]
exp fi (β [X] , x0 , z)
exp fi (β [Z] , x, z 0 )[X][Z]p(x0 ,z) = P
and
p
=
.
P
(x,z 0 )[X][Z]
exp(fi (β [X] , x00 , z))
exp fi (β [Z] , x, z 00 )
x00

z 00

The vector of parameters θ = (λ[X] , λ[Z] , β [X] , β [Z] ) of SAOMs is usually estimated using the method of moments (MoM), solving the equation





X(t1 ), Z(t1 ) = x(t1 ), z(t1 ) = 0 . (2)Eθ s X(t2 ), Z(t2 ) − s x(t2 ), z(t2 )Statistics s(X, Z) modeling influence (dependent variable: behavior) are distinguished from those modeling selection (dependent variable: network) by using
cross-lagged variables. Statistics suggested by (1), providing information aboutβ [X] and β [Z] , respectively, are
X [X]
X [Z]


si X(t2 ), Z(t1 ) andsi X(t1 ), Z(t2 ) .
ii

The solution of (2) is the MoM estimate for θ. It can be approximated using astochastic algorithm (Snijders, 2001) based on Monte Carlo approximation of the
expected values and the Robbins-Monro step (Robbins and Monro, 1951). The algorithm is implemented in the library RSiena (http://www.stats.ox.ac.uk/ snijders/siena/)of the R software.

Snijders et al.

2163

Generalized method of moments for SAOMs

It is natural that we would like to include as much information as possible in
the estimation process. When analyzing the co-evolution of networks and behaviors, additional information is provided by the non-cross-lagged statistics[V ]si (X(t2 ), Z(t2 )) for V = X, Z, where both dependent variables are taken at t2 .
Taking these into account may improve the estimation, especially when changes
in both the behavior and the set of outgoing ties are observed for a significant
proportion of actors. We refer to these changes as simultaneous changes. Here,
we propose to include the non-cross-lagged statistics in the estimation process
to get a better estimate for θ, following the generalized method of moments(GMoM; Hansen,
1982) estimation.For notational brevity, we denote (X, Z) =
X(t2 ), Z(t2 ) and (x, z) likewise.
Including more statistics than parameters results in an over-identified system of
moment equations. The GMoM estimate for θ is the value θb minimizing the distance between the expected values of the statistics and their sample counterpart:
F (X, Z; θ) = Eθ [s∗ (X, Z) − s∗ (x, z)]T W Eθ [s∗ (X, Z) − s∗ (x, z)](3)where s∗ (X, Z) is the vector containing the cross-lagged as well as non-crosslagged statistics, the expectation is conditional given X(t1 ), Z(t1 ) as in (2),and W is a positive semi-definite matrix. An optimal choice for W is the inverseof the variance-covariance matrix of s∗ (X, Z).
From (3) and the convexity of F (X, Z; θ), it follows that the estimate θb is thevalue solving∂
F (X, Z; θ) = Bθ Eθ [s∗ (X, Z) − s∗ (x, z)] = 0∂θwhere Bθ = ∂ Eθ [s∗ (X, Z) − s∗ (x, z)]T W /∂θ.

(4)

Equation (4) is equal to equation (2) except for the matrix Bθ . Therefore, θ̂ canbe computed using a modification of the stochastic approximation algorithm usedalso for the MoM. Main changes in the algorithm are related to the estimationand the inclusion of the matrix Bθ in the Robbins-Monro step. Based on theeconometric literature, we implemented both the efficient two-step and the iterated two-step GMoM estimators and tested their performance. However, here we
report only the results based on the iterated two-step estimator.

3

Simulations

Several simulations were conducted to investigate the relative efficiency of the
GMoM estimator with respect to the MoM estimator. A nested simulation design was used: For several specification of the SAOMs, 500 network panel data
were generated and estimated twice by both methods. Such a design has the advantage of accounting for the variation in both the simulations and the stochastic
algorithm.
For illustrative purposes, we consider a simple model specification based on empirical results deriving from the estimation of SAOMs in friendship networks andon the use of only one non-cross-lagged statistic as described in the following.

164

GMoM for estimating the parameters of SAOMs

The simulated network panel data comprises a binary directed network and abehavior with 5 ordinal categories, both simulated at 2 points in time, as well asa constant binary actor covariate. Those were generated using a SAOM specified
by basic effects describing the network structure (e.g. density and reciprocity), theimpact of an actor covariate (e.g. popularity and activity) and the distributionof the behavior (linear and quadratic effects). We also included in the model
the behavior similarity effect and the average similarity effect. These two model
homophily whose outcome is the similarity of two connected actors.
The similarity effect models selection, i.e. the tendency of actor to form ties withactors that are similar to themselves, whereas the average similarity effect models
influence, i.e. the tendency of actors which are tied to become similar in their[Z][X]
behavior. The corresponding cross-lagged statistics ssim and sav sim are defined,
respectively, as[X]ssim =Xxij (t2 )ij[Z]av sim =∆ − |zi (t1 ) − zj (t1 )|− simz∆1xij (t1 )x
j ij (t1 )

X

P
ij





∆ − |zi (t2 ) − zj (t2 )|− simz∆,

where ∆ is the range of the behavioral variable and simz is the average of∆−|zi −zj |the observed similarity scores.
∆
The non-cross-lagged statistic related to the statistics above is defined as:s∗sim =Xijxij (t2 )∆ − |zi (t2 ) − zj (t2 )|− simz∆.

and used here to compare the relative efficiency of the GMoM to the MoM.

4

Results

The root mean squared errors of the estimators were compared using a onesided Wilcoxon-signed ranked test (alternative hypothesis: RM SEGM oM <RM SEM oM ) to evaluate the relative efficiency of the GMoM to the MoM.
Table 1 shows the results for two models differing in the value of therate parameters, and consequently in the mean percentages of simultaneous changes (25% and 45% for lower and higher rates, respectively). The
comparison suggests that the GMoM outperforms the MoM and the gainin efficiency mainly affects the parameters related to the effects for thosea non-cross-lagged statistics was considered. The gain is due to both thesmaller deviations and the higher efficiency of the GMoM estimates, as
illustrated in Figure 1, which contrasts the absolute deviations and theestimated standard errors of the GMoM (y-axis) and the MoM (x-axis)
estimates for the parameters of the behavior similarity and the averagesimilarity effects.
More generally, the set of simulations we performed indicates that the gainin efficiency is negligible when i) the proportion of simultaneous changes israther low, and ii) the non-cross-lagged statistics are non-informative, i.e.,Snijders et al.
MoM
Est.
RMSE

GMoM
Est.
RMSE

3.50
-2.20
1.80
0.25
-0.27
0.67
-0.60
0.19
0.48
0.61
3.00

3.40
-2.34
1.85
0.21
-0.24
0.71
-0.66
0.23
0.59
0.73
3.40

0.84
0.80
0.35
0.10
0.21
0.34
0.44
0.51
0.39
0.36
1.83

3.41
-2.31
1.85
0.22
-0.24
0.72
-0.65
0.22
0.58
0.72
3.27

0.83
0.74
0.35
0.10
0.20
0.34
0.42
0.48
0.37
0.34
1.37

**

1.50
0.05
0.10
6.00

1.55
0.06
0.08
5.69

0.92
0.28
0.16
3.28

1.54
0.06
0.08
5.65

0.91
0.28
0.16
3.01

**

7.00
-2.20
1.80
0.25
-0.27
0.67
-0.60
0.19
0.48
0.61
3.00

6.66
-2.27
1.84
0.22
-0.23
0.71
-0.71
0.31
0.60
0.71
3.52

1.94
0.82
0.35
0.08
0.18
0.31
0.46
0.53
0.42
0.35
2.49

6.70
-2.24
1.84
0.22
-0.22
0.71
-0.70
0.29
0.58
0.70
3.36

1.81
0.67
0.32
0.08
0.17
0.30
0.40
0.44
0.33
0.28
1.39

**
**

2.00
0.05
0.10
6.00

2.00
0.07
0.08
5.34

1.28
0.25
0.13
3.36

2.00
0.06
0.08
5.41

1.24
0.25
0.13
3.00

*

θ
Selection
Rate
Density
Reciprocity
Transitive triplets
3-cyclesTransitive Ties
Outdegree popularity (sqrt)
Covariate-related popularity
Covariate-related activity
Same covariate
Behavior similarity
Influence
Rate
LinearQuadratic
Average similarity
Selection
Rate
Density
Reciprocity
Transitive triplets
3-cyclesTransitive Ties
Outdegree popularity (sqrt)
Covariate-related popularity
Covariate-related activity
Same covariate
Behavior similarity
Influence
Rate
LinearQuadratic
Average similarity

165

Sig.

*

**
**
**

**

TABLE 1. Simulation results, 2 waves for 50 actors: true value of the parameter(θ), mean parameter estimates (Est.), and root mean squared errors (RMSE).
The last column (Sig.) reports the significance of the Wilcoxon signed rank testcomparing the RMSE of the two estimators (* p-value < 0.05, **p-value < 0.01).

highly correlated with the cross-lagged statistics. In these cases, the MoM
is preferred to the GMoM since the MoM algorithm is faster. Furthermore,in accordance with applications of the GMoM in economics, we observedthat the algorithm based on the iterated two-step estimator is more prone
to instability when the non-cross-lagged statistics are non-informative.
Acknowledgments: Part of this research has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n. 319209.
References
Hansen, L. (1982). Large sample properties of generalized method of moments estimators. Econometrica, 50, 1029 – 1054.
Holland, P. W. and Leinhardt, S. (1977). A Dynamic Model for Social Networks. Journal of Mathematical Sociology, 5(1), 5 - 20.
Robbins, H. and Monro, S. (1951). A stochastic approximation method.

166

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0

(a) Behaviour similarity: absolute deviations

1

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2

3

(b) Average similarity: absolute deviations
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0

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(d) Average similarity: estimated standard errors

FIGURE 1. Comparison between the absolute deviations and the estimated standard errors of the MoM (x-axis) and the GMoM (y-axis) for the behavior similarity and the average similarity effects for the network panel data simulated withthe lower rates. The line in the plot represents the diagonal.

The Annals of Mathematical Statistics, 22, 400 - 407.
1
Snijders, T. A. B. (2001). The statistical evaluation of social network dynamics. Sociological Methodology, 31(1), 361 – 395.
Steglich, C. E. G., Snijders, T. A. B., and Pearson, M. (2010). Dynamic networks and behavior: Separating selection from influence. Sociological
Methodology, 40(1), 329 – 393.

Stochastic block models for social network
data: inferential developments
Francesco Bartolucci1 , Maria Francesca Marino2 , Silvia
Pandolfi1
12

Department of Economics, University of Perugia, Perugia, Italy
Department of Political Sciences, University of Perugia, Perugia, Italy

E-mail for correspondence: silvia.pandolfi@unipg.it
Abstract: Stochastic block models have known a flowering interest in the social network literature. They provide a tool for discovering communities andidentifying clusters of individuals characterized by similar social behaviors. In
this framework, full maximum likelihood estimates are not achievable due to theintractability of the likelihood function. For this reason, several approximate solutions are available in the literature. Here, we propose a new and more efficientapproximate method for estimating model parameters, which has a hybrid naturein the sense that it exploits different features of existing methods. The proposal
is illustrated by a Monte Carlo simulation study.
Keywords: Classification likelihood; Composite likelihood; EM algorithm; Random graphs; Variational inference.

1

Introduction

Stochastic Block Models (SBMs; Snijders and Nowicki, 1997, Nowicki and Snijders, 2001) represent an important tool of analysis in the social network literature when the focus is on discovering communities and clustering individuals
with respect to their social behavior. According to the SBM specification, eachindividual in the network belongs to one of k distinct blocks, corresponding to
the categories of a discrete latent variable, and the probability of observing aconnection between two units only depends on their block memberships. Despite
the simplicity of the model, Maximum Likelihood (ML) inference remains problematic due to the intractability of the likelihood function.
Some approximate solutions are available in the literature. These are mainly
based on classification likelihood (Choi et al., 2012), variational approximation
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

168inferential developments of SBM(Daudin et al., 2008), or on a composite likelihood approach based on all triples
in the network (Ambroise and Matias, 2012). In this work, we propose a new andmore efficient approximate method for estimating model parameters, which has
a hybrid nature as it is based on a classification likelihood but has features incommon also with full likelihood and with variational and composite likelihood
inference. We illustrate the potential of the proposed approach by an intensive
simulation study.
The reminder of this paper is structured as follows. The SBMs are introduced in
the following section, together with ML estimation of the model parameters andthe alternative approximate inferential solutions. In Section 3 we illustrate theproposed hybrid estimation method, whereas in Section 4 we describe the main
results of the simulation study.

2

Stochastic block models

In this section we briefly outline the main assumptions of the SBMs and we
introduce the alternative approaches for parameter estimation.

2.1

Model assumptions

Let Y denote a socio-matrix involving n individuals/nodes and whose genericelement, Yij , is equal to 1 if there is a tie between i and j and is equal to 0otherwise, with i, j = 1, . . . , n. We focus on undirected networks where Yij =Yji , i 6= j, with probability 1, although the extension to the directed case isstraightforward. Moreover, self cycles are excluded by design.
In order to analyze social interactions existing between the nodes in the network, SBMs assume the existence of k unobserved blocks, which are described by
the node-specific latent variables Ui , i = 1, . . . , n, having discrete support points
1, . . . , k and corresponding mass probabilities p(Ui = u) = πu , u = 1, . . . , k. Also,
SBMs postulate a local independence assumption between nodes: conditional on
the latent variables Ui and Uj , responses Yij are assumed to be independent
Bernoulli random variables with success probability ψui uj = p(Yij = 1|Ui =ui , Uj = uj ). Therefore the response variables only depend on the block memberships of individuals i and j.
Despite the simplicity of the above assumptions, ML inference for this class ofmodels remains problematic. This is because the manifest distribution of theentire network Y is obtained by marginalizing out all latent variables and thismarginalization is computationally tricky. In particular, we haveX
p(Y) =
p(Y | u)p(u),uwhere u denotes a realization of the random vector U = (U1 , . . . , Un )0 , andYYp(Y | u) =p(yij | Ui = ui , Uj = uj ),i<n j>i

p(u)

=

Y
i<n

πui .

Bartolucci et al.

169

Therefore, to estimate the model parameters collected in θ, approximate methods
that are alternative to full ML are needed, even dealing with small networks.

2.2

Maximum likelihood approachWith very small networks, the log-likelihood `(θ) = log p(Y ) may be maximized via the Expectation-Maximization (EM) algorithm, which is based on thefollowing complete data log-likelihood function related to the classification loglikelihood:XXlog p(yij | Ui = ui , Uj = uj ) πui πuj .
`∗cl (u, θ) =i<n j>iThe EM algorithm maximizes `(θ) by alternating two steps until convergence.
The E-step (Expectation) consists in computing the conditional expected valueof `∗cl (u, θ) given the observed data and the current value of the parameters. At
the M-step (Maximization) we update the parameter vector θ by maximizing
the expected value previously computed.

2.3

Available approximate estimation methods

Among the available approximate methods, it is important to mention the classification likelihood approach (Choi et al., 2012). In this framework, the realizationsof the random vector U are considered as fixed parameters to be estimated. Two
steps are alternated until convergence: C-step (Classification), which consists inmaximizing the classification likelihoodXX`cl (u, θ) =log p(yij | Ui = ui , Uj = uj )i<n j>iwith respect to u by iteratively moving nodes from one block to another as in a kmeans algorithm; M-step (Maximization), in which we update θ by maximizing`cl (u, θ) with u being fixed at the previous value.
An alternative estimation method is based on a variational approximation of
the model likelihood (Daudin et al., 2008), in which parameter estimates arederived by maximizing a lower bound of the log-likelihood function. This method
is computationally fast but may lead to non-optimal estimators and may sufferfrom local maxima solutions that strongly depend on the starting values of theestimation algorithm. Moreover, it may be quite slow to converge.
Finally, it is worth mentioning the composite likelihood approach introduced by
Ambroise and Matias (2012), see also Bartolucci et al. (2015), in which parameter
estimates are derived by maximizing a composite likelihood function based on theprobability of all triples in the network. This approach leads to an estimator withproperties similar to those of the standard ML estimator but it may suffer fromidentifiability problems. Even in this case, the algorithm may be slow to converge.

170

3

inferential developments of SBM

Proposed hybrid ML inference

The proposed hybrid ML method lies in between full ML and classification likelihood inference discussed in the previous section. More in detail, this method isbased on the following hybrid log-likelihood function:
XX
`hyb (ũ, θ) =log
p yi | Ui = ui , U(−i) = ũ(−i) πui ,i<nui

where yi is the vector of all observed responses yij , j 6= i, for unit i, ũ is arealization of U considered as a vector of fixed discrete parameters in the set{1, . . . , k}, and ũ(−i) is the corresponding subvector without the i-th element.
Moreover, p(yi | Ui = ui , U(−i) = ũ(−i) ) denotes the conditional distribution ofyi given the underlying latent variables.
The proposed method is based on alternating three steps until convergence. The
C-step (Classification) consists in maximizing `hyb (ũ, θ) with respect to ũ byiteratively moving nodes from one block to another as in a k-means algorithm.
At the E-step (Expectation) we compute the conditional expected value of thecomplete data log-likelihood function corresponding to `hyb (ũ, θ), and denotedby `∗hyb (ũ, θ), given the current value of θ and ũ. At the M-step (Maximization),
θ is updated by maximizing the expected value of `∗hyb (ũ, θ).
The inferential method we propose is related to the classification likelihood
method because we explicitly consider the partition defined by the latent variables as depending on parameters to be estimated. Moreover, it has features incommon with a composite likelihood method based on the response configuration of each node i. Finally, it relies on an optimization algorithm with structureand numerical complexity similar to that of the variational approach, while being
typically faster to converge.

4

Simulation study

To assess the performance of the proposed approach, we carried out a simulation
study in which this approach is compared with the available alternatives, thatis, the k-means algorithm, the variational, the composite, and the classification
likelihood approaches. We considered B = 100 samples generated from an SBM
with varying number of groups (k = 2, 3, 4, 5) and equal proportions (πu = 1/k,u = 1, . . . , k), varying sample sizes (n = 20, 50, 100), and varying intra-group(ψin ) and inter-group (ψout ) connectivity parameters. In particular, we assumed
three different models: high intra-group connectivity (M1, with ψin = 0.30 andψout = 0.03); high inter-group connectivity (M2, with ψin = 0.03 and ψout = 0.30);
no structure (M3, with ψin = 0.55 and ψout = 0.45). Then, we set ψui uj = ψin ×α
if ui = uj and ψui uj = ψout if ui 6= uj with α ∼ Unif(0.5, 1.5) for all models.
The agreement between the estimated and the true latent structure is evaluatedin terms of number of correctly classified nodes (#ccn) with respect to the true
partition.
With reference to the most sensible scenarios, Figure 1 reports the boxplots ofthe values of #ccn obtained in the B = 100 samples by the alternative inferential
approaches under comparison.

Bartolucci et al.

171

From the figure we observe that, for all methods under comparison, the qualityof clustering worsens as k increases, due to the higher uncertainty on the latent
structure, while it improves when the community structure in the data is moreevident, that is, when moving from model M3 to M1. Moreover, the proposed
approach seems to recover better the true clustering of the nodes in all simulation settings we considered, especially when dealing with more complex model
structures. Lastly, the hybrid ML algorithm we propose generally requires a lower
computational time to reach the converge and seems to be less sensitive to theadopted starting rule. In this respect, it represents an appealing alternative tothe available approximate ML methods for parameter estimation in the SBM
framework.
Further developments will be devoted to investigate the theoretical propertiesof the proposed method, with the aim of extending the approach also to thelongitudinal social network framework.

References
Ambroise, C. and Matias, C. (2012). New consistent and asymptotically normal
parameter estimates for random-graph mixture models. Journal of theRoyal Statistical Society: Series B, 74, 3 – 5.
Bartolucci, F., Marino, M.F., and Pandolfi, S. (2015). Composite likelihood inference for hidden Markov models for dynamic networks. MPRA Paper
No. 67242.
Choi, D.S., Patrick, J.W., and Airoldi, E.M. (2012). Stochastic blockmodels witha growing number of classes. Biometrika, 99, 273 – 284.
Daudin, J.J., Picard, F., and Robin, S. (2008). A mixture model for random graphs.
Statistics and Computing, 18, 173 – 183.
Nowicki, K. and Snijders, T.A.B. (2001). Estimation and prediction forstochastic blockstructures. Journal of the American Statistical Association,
96, 1077 – 1087.
Snijders, T.A. and Nowicki, K. (1997).
Estimation
andpredictionforstochastic blockmodels for graphs with latent block structure. Journal ofClassification, 14, 75 – 100.

172

inferential developments of SBM

●
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HybridKmeansVariational

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Hybrid

95

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HybridFIGURE 1. Number of correctly classified nodes (#ccn) under models M1, M2,
and M3.

A sequentially coupled non-homogeneous
dynamic Bayesian network model withsegment-specific coupling strengths
Mahdi Shafiee Kamalabad1 , Marco Grzegorczyk1
1

Johann Bernoulli Institute (JBI), Rijksuniversiteit Groningen, The Netherlands

E-mail for correspondence: m.shafiee.kamalabad@rug.nl
Abstract: We propose a generalised version of a non-homogeneous dynamic
Bayesian network (NH-DBN) model with sequentially coupled interaction parameters. Unlike the earlier model, our new model does not assume that the segmentspecific interaction parameters are coupled with the same coupling strength. Our
new model introduces segment-specific coupling strengths, and we show that thisgeneralization can lead to improved network reconstruction accuracies.
Keywords: Non-homogenous dynamic Bayesian networks; sequential-information
coupling; segment-specific coupling strengths, RJMCMC

1Introduction

In the field of systems biology non-homogeneous Dynamic Bayesian Networks(DBNs) have become a popular model class for learning regulatory networks fromgene expression data. In many applications the intensities of the regulatory processes (i.e. the network parameters) undergo temporal changes. The conventional
homogeneous DBN models, which assume that the network parameters stay constant over time, therefore often lead to biased results and erroneous conclusions.
Non-Homogeneous DBNs (NH-DBNs) combine homogeneous DBNs with multiple change point processes. Loosely speaking, a set of changepoints divides the
temporal data into disjunct segments and the data points within each segment
are modelled by separate DBNs. To allow for some information sharing among
segments it is often assumed that the same network structure applies to all segments and that only the network parameters are subject to temporal changes.
Often the number and locations of those changepoints are unknown so that they
have to be inferred together with the network structure and the segment-specificnetwork parameters.
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

174

Generalized sequential coupling scheme

The standard NH-DBN models assume a priori that the segment-specific network
parameters are independent, and thus, they do not allow for any information sharing: the network parameters have to be learned independently in each segment
and uninformative prior distributions are imposed.
A NH-DBN model with sequentially coupled network parameters was proposed in
Grzegorczyk and Husmeier (2012). The key idea is to take into account that the
network parameters often evolve gradually in time and that they are often similar
for neighbouring segments. Grzegorczyk and Husmeier (2012) therefore proposed
to use the posterior expectation of the network parameters of a segment h as priorexpectation for the next segment h + 1. For each segment h > 1 this yields aninformative network parameter prior, which is built from the Bayesian inference
result obtained for the preceding segment h − 1. In the model of Grzegorczyk andHusmeier (2012) there is one single coupling parameter for each network node andall node-specific network parameters are coupled with the same coupling strength
among all segments. A low (high) coupling parameter leads to peaked (vague)
network parameter priors for all segments, where each segment-specific prior iscentred around the posterior mean of its preceding segment. This is a potential
bottleneck of the approach, since the similarity of the segment-specific network
parameters can vary over time. There might be changepoints where the network
parameters change only slightly, so that an informative (peaked) prior should be
used, as well as changepoints with drastic network parameter changes, where an
uninformative (vague) prior would be more appropriate. To address this shortcoming we propose a novel generalised sequentially coupled NH-DBN model withsegment-specific coupling parameters. In the new model the coupling strengthsδh (h > 1) between two neighbouring segments h − 1 and h are segment-specific.
Our new ’generalised’ sequentially coupled NH-DBN model can be thought ofas a continuous version of the ’partially sequentially coupled’ NH-DBN model,
which we proposed one year ago at IWSM 2016; see Shafiee and Grzegorczyk(2016). In our earlier work we introduced discrete binary indicator variables δhwhich indicate for each segment h whether it is coupled to the preceding one(δh = 1; peaked prior around posterior of preceding segment) or uncoupled fromthe preceding one (δh = 0; vague uninformative priors around 0). The new model
replaces the segment-specific binary indicator variables δh ∈ {0, 1} (uncoupled
vs. coupled) by segment-specific continuous variables δh ∈ R+ , and thus models the coupling strengths between all neighbouring pairs of segments (h − 1, h)
continuously.

2

Methodological details

Let y be the response variable in a segment-wise linear Bayesian regression model.
Given a set of k covariates, π = {X1 , . . . , Xk }, we assume that the data canbe divided into H disjunct segments with segment-specific regression coefficient
vectors wh . Furthermore, let yh and Xπ,h denote the response vector and thedesign matrix for segment h, where Xπ,h includes a first column of 1’s for theintercept. We then have for h = 1, . . . , Hyh ∼ N (Xπ,h wh , σ 2 I)

(1)

Shafiee and Grzegorczyk

175

where σ 2 is the noise variance with σ −2 ∼ GAM (ν, ν) and ν = 0.01. While
the standard uncoupled NH-DBN uses independent Gaussian priors for all H regression coefficient vectors, P (wh ) ∼ N (0, δσ 2 I), the sequentially coupled modelfrom Grzegorczyk and Husmeier (2012) uses the prior(
N (0, δσ 2 I),
h=1(2)
P (wh ) =N (mh−1 , λσ 2 I) h > 1
where

−1 

mh = λ−1 I + XTπ,h Xπ,hλ−1 mh−1 + XTπ,h yh (h ≥ 1)
The prior of the first segment has a special parameter δ which is different fromthe coupling parameter λ, as the coefficients of the first segment h = 1 cannot be
coupled to a preceding segment and thus must get the uninformative mean vector
m0 = 0. Only for h > 1 the prior mean is the posterior mean of the preceding
segment E[wh−1 |yh−1 , Xπ,h−1 , λh−1 ] = mh−1 , where the noise variance σ 2 hasbeen marginalized out. In our new generalised sequentially coupled model wereplace equation (2) by:
P (wh ) = N (mh−1 , λh σ 2 I)

(3)where m0 = 0 and λ1 refers to the δ parameter in equation (2). The only difference between equations (2) and (3) is then that the single coupling parameter λfor h > 1 has been replaced by segment-specific coupling parameter λ2 , . . . , λH .
Our generalised model is identical to the original sequentially coupled model forH ≤ 2 and more flexible than the original model for H > 2. The wh posteriormeans of our model are

−1 

T
T
mh = λ−1
λ−1(h ≥ 1)h I + Xπ,h Xπ,hh mh−1 + Xπ,h yh
We follow Grzegorczyk and Husmeier (2012) and use the hyperpriors−1δ −1 ∼ GAM (α1 , β1 ), λ−1
1 ∼ GAM (α1 , β1 ), λh ∼ GAM (αh , βh ) (h > 2)with α1 = 2, β1 = 0.2, αh = 3, and βh = 3. The segment-specific regression
coefficient vectors, wh , the noise variance, σ 2 , and the segment-specific coupling
parameters, λh , can then be sampled from their full conditional distributions(Gibbs sampling steps)wh ∼ N (mh , σ 2 (λg I + XTπ,h Xπ,h )−1 )


1 1
k+1
T
,β +
[w
−
m
]
[w
−
m
]
λ−1
h
h−1
h
h−1
h ∼ Gam α +
2
2 σ2
H

σ

−2

T
1X
∼ Gam 0.01 + , 0.01 +(yh − m̃h )T Σ̃−1h (yh − m̃h )
2
2

!

h=1

where m̃h = Xπ,h m(h−1) , Σ̃h = I + λh Xπ,h XTπ,h , and T is the total number ofT T
data points. i.e. the length of the response vector y := (y1T , . . . , yH
) .

176

Generalized sequential coupling scheme

The marginal likelihood with all parameters, except for the λh hyperparameters,
integrated out can then be given in closed formp(yh |π, λh ) =1
Γ( Th2+ν )ν ν/2·· (ν + ∆2 )− 2 (T +ν)νT/2
Γ( 2 )
π h det(Σ̃h )1/2

where ∆2 = (yh − m̃h )T Σ̃−1h (yh − m̃h ) is the squared Mahalanobis distance, andTh is the length of the vector yh .
Inferring the covariate sets and the data segmentations:
The objective is to sample the covariate sets, π, and the data segmentations
from the posterior distribution. For that we use the Metropolis Hastings sampling moves, described in Grzegorczyk and Husmeier (2012). The covariate sets
π are modified by adding, deleting or substituting single variables from π, wherea priori every set π with up to three covariates is assumed to be equally likely
while the prior probability of sets with more than three covariates is set to zero(’fan-in equal to 3’). The adaptation of those moves to our model is straightforward. For the segmentations we again follow Grzegorczyk and Husmeier (2012)and employ a changepoint process where the distance between neighbouringchangepoints is assumed to follow a negative Binomial distribution with parameters p ∈ [0, 1] and r = 1. In our study we implement 9 different values:
p = 0.02, 0.025, 0.03, 0.05, 0.075, 0.1, 0.15, 0.2, 0.3. The changepoints are inferred
via changepoint birth, death and re-allocation moves, where unlike in Grzegorczyk and Husmeier (2012) and Shafiee and Grzegorczyk (2016), those moves here
become Reversible Jump Markov Chain Monte Carlo (RJMCMC) moves, as thenumber of continuous δh ∈ R+ parameters now varies with the number of segments H. We implement the RJMCMC variants of the three changepoint moves
in the standard way by re-sampling the involved δh parameters from their InverseGamma prior distributions. The determinant of the Jacobian transition matrix
is then equal to 1.
Networks: For a domain with n nodes we apply the regression model to eachnode y i (i = 1, . . . , n) separately. The potential regulator sets of y i are all subsets πi ⊂ {y 1 , . . . , y i−1 , y i+1 , . . . , y n } of the other nodes. As the interactions are
subject to a time delay, the segment-specific design matrices Xih are built fromthe values of the covariates at the preceding time points. Merging the n parent
sets gives an interaction network G := {π1 , . . . , πn }. There is an edge from nodey j to node y i in G if (and only if) y j ∈ πi .
Network reconstruction accuracy: With the MCMC sampling scheme, we
can generate samples of networks G r = {π1r , . . . , πnr }r=1,...,R , and we average
across those networks to obtain for each individual
i (j, i ∈ {1, . . . n} :PR edge j →
r
1
I
(G
),
where Ij→i (G r ) =j 6= i) a marginal posterior probability êj,i = R
j→i
r=1
r
r
1 if j ∈ πi , and Ij→i (G ) = 0 otherwise. When the true edges are known,ei,j ∈ {0, 1}, the network reconstruction accuracies can be quantified in terms ofareas under the precision-recall curves (AUPRC).

Shafiee and Grzegorczyk

177

FIGURE 1. Network reconstruction accuracy for S. cerevisiae (yeast).
AUPRC histograms for the three NH-DBN models. The white bars refer to the
NH-DBN model with sequentially coupled parameters from Grzegorczyk andHusmeier (2012), the grey bars refer to the standard uncoupled NH-DBN model,
and the black bars refer to the generalised model, proposed here. There is a histogram for each of 9 different hyperparameters of the negative Binomial prioron the distance between neighbouring changepoints. The heights of the bars are
the average AUCPR values, averaged across 10 independent MCMC simulation
results and the error bars indicate standard deviations.

3

Data

Cantone et al. (2009) synthetically designed a network of five genes in Saccharomyces cerevisiae (yeast) and measured expression levels of these genes in vivo
with quantitative real-time PCR at 37 time points over 8 hours. In about the
middle of this time period, they changed the environment by switching the carbon source from galactose to glucose. We pre-process those data as described in
Grzegorczyk and Husmeier (2012), including a gene-specific standardization tomean 0. The goal of our empirical study in Section 4 is to infer the true network,
shown in in Figure 1, from the data.

4

Yeast network reconstruction

We show that the novel generalised NH-DBN model with segment-specific coupling parameters reaches a higher network reconstruction accuracy than the standard uncoupled NH-DBN model and the original sequentially coupled NH-DBN
model, proposed by Grzegorczyk and Husmeier (2012). To this end, we crosscompared the network reconstruction accuracies of those models on the yeast
gene expression data set, described in Section 3. For all three models we ran
MCMC simulations with 9 different parameters p of the negative Binomial prior(on the distances between changepoints), so as to compare the models for varying
numbers of changepoints: Higher p values lead to more changepoints and thus
make the information-coupling scheme more important. We ran each MCMC simulation for R = 100, 000 iterations, and our convergence diagnostics suggested
that this simulation length is sufficient for all three models. The results of ourempirical study are summarized in Figure 1. It can be seen that the new model,
overall, shows the best performance. It performs significantly better than thetwo competing models for high p parameters (i.e. when many changepoints areinferred). For low p parameters the performance between the two sequentiallycoupled model does not differ, since the two models do not differ if there is only
up to one single changepoint per gene (H ≤ 2 segments).

178

5

Generalized sequential coupling scheme

Conclusion

Our results suggest that the new NH-DBN model with segment-specific couplingstrength parameters can reach a significantly better network reconstruction accuracy than the two competing NH-DBN models. The new model can be thoughtof as a continuous version of the partially sequentially coupled NH-DBN model,
proposed in Shafiee and Grzegorczyk (2016). The focus of our future research
will be on cross-comparing those two models systematically and on determining
their relative merits and shortcomings.

References
Cantone I. et al. (2009) A Yeast Synthetic Network for In Vivo Assessment ofReverse-Engineering and Modeling Approaches Cell, 137, 172-181.
Grzegorczyk, M., and Husmeier, D. (2012). A non-homogeneous dynamic Bayesian
network with sequentially coupled interaction parameters for applications
in systems and synthetic biology. Statistical Applications in Genetics andMolecular Biology, 11(7), online article.
Shafiee Kamalabad, M. and Grzegorczyk, M. (2016). A non-homogeneous dynamic
Bayesian network model with partially sequentially coupled network parameters. Proceedings of the International Workshop on Statistical Modelling(IWSM2016), vol. 1, 139-144, Rennes, France.

Multiple imputation for high-dimensional
data using sequential penalized regression
Faisal Maqbool Zahid1 , Christian Heumann1
1

Department of Statistics, Ludwig-Maximilians-University Munich, Germany

E-mail for correspondence: faisal-maqbool.zahid@stat.uni-muenchen.de
Abstract: Missing data occurs in almost every field of research. Multiple imputation (MI) is a statistical technique which has become increasingly popular
to generate multiple data sets without missing values which can then be analyzed using the standard statistical methods for complete data.The sequential
regression imputation, building one model for each variable with missing data,
is a practical approach under MAR to get multiply imputed data sets. The existing algorithms and software tools either perform poorly or fail to respond forhigh-dimensional data structures. The question of the best strategy for multiple
imputation for high-dimensional data is still not clearly answered in the literature.
In this paper we are proposing an MI technique based on sequential penalized
regression to address this issue. The proposed technique uses L1 and L2 penalties and thus gives a penalized version of the sequential regression approach withhigh-dimensional data. The performance of the proposed technique is examinedin a simulation study with covariates from different distributions. Mean Squared
Imputation Error (MSIE) is used to study the imputation performance of the
proposed algorithm as compared to the other existing approaches.
Keywords: Missing data; Multiple imputation; Sequential regression imputation; Penalization; Conditional distribution; High-dimensional data.

1

Introduction

Missing data occurs in almost every field of research. Multiple imputation (MI)has become increasingly popular for imputing the missing data in recent years due
to its flexibility. Multiple imputation replaces the missing value with more than
one plausible values drawn from their predictive distributions conditional on theobserved data. As a result, one gets multiple imputed data sets which can thenbe analyzed independently using standard methods for complete data and reThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

180Multiple imputation for high-dimensional data

sults from all imputed data sets are combined using Rubin’s rule (Rubin (1987)).
Harel and Zhou (2007) and Horton and Kleinman (2007) provide a nice reviewof theory and implementation of different MI approaches with a comparison ofdifferent software packages for MI. In case of high-dimensional data the numberof variables p can be close to the or even greater than the sample size n. In suchcase the usual likelihood estimates in the imputation model may not converge oreven do not exist for p > n case. The use of regularization techniques becomes
indispensable in such cases with high-dimensional data. There is still a gap in
the literature that needs to be filled relating to the development of efficient MI
algorithms for high-dimensional data. Although Zhao and Long (2016) proposed
Bayesian lasso regression for multiple imputation with normally distributed data,
they considered only one variable with missing data in their simulation setting.
This approach is also computationally very expensive and even becomes practically infeasible with an increasing number of variables with missing data. In
this paper we are proposing regularized regression for model selection and parameter estimation for the imputation model. Each variable with missing values
is assumed to have a different distributional form and is imputed with its own
imputation model using L1 penalty (for model selection) and if needed also L2
penalty (for fitting the model with selected covariates).

2

Regularization

Regularization methodsPthat are derived from MLE are based on the penalizedλβ ∗) = nββlog-likelihood lp (βi=1 li (β ) − 2 J(β ).
Pβ ) = pj=1 βj 2 .
Ridge Regression: Ridge regression uses the quadratic penalty J(βIt shrinks the parameter estimates towards zero and each other. The shrinkage
parameter λ ≥ 0 controls the amount of shrinkage. Ridge regression produces
biased estimates as β̂ ∗ = (XT WX + λI)−1 XT W D−1 [y − µ ], with cov(β̂ ∗ ) =Σ̂(β̂ ∗ ) = (XT WX + λI)−1 (XT WX) (XT WX + λI)−1 . Here mean µi is relatedΣ̂
to the linear predictor ηi = xTi β as µi = h(ηi ). D and W are n × n diagonal
matrices. The ith diagonal element of D is Di = ∂h(ηi )/∂ηη . The ith element of
W is given as wi = σi2 = var[h(ηi )] for i = 1, . . . , n.
Lasso Regression:P
Lasso (least absolute shrinkage and selection operator) usesβ ) = pj=1 |βj |. Lasso penalty set some coefficients exactly equal
L1 penalty J(βto 0 with large enough value of λ and hence provides a parsimonious model withp >> n. The lasso penalty will perform imputation model selection for highdimensional data in our algorithm.

3

MI using regularization

Let Xn×p be a data matrix consisting of p variables (complete or with missing
values) corresponding to a set of n subjects. The variables in the data matrix can
assume different distributional forms. Let all the variables in the data matrix be
comp
miss
divided into two mutually exclusive groups as X = (Xmiss
n×q , Xn×(p−q) ) where X
compcontains q, out of p, variables with some missing values and Xn×(p−q) covers restof the p − q complete variables without any missing information. Here q may beZahid FM and Heumann C

181

equal to p in which case X = Xmiss . The proposed algorithm can be described asfollows:• Step 1: Initialization
Repeat Step 2 till convergence• Step 2: Convergence, for j = 1, 2, . . . , qa) Fit xmiss∼ X = (Xcomp , Xmissj−j ) using L1 penalty.
β from xmissb) compute β̂∼ Xselect .
jµ = Xselectβ.
c) update missing values of xmiss
with µ̂
mj ×p · β̂jRepeat step 3 for m times with X achieved at convergence.
• Step 3: Imputation, Repeat for j = 1, 2, . . . , q,a) perform steps 2(a,b) to compute Σ̂
β̂).
Σ̂(β̂
β̂)) to impute theΣ̂(β̂β̂, Σ̂
b) compute β̂ new = a random sample from MVN(β̂missing values with select
if xmiss
is normal,j Xmj ×p · β̂ new + noise
missis binary, and))ifxrbinom(h(Xselect
·
β̂
mj ×p
j
new

 rpois(h(Xselect · β̂
missis a count variable.
if xj
mj ×p
new ))c) update Xmiss
n×q .

4

Simulation study

A simulation study was performed with S = 100 data sets using sample sizen = 100 and p = 200. The covariates are drawn from a p-dimensional multivariate
normal distribution with mean 0 and ρ|j−k| (with ρ = 0.8) covariance among xj
and xk . The MAR (Missing at
TABLE 1. Overall MSIE for imputed values with proposed algorithm, mice and VIM. The
MIlasso1 represents the results when step 3 of algorithm is also used for the convergence.
The numbers within brackets represent number of samples (out of 100) for which a particularalgorithm didn’t respond. The bold face values represent the best results for a particularapproach.

miss %
10
20
30
40
60

MIlasso1
810.661642.06
2488.913347.97(14)5025.75

MIlasso2776.01
1568.482374.673203.18(14)4848.28

mice
1370.292415.59
3227.52
3966.40(50)5469.01

VIM
1305.942656.38
4060.235553.46(20)8778.00

Random) mechanism was considered with 10%, 20%, 30%, 40%, and 60% missing
values in each of q = 100 covariates selected at random. One third variables wereMultiple imputation for high-dimensional data
missing = 20%

●
●

●
●

log(MSIE)
7.5 7.7 7.9log(MSIE)
6.6 6.8 7.0 7.2

missing = 10%

●

●

●
●
●

●
●

●

Ila
M

V
IM
IM
V

ss

IM

e

●

V

e
m
ic

o2
ss
Ila
M

Ila

ss

o1

●

●

●

o2

●

ss

●

Ila

●

●

M

●

●

o1

●

M

ic
e

missing = 60%

log(MSIE)
8.4 8.6 8.8 9.0 9.2

missing = 30%

log(MSIE)
7.8 8.0 8.2 8.4

m

Ila
ss
o2

M

M

Ila
ss
o1

IM
V

ic
e
m

ss
o2
M
Ila

M
Ila

ss
o1

7.3

●

m
ic

182

FIGURE 1. Simulation Study: Box plots for log(MSIE)

binary and one third were taken as count variables randomly. The performanceof proposed algorithm is compared with existing software packages mice, VIM andAmelia. The Mean Squaredh Imputation
Error (MSIE)oiis used for comparison andP
P n Pmj ∗1
is given as: MSIE = S1 S m(xij − xij )2 , where x∗ij represents thei=1jith imputed value for observed value xij of variable xj . The results are shownin Table 1. The values within brackets here show the number of samples forwhich a particular algorithm didn’t work. The results of Amelia are not shownin the table because it did not respond in most cases because of its sensitivity to
the correlation structure and high dimension. The results of log(MSIE) are alsopresented in the form of box plots in Figure 1. The MIlasso1 represents the results
when step 3 of algorithm is also used for the convergence. The box plots show thatour proposed algorithm performs better than its competitors. The MSIEs werealso splitted according to the distribution of the missing variables to examine how
well the different algorithms impute under different distributional forms. These
results of such split, not shown here, also showed consistent performance in favorof the proposed algorithm.

References
Rubin, D.B. (1987). Multiple Imputation for Nonresponse in Surveys New York,
NY: Wiley.
Horton, N. J., Kleinman, K. P. (2007) Much ado about nothing: A comparisonof missing data methods and software to fit incomplete data regression
models. The American Statistician, 61, 79 – 90.

Zahid FM and Heumann C

183

Harel, O. and Zhou, X. (2007) Multiple imputation: Review of theory, implementation and software. Statistics in Medicine, 26, 3057 – 3077.
Zhao, Y. and Long, Q. (2016) Multiple imputation in the presence of high-dimensional
data. Statistical Methods in Medical Research , 25 (5), 2021 – 2035.

REML in nonlinear mixed-effects models
with heavy-tailed distributions
José Clelto Barros Gomes1 , Cibele Maria Russo2
1
2

Departamento de Estatı́stica, Universidade Federal do Amazonas, Brazil
Departamento de Matemática Aplicada e Estatı́stica, Instituto de Ciências
Matemáticas e de Computação, Universidade de São Paulo, Brazil

E-mail for correspondence: clelto@ufam.edu.br
Abstract: This work proposes the estimation of nonlinear mixed-effects models for continuous data. We present a restricted maximum likelihood estimator(REML), based on the integration of the fixed effects and a stochastic simulation
procedure. A comparison of REML estimator with maximum likelihood (ML)estimator is presented. Random effects and errors are assumed to follow normal,slash or Student-t distributions. A pharmacokinetic dataset are used to illustrate
the methods.
Keywords: Nonlinear mixed models, restricted maximum likelihood, MCEM
algorithm

1

Introduction

Nonlinear mixed-effects models (NLME) aim to model the nonlinear relationshipbetween the response variable and the covariates through the parameters. The
nonlinear relationship can occur either in fixed or random effects or only in oneof them. Russo (2009), for example, worked with a nonlinear marginal model,
where random effects are added linearly to the model. Mixed-effects models areusually proposed for problems with correlated data, such as longitudinal or repeated measures data, with applications in several areas such as epidemiological,
pharmacokinetic industry, economics and agriculture. Several examples can befound in Pinheiro and Bates (2000). These models usually bring a great challenge in the parameters estimation of the fixed effects and variance components.
Most authors use the ML estimate or the REML estimate. Unlike the ML estimates, the REML is preferred since it produces less biased estimates for variancecomponents. It takes into account the degrees of freedom lost when estimatingThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Gomes and Russo

185

the fixed effects parameters, which does not occur with the ML estimator. This isespecially more evident in small samples, since for large samples the estimates do
not differ much. The development of REML will have a formulation as proposedby Meza et al. (2007) and Monte Carlo Expectation - Maximization (MCEM)
algorithm will be used to estimate the parameters.

2

The model formulation

The scale mixture normal (SMN) distributions have a stochastic representation
Y = µ + κ(U )1/2 Z,
where Z ∼ Nn (0, Σ) is independent of the mixture variable U ∼ H(u; ν) with H
a cumulative distribution function that specify the SMN model, and ν is a scalar
or vector parameter indexing the distribution of U , κ(U ) is the weight function.
The NLME can be written in two stage. The first stage consists of n nonlinear
regression models
yi = g(ϕi , Xi ) + i , i = 1, . . . , n,(1)where yi (ni × 1) denotes the response vector for subject i, Xi is a matrix of
explanatory variables, n is the number of subjects, ni the number of observationsof subject i and i the within-subject errors. In the second stage the fixed andrandom effects are included to the model through the vector ϕi (ni × 1) that canbe written linearly by
ϕi = Ai β + bi ,(2)
where β (p × 1) denotes the vector of fixed effects, bi (q × 1) denotes the vectorof random effects and Ai (q × p) is a known design matrix . From (1) and (2) we
assume


 
 


i
0
Σi 0
∼ SMNni +q
,
;H
,
bi
0
0 D
where D and Σi are positive-definite dispersion matrices. Here we assume D =D(τ ) = diag(τ ) is a diagonal matrix and let τ = (τ1 , . . . , τq )> be the elementsand Σi = σ 2 Ini , with σ 2 > 0 a scalar. For the sake of simplicity we use g(β, bi )to represent g(ϕ, Xi ) = g(Ai β + bi , Xi ). We assume to errors and random effects
the normal, Slashν and Student-tν distributions.

3

Estimation

Assuming a vague prior for the fixed effects β, REML estimation to variancecomponents, τ and σ 2 , can be obtained integrating out the fixed effects, besides
integrate in b, given by
Qn R ∞ R R
f (y; τ , σ 2 ) =φ (yi |g(β, bi ), κ(ui )Σi )i=1 0
Rq Rp ni×φq (bi |0, κ(ui )D) dbi dβ dH(u; ν),
where φn (·|µ, Σ) denotes n-dimensional normal probability density function withparameters µ e Σ. The complete data of the model are given by yc = (y, β, b, u),

186

REML in nonlinear mixed-effects models

and its likelihood function can be written as
Qn∗
Lc (θ ∗ |yc ) =i=1 p(yi , β, bi , ui |θ )Qn

i=1 f (yi |β, bi , ui , σ=∗

2)f (bi |ui , τ )h(ui |ν),2

where θ = (τ , σ ). The complete likelihood function is given by`c (θ ∗ )

=
=

n
P

`(θ ∗ ; yi , β, bi , ui )

i=1
n
P

{log f (yi |β, bi , ui , σ 2 ) + log f (bi |ui , τ ) + log h(ui |ν) + C},

i=1

C assumes a constant value.
To estimate the parameters we use MCEM algorithm, which facilitates the inference and consists in two steps. In the E Step we obtain the expectation in
Q-function
Q(θ ∗ |θ ∗(t) ) = E[`c (θ ∗ )|y; θ ∗(t) ]and in the M Step we want to find θ ∗(t+1) such that maximize Q(θ ∗ |θ ∗(t) ),Q(θ ∗(t+1) |θ ∗(t) ) ≥ Q(θ ∗ |θ ∗(t) ),the two steps are repeated till convergence. For more details of this algorithm see
Wang (2007).
The Step E of the ith individual in the (t + 1)th iteration can be written as
Qi (θ ∗ |θ ∗(t) )=E[`(θ ∗ ; yi , β, bi , ui )|yi , θ ∗(t) ]

=

R∞R
0

Rq

R

Rp

{φni (yi |g(β, bi ), κ(ui )Σi )×φq (bi |0, κ(ui )D)f (β, bi , ui |yi , θ ∗(t) )} dβ dbi dui .
One can use the Gibbs sampler algorithm with Metropolis-Hastings steps togenerate samples of [β, bi , ui |yi , θ ∗(t) ] by sampling of the full conditional distributions from [ui |β, bi , yi , θ ∗(t) ], [bi |β, ui , yi , θ ∗(t) ] and [β|bi , ui , yi , θ ∗(t) ].
(1)

(1)

(m )

(m )

Assuming that {(β (1) , bi , ui ), . . . , (β (mi ) , bi i , ui i )} is a random sample ofsize mi from [β, bi , ui |yi , θ ∗(t) ], the E Step at the (t+1)th iteration can be writtenasnPQ(θ ∗ |θ ∗(t) ) =Qi (θ ∗ |θ ∗(t) )
i=1 "
#
mi
n
P
P
(j)
(j)
∗
(j)
1
=
`(θ ; yi , β , bi , ui )
mi
i=1
j=1


m
(j)
n P
i
P
κ−1 (ui )
(j)
(j)
1
∝
− n2i log(σ 2 ) −kyi − g(βi , bi )k2
mi
2σ 2
i=1 j=1


mi
(j)
n P
P
κ−1 (ui ) (j)>−1 (j)
1
1+−
log(|D|)
−
b
D
b
.
i
i
mi
2
2
i=1 j=1(3)
In M Step, the REML for the variance components, which are solutions from (3)are given bynσb2(t+1) =mii
1 X X 1 h −1 (j)(j)(j)κ (ui )kyi − g(βi , bi )k2 ,N i=1 j=1 miGomes and Russo
and

187

mi
n X

i
X
1 h −1 (j)(j) (j)>b (t+1) = 1
Dκ (ui )diag bi bi.
n i=1 j=1 mi

b = E(β|y, θb∗ ).
The fixed effects in the context of REML can be obtained by β
The standard error estimates follow the expressions developed in Louis (1982),
with the observed information matrix written as



 2
∂`(θ|y, β, b, u)∂ `(θ|y, β, b, u)
−
V
ar
.
−E
∂θ∂θ >
∂θ
b∗
b∗
θ=θ
θ=θ
The expectation and variance are computed with respect to f (β, bi , ui , |yi ,θ ∗(t) ).

4

Application

As an illustration, consider the study of the pharmacokinetics of the drug theophylline used in the treatment of asthma (Pinheiro e Bates, 2000). The drug
was given orally to twelve patients, whose blood samples were collected in 11
times points after administration, within a maximum of 25 hours, and the serum
concentration of the substance (in mg/L) was measured (Figure 1). The firstorder compartment model is usually reparameterizated in terms of the logarithm of the clearance and constant rates. Thus, the model can be written as:
exp(lKe +lKa −lCl )Ct = Dose{exp[− exp(lKe )t] − exp[− exp(lKa )t]}, where Ct deexp(lKa )−exp(lKe )notes the concentration observed at the time t (mg/L), t is the time (in hours),
Dose is the dose, lKe = log(Ke ), lKa = log(Ka ) and lCl = log(Cl ). Ke is theconstant elimination rate (1/hour), Ka is the constant absorption rate (1/hour)which describes how the drug is absorbed from the intestine into the bloodstream,
Cl is the clearance rate (L/kg) representing the volume of blood from which thedrug is eliminated per unit of time.
REML and ML estimates for variance components parameters are presented inTable 1. Note that the estimates are close, and we highlight the REMLs of thevariance components, which are generally larger than the ML estimates. A basedREML fixed effects estimates are also shown as well as the ML estimator.

5

Discussion and remarks

REML and ML estimates to variance components in NLME were obtained supposing to the within-subject errors and random effects are normal, Slash andStudent-t distributed. Here, we set four degrees of freedom for Student-t andSlash. We run the MCEM algorithm for an initial 11,000 iteration, in the endof the process we discarded half of them and thinning it in 100 to obtain theconvergence. The process to REML was slower than ML to converge. In future
research we will investigate REML estimator by Monte Carlo simulation.

Acknowledgments: The first author thanks to Fundação de Amparo à Pesquisa
do Estado do Amazonas - FAPEAM, Brazil. Second author thanks to Fundaçãode Amparo à Pesquisa do Estado de São Paulo - FAPESP, Brazil.

188

REML in nonlinear mixed-effects models

●
●
●

10

●
●
●

●
●

Concentration (mg/L)

●

8

●
●
●
● ●
●●
● ●
●
●● ●
●
●
●
●● ●
● ●

6
●
●
●
●
●
●

4

●
●

●
●
●

●
●

●

●
●
●

●
●

●
●
●

●
●
●
●
●
●

●

●
●

●
●

●
●
●

●
●
●
●

●

●
●
●
●

●
●
●
●
●

●
●

●
●
●
●
●
●
●
●
●
●

●
●
●
●●
●
●

2

●

●
●
●
●

●

●

●
●
●●●
●
●●
●

●

●●

0

●
●
●

0

5

10

15

20

25

Time (h)FIGURE 1. Serum concentration (mg/L) in the times of the drug administration
to twelve patients.

TABLE 1. REML and ML estimates (standard errors) for theophylline serum
concentration data
Par
ML
lKe
lKa
lCl
σ2
τ1
τ2
τ3

NormalREML-2.460(0.050)0.401(0.056)-3.206(0.035)0.501(0.063)0.001(< 0.001)
0.441(0.007)0.029(0.007)-2.458(0.057)0.493(0.058)-3.229(0.039)0.504(0.064)0.001(< 0.001)
0.490(0.007)0.031(0.007)

Student4
ML
REML-2.434(0.040)0.342(0.046)-3.137(0.028)0.296(0.040)0.001(< 0.001)
0.509(0.005)0.023(0.006)-2.460(0.047)0.547(0.049)-3.230(0.032)0.313(0.043)0.001(< 0.001)
0.557(0.006)0.027(0.006)ML

Slash4REML-2.443(0.047)0.414(0.052)-3.195(0.032)0.322(0.043)0.000(< 0.001)
0.368(0.005)0.020(0.005)-2.457(0.053)0.473(0.055)-3.222(0.036)0.327(0.043)0.001(< 0.001)
0.416(0.005)0.023(0.005)

References
Louis, T. A. (1982). Finding the observed information matrix when using the
EM algorithm. Journal of the Royal Statistical Society. Series B (Methodological), 226 – 233.
Meza, C., Jaffrézic, F., and Foulley, J.L. (2007). Reml estimation of variance parameters in nonlinear mixed effects models using the saem algorithm. Biometrical Journal, 49, 876 – 888.

Gomes and Russo

189

Pinheiro, J.C. and Bates, D.M. (2000). Mixed-Effects Models in S and S-Plus.
New York: Springer.
Russo, C.M., Paula, G.A., and Aoki, R. (2009). Influence diagnostics in nonlinear mixed-effects elliptical models. Computational Statistics and Data Analysis, 53, 4143 - 4156.
Wang, J. (2007). EM algorithms for nonlinear mixed effects models. Computational Statistics & Data Analysis, 51, 3244 – 3256

Parameter Inference in the Pulmonary
Circulation of Mice
L. Mihaela Paun1 , M. Umar Qureshi2 , Mitchel Colebank2 ,Mansoor A. Haider2 , Mette S. Olufsen2 , Nicholas A. Hill1 , Dirk
Husmeier1
1School of Mathematics and Statistics, University of Glasgow, Glasgow G12
8SQ, UK
2
Department of Mathematics, NC State University, Raleigh, NC 27695, USA
E-mail for correspondence: l.paun.1@research.gla.ac.uk
Abstract: This study focuses on parameter inference in a pulmonary blood circulation model for mice. It utilises a fluid dynamics network model that takes
selected parameter values and aims to mimic features of the pulmonary haemodynamics under normal physiological and pathological conditions. This is of medical
relevance as it allows monitoring of the progression of pulmonary hypertension.
Constraint nonlinear optimization is successfully used to learn the parametervalues.
Keywords: Pulmonary hypertension; Parameter Inference; Constraint Nonlinear Optimization; Partial Differential Equations; Windkessel model.

1

Introduction

Pulmonary hypertension (PH) is a leading cause of right heart failure. It involves
vascular remodelling including stiffening of the large and small arteries. Clinically, PH is diagnosed by analysing blood pressure (BP) measured invasively inthe large pulmonary arteries. However, key parameters, including arterial stiffness, cannot be measured in vivo. This creates the need for methods to estimateparameters indirectly from the measured haemodynamic blood flow and pressure data. This study uses a 1D fluid dynamical network model that predicts
blood flow and pressure in the large pulmonary arteries (for details see Qureshi
et al., 2017). The model is used to predict blood flow and pressure in healthy
and hypoxic mice, for which data were acquired invasively (Tabima et al., 2012).
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Paun et al.

191

The method discussed here is not specific to mice, but can easily be extended
to analysis of similar data from humans for whom repeated invasive procedures
are required for diagnosis and treatment. The ultimate goal, and hence the motivation behind inferring the parameters, is to minimise the number of invasive
procedures for PH patients, as well as to assist clinicians in devising better treatment strategies. Thus, this study focuses on inference of key parameters pertinent
to disease detection and treatment. We show that using this statistical method,
BP prediction is improved in healthy and hypoxic mice compared to the reference
prediction obtained by using the best parameter guesses in Qureshi et al. (2017).
This leads to enhanced reliability of key parameter estimates obtained using themodel.

2

Mathematical Model

The 1D fluid-structure model is derived from the incompressible axisymmetric
Navier–Stokes equations for a Newtonian fluid, coupled with a constitutive wallmodel predicting stiffness of the blood vessels. In addition, assuming that thevessels are cylindrical and the wavelength is significantly longer than their radii,
conservation of mass and momentum give
∂A
∂q
+
= 0,
∂t
∂x

∂q
∂ q2
A ∂p
2πµr q
+
+
=−
,
∂t
∂x A
ρ ∂x
δ A

(1)

where x (cm) and t (s) are the axial and temporal coordinates, p (mmHg) isblood pressure, q (ml/s) is blood flow rate, A (cm2 ) is the cross-sectional area,δ = 40 µm is the thickness of Stokes-layer in velocity profile, ρ = 1.055 g/ml is
the blood density and µ = 0.0528 cm2 /s is the viscosity. Assuming the arterial
walls are homogeneous, isotropic and thin, the pressure-area relation is given by
!
r
r
A0
4 Eh
A ∂p
2 Eh A0p − p0 =1−=⇒ c2 (p) =
=
,
(2)
3 r0
A
ρ ∂A
3ρ r0
A
where c (cm/s) is the wave speed, A0 is the vessel cross-sectional area and r0 (cm)the vessel radius at the pressure p0 .
The arterial network geometry, including length and radii for the 13 largest vessels
in the pulmonary vasculature is obtained from a micro CT image of a healthy
mouse lung (see Figure 1(a)). To solve the equations, boundary conditions are
specified at the inlet and outlet vessels in the network. The system is driven by
imposing an invasively measured flow profile at the inlet of the main pulmonary
artery (MPA), while conservation of blood flow and continuity of pressure are
enforced across the bifurcations. For the outflow boundary conditions, 3-element
Windkessel models (two resistors R1 , R2 and a capacitor C) are attached tothe outlet of each of the seven terminal arteries in the network. The outflow
boundary conditions account for the lumped effects of pulmonary haemodynamics
beyond the truncated network of large arteries. The Windkessel model relates flow
and pressure in the time domain over a cardiac cycle of length T via the input
impedance Z(ω) by:Z
R2
1 T
Z(ω) = R1 +=⇒ q(L, t) =p(L, t − τ )Z(τ )dτ,(3)
1 + iωCR2
T 0

192

Parameter Inference in the Pulmonary Circulation

where R1 and R2 are the proximal and distal vascular resistances beyond eachtruncated artery, R1 + R2 is the total resistance and C is the total compliance ofthe vascular bed. The model takes a number of parameters as input and predicts
the flow and pressure at different locations along the large pulmonary arteries.
One of these parameters is the arterial stiffness, which significantly increases
during the PH and which can be expressed as follows: Eh= f1 exp(f2 r0 ) + f3 ,r0
where E and h are the Young’s modulus and thickness of the arterial wall, andf1 (g cm/s2 ), f2 (cm−1 ), f3 (g cm/s2 ) are the material parameters.

3

Methodology

Let the statistical model be defined by yi = f (xi ; θ) + i , where yi ∈ y are thenoisy measured flow and pressure, f (.) describes the system behaviour that comesfrom numerically solving the fluids model, θ are the parameters that we wish toinfer from the observed flow and pressure and they are bounded, xi ∈ x denoteother input variables and  are the errors, which we assume are i.i.d and follow a
Gaussian distribution. The objective function to be minimised using Constraint
Nonlinear Optimization is the Residual Sum of Squares,X
RSS =(yi − f (xi ; θ))2 .
(4)i

Under the assumption of i.i.d. Gaussian errors, the log likelihood of the datatakes the form
√
RSS
.
(5)
log(L) = −n log 2πσ 2 −2σ 2A Sequential Quadratic Programming (SQP) gradient-based method is used tominimise the RSS (Boggs et al., 2000).

4Simulations

Simulations are set up to mimic experimental waveforms, which are recorded in
the MPA in healthy and hypoxic mice (Tabima et al., 2012). The parameter set
to be inferred initially includes θ = ( Eh
, r1 , r2 , c1 ), where r1 , r2 are resistancesr0(r1 = (1 − 0.5r1 )R01 , r2 = (1 − 0.5r2 )R02 , R01 and R02 are nominal resistances)and c1 capacitance (C = (1−0.5c1 )C0 , C0 is nominal capacitance) used to predictis the elastance used to predict stiffness
parameters assigned at the outlet and Eh
r0
in all vessels. The parameter set is subsequently extended to include a taperingfactor, ζ, for the large vessels in the network, as there was evidence of vessel radii
decreasing along their length, but this was not quantified during the segmentation
process. Since the parameters are on different scales ( Eh
∈ [2.424 × 105 , 6.85 ×
r0
610 ], r1 , r2 , c1 ∈ [−2, 2], and ζ ∈ [0, 1.2]), to avoid having an ill-conditioned
problem induced by a high condition number in the Hessian matrix, we rescale
the parameters to have the same order of magnitude (Yang et al., 2010). Certain
parameter configurations violate the model assumptions; these are marked by
setting RSS to a high value (1010 ). The RSS is calculated for pressure and we
aim to find the set of parameters that minimise the RSS. The initial parametervalues are uniformly drawn from a Sobol sequence to ensure a good coverage ofPaun et al.

193

the multidimensional parameter space (Bratley et al., 1988). The algorithm isiterated until it satisfies the convergence criterion, i.e. |θi − θi+1 | < 10−11 . One
forward simulation of the mathematical model takes 13 seconds to complete.
The optimization problem required 3 hours on average to reach convergence ofparameter estimates.

5

Results and Discussion

Regardless of the initial value, the algorithm converged for both the healthy andthe hypoxic mouse studied. Figure 1 shows our optimised pressure waveform, plotted alongside the measured and the reference pressure for the 4D optimization
problem. Panel (d) shows the pressure fit for the hypoxic mouse. The optimized fit
predicts data better than nominal parameter values, supported by a significantly
smaller RSS than the one between the reference and the measured pressure (panel(b)). For the healthy mouse (panel (c)), the simulated pressure closely follows themeasured pressure except near the peak, where an offset is registered. Nevertheless, in this case too, a clear improvement is achieved over the reference pressure.
We hypothesise that this peak shift is a consequence of (i) the model specifying the elastic behaviour of the blood vessels and/or the boundary conditions,(ii) uncertainty of the geometry measurements which are not specific to a given
mouse, (iii) a combination of (i) and (ii). The overall model prediction appears
better for the hypoxic than the healthy mouse. When the tapering parameter
(a)

300

1

3

4
7

5

6

13

10

Optimised
Reference

11

135

100

8
9

328

200

RSS

2

(b)

47

12

23

(c)

20

Measured
Reference
Optimised

15

Healthy1000.05Time (s)0.1

Healthy

35

Pressure (mmHg)

Pressure (mmHg)

0

Hypoxic(d)Measured
Reference
Optimised

30
25

Hypoxic

20
15
10

0

0.05

0.1

Time (s)FIGURE 1. (a) The arterial network for the fluid dynamical model, (b) Comparison of RSS between reference and optimised pressure simulations, (c) & (d)Comparison of simulated pressure using reference and optimised parameters values for the healthy and hypoxic mice.
is included in the analysis, a reduction of 31% is registered in the RSS for thecontrol mouse. These results are summarised in Figure 2. Panel (b) illustrates the194

Parameter Inference in the Pulmonary Circulation

(a)
135

Optimised 4D
Optimised 5D

RSS

150

100

93

50

0

Healthy

Pressure (mmHg)

optimised pressure waveform for the 5D problem plotted alongside that for the
4D case and the measured pressure is superimposed. While the deviation fromthe measured data has decreased, the offset in the peak value is still present. In
order to select between the two competing models, the 5D model, which includes
a tapering parameter, and the 4D one, model selection using the Akaike Information Criterion - AIC (Akaike, 1971) and Bayesian Information Criterion - BIC(Schwarz, 1978) can be performed. However, the estimate of the error variance,σ 2 needed to calculate the log likelihood of the data (5) is not available1 . Hence,
we take the inverse approach and calculate what error variance would make us
favour the 5D model over the 4D model. Calculations indicate that if σ 2 < 21, i.e.
signal-to-noise ratio, SN R > 2.40, then the 5D model is preferred according to
the AIC; if σ 2 < 6.06, i.e. SN R > 8.30, then the 5D model is favoured according
to the BIC.
(b)

20

15

Measured
Optimised 4D
Optimised 5D

Healthy

10
0

0.05

0.1

Time (s)FIGURE 2. (a) Comparison of RSS between the optimised pressure simulations
for the healthy mouse, (b) Comparison of simulated pressure using 4D and 5D
optimised parameters values for the healthy mouse.
In conclusion, parameters have successfully been inferred for this fluid-structure
model. Future work will include improvements in the model to capture a more
realistic elastic behaviour of the vessel wall. This may address the alignment issue
observed near the peak pressure for the healthy mouse by better controlling thesteepness of the pressure. Additionally, to account for the uncertainty associatedwith the geometry measurements, we will estimate them as part of the inference
procedure. Finally, we also aim to apply the statistical methods presented here
to a population of mice, as well as to data from human patients.

Acknowledgments: This work is part of the research programme of the Centre for multiscale soft tissue mechanics with application to heart & cancer (SofTMech), funded by the Engineering and Physical Sciences Research Council (EPSRC) of the UK, grant reference number EP/N014642/1. Olufsen, Haider andQureshi were supported by National Science Foundation (NSF-DMS # 1615820).
Data were made available by N. Chesler, Department of Biomedical Engineering,
University of Wisconsin, Madison.
1
In principle, we could infer the variance, but due to the slight model mismatch apparent from Figures 1 and 2, the results would be misleading.

Paun et al.

195

References
Akaike, H. (1971), Information theory and an extension of the maximum likelihood principle. In: Proc. 2nd International Symposium on Information
Theory, Tsahkadsor, Armenia, USSR. 267 – 281.
Boggs, P.T. and Tolle, J.W. (2000), Sequential quadratic programming for largescale nonlinear optimization. Journal of Computational and Applied Mathematics, 124(12), 123 – 137.
Bratley, B. (1988), Algorithm 659: Implementing Sobol’s Quasirandom Sequence
Generator. ACM Trans Math Softw, 14(1), 88 – 100.
Qureshi, M.U. et al. (2017), Simulating effects of hypoxia on pulmonary haemodynamics in mice. In: Proc CMBE. 271 – 274.
Schwarz, G. (1978), Estimating the Dimension of a Model. Ann. Statist., 2(6),461 – 464.
Tabima, D.M. et al. (2012), Persistent vascular collagen accumulation alters hemodynamic recovery from chronic hypoxia. J. Biomech, 45(5), 799 – 804.
Yang, K.W. and Lee, T.Y. (2010), Heuristic scaling method for efficient parameter estimation. Chemical Engineering Research and Design, 88(56), 520 –528.

The Pairwise Expectation Maximization
Algorithm for Fitting Parameter-Driven
Models
Xanthi Pedeli1 , Cristiano Varin1
1

Ca’ Foscari University of Venice, Italy

E-mail for correspondence: xanthi.pedeli@unive.it
Abstract: The likelihood function is often intractable in discrete-valued timeseries models. Several simulation-based methods to approximate the likelihood
have been proposed in the literature. Although simulation methods may makethe estimation problem feasible, they are often computationally intensive and thequality of the numerical approximations may be difficult to assess. In this paper
we employ a pairwise likelihood approach, that is a composite likelihood basedon pairs of observations, for the estimation of parameter-driven models for timeseries of counts. Maximization of the pairwise likelihood is carried out with apairwise version of the expectation maximization algorithm.
Keywords: Expectation maximization algorithm; Pairwise likelihood; Parameterdriven models; Surveillance; Time series of counts.

1

Pairwise fitting of parameter-driven models

Parameter-driven models (Cox, 1981) are often considered to extend generalized linear models for time series analysis of counts. The observations Yt are assumed to be independent random variables conditionally on a latent process Ut ,t = 1, . . . , n. The conditional mean of the observations is E(Yt |Ut ) = g(xTt β + Ut ),where g(·) is a link function, xt is a vector of covariates with associated regression parameters β and Ut is an autoregressive and moving average process. For
these models, maximum likelihood estimation is usually intractable because thelikelihood is the n-dimensional integral
Z YnL(θ; y) =f (yt |ut ; θ)f (u1 , . . . , un ; θ)du1 . . . dun(1)
Rn t=1This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Pedeli and Varin

197

where y = (y1 , . . . , yn )T and the model parameter θ includes the regression parameters β and the parameters of the latent process. Several simulation-based
strategies for approximate likelihood inference have been suggested in the literature, see, for example, Davis and Dunsmuir (2016) and the references therein.
In this paper, we focus on an alternative non-simulation approach based on theidea of pairwise likelihood (Varin et al., 2011) that allows for the replacement of
the n-fold integration in (1) with a set of two-dimensional integrals. We consider
estimators obtained by maximizing the pairwise likelihood of order d that involves
only pairs of observations that are far apart no more than d lags,
n
d Z
Y
Y
L(d)
f (yt−l |ut−l ; θ)f (yt |ut ; θ)f (ut−l , ut ; θ)dut−l dut .
p (θ; y) =t=l+1 l=1

R2

It can be shown that the distance d that maximize the efficiency of the estimators derived from the pairwise likelihood depend on the order of the latent
autoregressive and moving average process.
Instead of a direct maximization of the pairwise likelihood, we develop a pairwise
version of the expectation maximization algorithm. The expectation step of thealgorithm requires to compute the conditional expected value of a set of bivariate
complete log-likelihoods,
Q(θ|θ̂(i) ) =

n
d
X
X

n
o
E log f (yt−l , yt , Ut−l , Ut ; θ) yt−l , yt ; θ̂(i) .

t=l+1 l=1

The above function can be efficiently approximated with double Gauss-Hermite
quadrature. The maximization step is replaced with a conditional maximizationof each model parameter with the other parameters fixed at their previous values.
The virtue of the conditional maximization is that simple closed-form expressions
for the maxima are available, thus making the implementation of the algorithm
particularly convenient.
Simulation studies not shown in this short paper indicate that the pairwise fitting
method compare well in terms of statistical and numerical efficiency with popular
approaches for inference in parameter-driven models implemented in R (R Core
Team, 2016).

2

Application

We illustrate our fitting method using the meningo.age data set included in the
R package surveillance (Höhle et al., 2016). The data concern monthly countsof meningococcal infections in France during the period 1985–1997 subdivided
into four age groups, see Figure 1. The total number of monthly observations is
n = 156.
The standard analysis of this type of surveillance data assumes that infection
counts Yt for each age group are marginally distributed as independent Poisson random variables with mean exp(ηt ) specified in way to account for annual
seasonality and linear trend,




t
t
t
ηt = β0 + β1 cos 2π+ β2 sin 2π
+ β3
.
(2)
12
12
156

198

Pairwise Expectation Maximization
age<1

age1-5age5-20age>2030

20

Number of infections

10

0

30

20

10

0
1985

1990

1995

1985

1990

1995

Date

FIGURE 1. Monthly counts of meningococcal infections in France for the period
1985–1997. Source: R package surveillance (Höhle et al., 2016).
In order to handle for the presence of serial correlation, we use the pairwise
expectation maximization algorithm for fitting the parameter-driven model thatassumes that Yt follows a Poisson distribution with mean exp(ηt + Ut ), where thelinear preditor ηt is specified as in (2) and Ut is an autoregressive model of orderone.
The parameter estimates and the corresponding standard errors obtained withthe standard analysis and the parameter-driven model are displayed in Table 1.
Results of the standard analysis indicate significant seasonality and decreasing
linear trend for all age groups. The fitted parameter-driven models confirm theconclusions of the standard analysis for all age groups apart from infants (age< 1), where there is significant autocorrelation and no evidence of a decreasing
trend of infections. The application illustrates how ignoring serial correlation insurveillance data can lead to misleading inference about sensible quantities suchas trends.

Acknowledgments: This project has received funding from the European
Union’s Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreement no. 699980.
References
Cox, D.R. (1981) Statistical analysis of time series: some recent developments.
Scandinavian Journal of Statistics 8, 93–115.

Pedeli and Varin

199

TABLE 1. Parameter estimates (standard errors) for models fitted to the montlhy
counts of meningococcal infections in France for the period 1985–1997.

Standard Poisson regression
age < 1
β0
β1
β2
β3

β0
β1
β2
β3
φ
σ

age 1 − 5

age 5 − 20

age > 20

1.64 (0.07)
0.17 (0.06)
0.37 (0.06)−0.43 (0.13)

2.37 (0.05)
0.16 (0.04)
0.31 (0.04)−0.75 (0.10)

2.71 (0.05)
0.12 (0.04)
0.26 (0.04)−0.71 (0.09)

1.95 (0.06)
0.21 (0.05)
0.43 (0.05)−0.31 (0.11)

age < 1age 1 − 5

age 5 − 20

age > 20

1.60 (0.13)
0.16 (0.08)
0.37 (0.07)−0.42 (0.25)
0.73 (0.35)
0.19 (0.07)

Parameter-driven

2.33 (0.07)
0.15 (0.05)
0.31 (0.06)−0.72 (0.12)
0.32 (0.37)
0.21 (0.06)

2.67 (0.07)
0.11 (0.05)
0.25 (0.05)−0.69 (0.11)
0.21 (0.24)
0.23 (0.04)

1.93 (0.07)
0.21 (0.05)
0.43 (0.05)−0.28 (0.13)
0.48 (0.43)
0.13 (0.06)

Davis, R.A. and Dunsmuir, W.T.M. (2016). State space models for count timeseries. In: Handbook of Discrete-Valued Time Series, Chapman & Hall/
CRC.
Höhle, M., Meyer, S. and Paul, M. (2016). surveillance: temporal and spatiotemporal modeling and monitoring of epidemic phenomena. R package version 1.13.0.
R Core Team (2016). R: A language and environment for statistical computing.
R Foundation for Statistical Computing, Vienna, Austria.
Varin, C., Reid, N., and Firth, D. (2011). An overview of composite likelihood
methods. Statistica Sinica 21, 5–41.

Estimating abrupt change models withcovariate-dependent changepoint
Salvatore Fasola1 , Stefania La Grutta1 , Vito M.R. Muggeo2
1
2

Institute of Biomedicine and Molecular Immunology - CNR, Palermo, Italy
Department of Economics, Business and Statistical Sciences, Palermo, Italy

E-mail for correspondence: salvatore.fasola@ibim.cnr.it
Abstract: We discuss the problem of estimating a changepoint expressed as alinear function of covariates in piecewise constant regression models. Parameters
involved in the change-point function are unbounded, therefore conventional gridsearch techniques result to be unfeasible for estimation. We propose an iterative algorithm which is able to work in this framework, and illustrate it by anapplication to a real dataset.
Keywords: Linear changepoint; Model linearization; Iterative algorithm.

1

Introduction

Changepoint detection is a commonly addressed problem, especially in Econometrics, Biology, Genetic and Bioinformatics. A piecewise constant regression
model with a single changepoint can be expressed asµi = β0 + β1 I(xi > ψ),(1)where the mean value µ of the response variable Y changes abruptly at the value
ψ of the covariate x. Most of the algorithms and applications developed are concerned with estimation of the number and locations of several changepoints. The
most widespread approach is grid search based on dynamic programming (Jackson et al., 2005), which can yield the global solution with a O(n) computational
cost for any number of breaks (Maidstone et al., 2016).
In this paper we consider the problem of possible heterogeneity in the changepoint, which means to replace in (1) ψ with ψi . Some papers address the problem
by introducing random effects in the model (Muggeo et al. 2014, Jackson andSharples 2004). However, no papers address the problem of modelling possible
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Fasola et al.

201

heterogeneity due to covariates, namely when ψi = ψ(vv i ; θ). Grid search algorithms are difficult to use, since the parameter θ has unbounded support. In thispaper we consider an abrupt change model with covariate-dependent changepoint,ψ(vv i ; θ) = θ0 + θ1 vi and introduce a heuristic, iterative algorithm for estimation.

2

Methods

We consider the following model where the mean level of the response Y exhibits
a jump at given values ψ of the explanatory variable x. However, ψ is supposed
to depend linearly on a single additional covariate v:µi = β0 + β1 I(xi > θ0 + θ1 vi ).

(2)

If the point (v, x) lies below the straight line ψ = θ0 + θ1 v, the mean level of Y is
β0 , while it shifts instantaneously to β0 + β1 above. Estimating (2) by searchingfor θ̂1 and θ̂2 on a grid is obviously very difficult, since these parameters do nothave a finite support. We therefore propose to use the identity
I(xi > θ0 + θ1 vi ) =1 xi − θ0 − θ1 vi
1+
2 |xi − θ0 − θ1 vi |2

(3)

for xi 6= θ0 + θ1 vi , which substituted in (2) gives
1 xi − θ0 − θ1 vi
1µi = β0 + β1+
2 |xi − θ0 − θ1 vi |
2


1
xi
1
+
+
= β0 + β1
2 |xi − θ0 − θ1 vi |
2




1
1
vi
1
+ (−β1 θ0 )+ (−β1 θ1 )2 |xi − θ0 − θ1 vi |2 |xi − θ0 − θ1 vi |=β0 + β1 zi + γ0 wi0 + γ1 wi1 ,

(4)

where
γ0 = −β1 θ0and

γ1 = −β1 θ1 .

(5)

Note the ‘working’ covariates

1
xi
1
+
,
2 |xi − θ̃0 − θ̃1 vi |
2


1
1
wi0 =
,
2 |xi − θ̃0 − θ̃1 vi |


zi =

and


wi1 =

1
vi
2 |xi − θ̃0 − θ̃1 vi |enter model (4) linearly, with θ̃0 and θ̃1 meaning approximate values.
Formulas above suggest the following simple iterative algorithm:
1. choose starting values θ̃0 and θ̃1 ;

(6)

202

Estimating a covariate dependent changepoint

2. compute the working covariates (6);3. estimate the working linear model (4) and extract β̂1 , γ̂0 and γ̂1 ;4. update the parameter values via θ̂0 = −γ̂0 /β̂1andθ̂1 = −γ̂1 /β̂1 ;5. set θ̃0 = θ̂0 and θ̃1 = θ̂1 and iterate 2 to 4 until convergence.
A possible strategy for initializing the algorithm is to set starting values θ̃1 =0, and θ̃0 = x̄, where x̄ is the sample mean of the xi s. Unfortunately, simple
application of the aforementioned algorithm will not work properly. In fact, thelikelihood typically exhibits many local optima. Besides, denominators of theworking covariates (6) go to zero when xi ≈ θ̃0 + θ̃1 vi , namely when points(vi , xi ) are close to the approximate straight line. In the next section we illustrate
a simple point scaling that improves feasibility of the algorithm.

3

Scaling v and x values

The idea is moving points (vi , xi ) away from the straight line x = θ̃0 + θ̃1 v. Note
that, when θ̃1 = 0, we can simply scale the covariate x, since the approximatechange-point θ̃0 is unique. Therefore, we operate a rotation of the points (vi , xi )to reduce to this suitable situation. At this aim, we consider the angle definedby the approximate straight line, namely ρ̃ = arctan(θ̃1 ), and apply the following
transformation:
 ∗ 


vi
vi
= Λ̃
,
(7)
∗
xi
xi
where Λ̃ is the rotation matrix 
cos(ρ̃)λ̃11 λ̃12=Λ̃ =− sin(ρ̃)λ̃21 λ̃22

sin(ρ̃)cos(ρ̃)
.

(8)

Rewriting the rotated covariate x∗ according to (7) gives
x∗

=

θ̃0∗

λ̃21 v + λ̃22 x

=

θ̃0∗

λ̃22 x=θ̃0∗ − λ̃21 v

andso that

x=

λ̃21
θ̃0∗
−
v,
λ̃22
λ̃22

θ̃0∗
λ̃22

and θ̃1 = −

λ̃21
,
λ̃22

(9)

θ̃0∗ = λ̃22 θ̃0 and θ̃1∗ = 0.

(10)

θ̃0 =
and, conversely,

Note that the slope of the rotated line θ̃1∗ is 0 by construction, and this makes
easier to operate the scaling. We use a scaling factor c ∈ (0, 1) and compute alower, θ̃0∗− , and a upper, θ̃0∗+ , ‘threshold’ value:θ̃0∗− = θ̃0∗ − c(θ̃0∗ − x∗(1) ),θ̃0∗+ = θ̃0∗ + c(x∗(n) − θ̃0∗ ).

Fasola et al.

203

We therefore consider the standard linear trasformation
x0i = x∗(1) + (x∗i − x∗(1) )(1 − c)(11)x0i = θ̃0∗+ + (x∗i − θ̃0∗ )(1 − c)(12)for x∗i ∈ [x∗(1) , θ̃0∗ ], andfor x∗i ∈ (θ̃0∗ , x∗(n) ]. Figures 1 and 2 show the scalement mechanism through an
example on a set of n = 100 randomly generated covariate pairs, assuming v ∼N (2, 32 ) and x ∼ N (5, 62 ). Supposing the starting straight line given by xi =8 + 2vi (Figure 1, left panel), the rotated straight line, on the rotated points,
is x∗ = 3.58 (Figure 1, right panel). The left panel in Figure 2 represents therotated points without scaling x∗ , namely when c = 0, while the right panel
shows the effect of a scaling factor c = 0.1; note the scaling induces a point-freeinterval (dashed lines) in the neighbourhood of θ̃0∗ (solid line). Finally, we use x0 ,v 0 = v ∗ and the rotated straight line to compute auxiliary covariates (6) and fit
the working linear model (4).
We stress that, despite the multiple transformations induced to the covariates,
estimates of the mean levels β0 and β1 are substantially unaffected, while thestraight line parameter estimates, of course, are. In particular, the rotation only
induces a reparametrization according to (9) and (10), while the scaling should
also favour to skip some spurious optima. Note that c should be reduced (halved
for example) throughout iterations any time the likelihood decreases.

4

Real data example

●

−5

0

5

v

5
0
−5

●

●

●

x*

5

●
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●●
●
● ●

~
θ0 = 8~
θ1 = 2

−5

0

x

10 15

To illustrate, we apply the proposed algorithm to the airquality dataset shippedwith the R environment. The dataset consists of 154 daily observations concerningsome air quality values in New York from May 1, 1973 to September 30, 1973.
Tropospheric ozone is an atmospheric pollutant, and its concentration represents

●

●
● ●●
●● ●●●●● ●●● ●● ●●● ●
●
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●
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●
● ●
● ●
●●
●
●
~
●●
θ 0* = 3.58
● ●●~
θ 1* = 0
●●●
●

−5

0

5

10

15

v*

FIGURE 1. Example of starting straight line in a toy dataset (left panel) androtated points and straight line (right panel).

Estimating a covariate dependent changepoint

●

● ●●
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0

5

10

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5● ● ●● ●
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0

c=0

−5

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−5

0
−5

x*

5

●
●

x'

204

c = 0.1

15

−5

0

v*

5

10

15

v'

FIGURE 2. Only rotated (left panel) vs rotated and scaled data (right panel).

80

●● ●
●●
●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●● µ = 26.18

●●●

●●

●
● ●

70

●
● ●
● ● ●
● ●●●●●●●●●●● ● ● ●

●●

●●
●
● ●●

60

temperature90●^ = 84●µ●●● ●●●●●● ●● ●
●
●●●
●●
●
●●
^
●
● ●●

●

5

10

●

●

●●

15

●
●

20

wind speed

FIGURE 3. Airquality dataset: wind-temperature scatterplot with point size
proportional to the ozone level and estimated linear change-point (dashed line).

a common variable of interest in environmental science. The ozone levels may
depend on many factors, including atmospheric agents, such as temperature andwind. Therefore we model the mean levels of ozone (Y , parts per billion) as afunction of temperature (x, degrees Fahrenheit) and wind (v, average speed inmiles per hour). Figure 3 displays the wind-temperature scatterplot with point
sizes proportional to the ozone levels for the n = 116 complete records. From avisual inspection the concentration of the pollutant seems to increase abruptly
in the top-left region of the plot; in particular, a threshold temperature value
appears to be approximately 80 degrees. Estimating model (2), assuming the Yi s
to be independent and Gaussian, could yield additional information.
To initialize the algorithm we choose ψ̃ = 80 (θ̃1 = 0) as starting guess. The fitted
regression equation is µ̂i = 26.18 + 57.82 I(xi > 72.83 + 1.24vi ). Figure 3 displays
the estimated straight line on the scatterplot. The mean ozone level appears toincrease abruptly from about 26 to 84 p.p.b as temperature goes beyond someFasola et al.

205‘critical’ value. In the absence of wind, the ‘critical’ value is about 73 degrees.
Wind seems to have a ‘positive’ effect in limiting the ozone levels: in fact, as thewind increases, temperature has to increase further to cause a ‘jump’ in the ozone
levels. The BIC provides good evidence supporting the choice of this regression
model.

References
Jackson, B., Scargle, J.D., Barnes, D., Arabhi, S., et al. (2005). Hierarchical generalized linear models. An algorithm for optimal partitioning of data on aninterval, 12, 105 – 108.
Maidstone, R., Hocking, T., Rigaill, G., and Fearnhead, P. (2016). On optimal multiple changepoint algorithms for large data. Statistics and Computing, 27,
1 – 15.
Muggeo, V.M.R., Atkins, D.C., Gallop, Robert J., and Dimidjian, S. (2014). Segmented mixed models with random changepoints: a maximum likelihood
approach with application to treatment for depression study. Statistical
Modelling, 14, 293 – 313.
Jackson, C.H., and Sharples, L.D. (2004). Models for longitudinal data with censored changepoints. Journal of the Royal Statistical Society: Series C (Applied Statistics), 53, 149 – 162.

Identifying influential observations incomplex Bayesian mediation models
Šárka Rusá1 , Arnošt Komárek1 , Emmanuel Lesaffre2 , Luk
Bruyneel3
1

Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic2
Leuven Biostatistics and Statistical Bioinformatics Centre, University of Leuven, Belgium
3
Leuven Institute for Healthcare Policy, University of Leuven, Belgium
E-mail for correspondence: rusa@karlin.mff.cuni.cz
Abstract: Although increasingly complicated (moderated) mediation models arebeing employed in practice, most of existing mediation literature has not dealt
with model diagnostics. We propose a Bayesian approach to the detection of influential observations (or sets of observations). Importance sampling with weights
which take advantage of the dependence structure in mediation models is utilized in order to estimate the case-deleted posterior means of the parameters.
The method is applied to the ordinal measurements of patients’ willingness torecommend hospitals collected on patients in a large European study to answer
the research question whether the outcome depends on recorded system-levelfeatures in the organization of nursing care, and whether the related effect is mediated by two measurements of nursing care left undone and possibly moderated
by nurse education.
Keywords: Bayesian mediation models; Influential observations, Importancesampling.

1

Introduction

Recent advances in the literature on mediation models extended the simple threevariable (outcome, mediator, regressor) mediation model to complex models withmultiple mediators, non-continuous outcome, multi-level setting and other generalizations. The Bayesian approach, which we will also adopt in this paper, isoften utilized in such complicated models because the generalizations are more
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Rusá et al.

207

straightforward. However, no general model diagnostic tool for identification of
influential observations is readily available for these models and we deem thedevelopment of such instruments as necessity. In this paper, we propose the useof importance sampling with weights which take advantage of the dependence
structure in mediation models in order to estimate the case-deleted posteriormeans of the model parameters and other characteristics hence allowing for identification of influential observations. Suitable importance weights can be used forleave-one-out diagnostics, as well as leaving out all observations from a specifichospital. The methods shown here on one specific mediation model can be usedin the same fashion in any mediation model with or without latent variables.

2

The RN4CAST Study and the Research Question

The Registered Nurse Forecasting (RN4CAST) study (Sermeus et al., 2011) isa cross-sectional survey of patients and nurses conducted in 11 European countries, in which the patients and nurses are further clustered in hospitals and nursing units. The data collected in 2009 – 2010 during this FP7-funded project contain information on various hospital characteristics such as nurse staffing, nurse
education, number of beds, etc. On top of that, the nurses were interviewed on
various aspects of the nursing care such as their well-being, satisfaction with theirjob, their willingness to recommend the hospital and overtime work. The patientsprovided information on their satisfaction with hospital care and hospital rating.
In our previous work (Rusá et al., 2017), we presented a multi-level moderated mediation model with ordinal outcome in order to evaluate on how thepatients’ willingness to recommend the hospital (patient’s satisfaction) relates
to the system-level features in the organization or nursing care and whether the
association is mediated by two measurements of nursing care left undone andpossibly moderated by nurse education. In this paper, we aim to explore if there
exist hospitals which are influential with respect to such a model. In particular,
we want to investigate the change in the indirect effects.

3

Multi-Level Moderation Mediation Model with
Ordinal Outcome

We denote the patient outcome (patients’ willingness to recommend the hospital)
by Yijk which represents the value of the outcome measured on an ordinal scale
0, . . . , L on the i-th patient from hospital j and country k. It is assumed that?
there exist unknown thresholds α = (α1 , . . . , αL )T , and a latent variable Yijk
such that
Yijk = l,iff?
αl < Yijk≤ αl+1 ,l = 0, . . . , L,−∞ = α0 < α1 < . . . < αL < αL+1 = ∞.
The purpose of the model studied in Rusá et al. (2017) was to ascertain whether
there was a significant effect of nurse staffing, the quality of nurse work environment and other hospital characteristics (Xjk ) on the patient outcome (Yijk )and if it was mediated by two mediators of care left undone (M1,jk and M2,jk ).

208

Identifying influential observations in complex Bayesian mediation models

Moreover, we assessed the moderating effect of nurse education (these terms areincluded in the covariate vector Xjk ).
In order to deal with the multi-level structure of the RN4CAST data, we included
the hospital-level random effects (Ujk ). Since countries were not sampled but
chosen and there are just a handful of countries, we have considered their effects
as fixed. Consequently, the considered model equation for the latent outcome
variables had the following form:
?
>
>
Yijk
= βk + Xjk
βX + Mjk
βM + Ujk + εijk ,>Mt,jk = γt,k + Xjk
γt,X + ξt,jk ,
i.i.d.
2
Ujk ∼ N 0, σhospital,ξjk = ξ1,jk , ξ2,jk

t = 1, 2,

(1)
(2)


i.i.d.
εijk ∼ N 0, σε2 ,> i.i.d.
∼ NT 0, Σξ .

We assume that Ujk , εijk , ξ1,jk , ξ2,jk are mutually independent. For βk , βX , βM ,γ1,k , γ2,k , γ1,X , γ2,X , we used vague normal priors. As for Σξ , we will parametrize2, the variance of ξ2,jk denoted byit using the variance of ξ1,jk denoted by σξ12σξ2 and their correlation ρ. For the standard deviations σhospital , σε σξ1 , σξ2 ,we specify improper uniform priors on (0, ∞). A noninformative prior was considered for ρ: ρ ∼ U (−1, 1) (a uniform distribution on (−1, 1)). The following
noninformative prior was used for the thresholds (Song and Lee, 2012, p.117):p(α2 , . . . , αL−1 ) ∝ C, for α1 < α2 < . . . < αL−1 < αL . The samples fromthe posterior distributions were obtained using the Stan software. The inference is based on MCMC methods, mainly the Hamiltonian Monte Carlo andthe Metropolis algorithm.

4

The Detection of Influential Observations

Besides the estimation of the parameters in our model, it may be of interest toassess the influence of some (sets of) observations on the parameter estimates. In
case of our application, it makes sense to evaluate the change in the estimates ofindirect effects based on the data without a specific hospital. To this end, importance sampling has been utilized in the literature for different sorts of models,see e.g. Bradlow and Zaslavsky (1997). We have generalized this approach to ourmoderated mediation model with ordinal outcome as sketched below.
Let θ1 and θ2 , respectively, be the vectors containing all the model parametersfrom equations (1) and (2) including the augmented data, respectively. The importance weights needed to evaluate the influence of a hospital j in a country k
are defined as



−1
?
?
?
wjk
(θ, Y , m) = p Yjk |Yjk
, α p Yjk
|θ1 , Ujk p (mjk |θ2 )
,
(3)
?
?
where θ = (θ1 , θ2 ) and Yjk
contains all latent data Yijk
in hospital j in a coun?
try k. Further, m and Y , respectively, contain all the mediator measurements?
mjk and all latent data Yjk, respectively. Itcan be shown that we can use the(n)?
standardized importance weights wjk ∝ wjk
θ (n) , Y , m , n = 1, . . . , N to esti-

mate the hospital-deleted posterior mean of some function h(θ) using the MCMC

Rusá et al.

209

P
(n)
(n)
)
sample θ (n) from the corresponding posterior distribution as N
n=1 wjk h(θand the value of the cumulative distribution function of h(θ) without hospital j
P
(n)
(n)
) ≤ a}.
in country k at a point a as N
n=1 wjk I{h(θHowever, it turns out that such weights might be too unstable which could leadto erroneous conclusions. This motivates us to define more flexible weights which?
allow us to choose a suitable function g Yjk, Y , m such that some terms in (3)?
almost cancel out when we multiply the original weights (3) with g Yjk,Y ,m .
To be more specific, let



−1
?
?
?
(θ, Y , m) = p Yjk |Yjk
, α p Yjk
|θ1 , Ujk p (mjk |θ2 )w(g)jk?
g Yjk,Y ,m ,
(4)for some g (θjk , Y , m) > 0. Then we can suggest more appropriate and stable
importance weights, e.g. we can set


?
?
g Yjk
, Y , m = p Yjk|µ̄jk ,
(5)

0.20
0.15
0.10
0.00

0.05

Case−deleted estimate

0.25

0.30

>
>
where µ̄jk is the posterior mean of µjk = βk + Xjk
βX + Mjk
βM + Ujk .

1

4

7

11

15

19

23

27

31

35

39

43

47

51

55

59

Index of hospital

FIGURE 1. The estimate of the indirect effect of nurse working environment on
the patient outcome without hospital j in Belgium (red points) and the credibleintervals (whiskers) compared to the original point estimate (black solid line) andCI (dashed lines).

210

Identifying influential observations in complex Bayesian mediation models

Let us note that the estimate computed with the importance weights (4) (that
P
(n)
(n)
)) still converges in probability to the case-deleted posterioris N
n=1 w(g)jk h(θmean of h(θ).

0.20
0.15
0.10
0.00

0.05

Case−deleted estimate

0.25

0.30

0.35

To illustrate the method described above, we computed the hospital-deleted estimates of the indirect effect of nurse working environment for all Belgium hospitals. The hospital-deleted estimates based on the importance weights (4) withg defined as in (5) and the original estimate are depicted in Figure 1 together
with the corresponding 95%-credible intervals. We can conclude that no estimateof the indirect effect is excessively influenced by any Belgium hospital. Similarly,
we can estimate the country-deleted indirect effect with the importance weights
given by the product of the importance weights corresponding to the hospitals in
the the country. The estimates of the country-deleted indirect effects are shownin Figure 2. We conclude that none of these estimates differ from the posteriormean based on the whole dataset considerably so the model is stable with respect
to the influence of particular hospitals and countries.

BE

CH

DE

ES

FI

GR

IE

PL

Country

FIGURE 2. The estimate of the indirect effect of nurse working environmenton the patient outcome without country k (red points) and the credible intervals (whiskers) compared to the original point estimate (black solid line) and CI(dashed lines).

Rusá et al.

211

Acknowledgments: The work was supported by the research grants GAUK
305–10/250689 and GACR 15-04774Y. Support from the Interuniversity Attraction Poles Programme (IAP-network P7/06), Belgian Science Policy Office, isalso gratefully acknowledged.

References
Bradlow, E.T. and Zaslavsky, A.M. (1997). Case Influence Analysis in BayesianInference. Journal of Computational and Graphical Statistics, 6(3), 314 –331.
Rusá, Š. , Komárek, A., Lesaffre, E., Bruyneel, L. (2017). Multi-level moderation
mediation model with ordinal outcome. Submitted.
Sermeus, W., Aiken, L. H., Van den Heede, K. et al. (2011). Nurse forecasting inEurope (RN4CAST): Rationale, design and methodology. BMC Nursing,10(6), 643 – 664.
Song, X. Y. and Lee, S.Y. (2012). Basic and Advanced Bayesian Structural Equation Modeling: With Applications in the Medical and Behavioral Sciences.
Chichester, West Sussex: John Wiley.

Penalized complexity priors for varying
coefficient models
Massimo Ventrucci1 , Maria Franco-Villoria2 , Håvard Rue3
1

University of Bologna, Italy
University of Torino, Italy
3
KAUST, Saudi Arabia
2

E-mail for correspondence: massimo.ventrucci@unibo.it
Abstract: We discuss an illustrate the use of penalized complexity priors forvarying coefficient models, introducing a natural base model choice that corresponds to a constant coefficient (linear model).
Keywords: Varying Coefficient models, PC prior; INLA; RW1.

1

Introduction

Varying coefficient models (VCMs, Hastie and Tibshirani (1993)) are useful in thepresence of an effect modifier, a variable that “changes” the effect of a covariate
of interest on the response. In practice, VCMs gain flexibility with respect tostandard linear models by allowing one or more regression coefficients to varyover a covariate such as time or space. Consider the simple case where there aren observational units indexed by i = 1, . . . , n and one covariate xi whose effect on
the response yi depends on another variable zi ; the latter could be a continuous
variable (e.g. temperature) or a time/space index (day, region, etc). Assuming yi
belonging to the exponential family, the linear predictor of a generalized VCM isηi = α + f (zi )xi

i = 1, . . . , n.

(1)

We follow a Bayesian hierarchical framework where the varying coefficient f (zi ),z1 ≤ ... ≤ zn , in Eq. (1) is described by a vector of random effects θ =(θ1 , . . . , θn )T distributed at prior as a Gaussian Markov Random Field (GMRF,
Rue and Held (2005)). A GMRF is a multivariate normal distribution with mean
vector µ and a sparse precision Q(τ ) that depends on some hyper-parameters τThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Ventrucci et al.

213

and whose non zero pattern specifies conditional dependencies among neighbouring random effects.

2

PC priors for varying coefficient models

Model (1) turns into a generalized linear regression model when the varying
coefficient f (z) is constant over z. We denote this model as the base model.
Therefore, the VCM in equation (1) can be seen as a flexible extension of thebase model. If we consider f (z) in terms of the vector of random effects θ =(θ1 , . . . , θn )T introduced in Section 1, the base model can be obtained setting thehyper-parameters τ to a particular value (in Section 2.1 an example is discussed).
Elicitation of priors for precision parameters is a long standing topic in the literature on hierarchical Bayesian models. Simpson et al. (2016) recently introduced
a new framework for building priors in Bayesian hierarchical additive models,denoted as Penalized Complexity (PC) priors. PC priors are computed based on
specific principles in which a model component is seen as a flexible parametrization of a base model. The idea is to penalize model complexity, defined in termsof distance from the base model, in such a way that the base model is favoured
unless the available data support a more flexible one. Let ξ denote the flexibility
parameter and the base model be at ξ = 0, the complexity introduced by ξ > 0
is measured using the Kullback-Leibler divergence (Kullback and Leibler, 1951),

Z
f (x; ξ)KLD(f ||g) = f (x; ξ) logdx(2)g(x; ξ = 0)
for flexible model f and base model g. The PC prior is defined as an exponential distribution on the (transformed) KLD distance scale d(ξ) = d(f ||g) =p
2KLD(ξ),π(d(ξ)) = λ exp(−λd(ξ))(3)
The PC prior in the original parameter scale ξ follows by a change of variabletransformation. For more details on PC priors we refer to the paper by Simpson
et al. (2016). In the next section we focus in particular on the PC prior for the
precision of a random walk, which is a suitable smoothing prior in the context of
VCM.

2.1

The random walk caseA useful parametrization for the varying coefficient in Model (1) is θi = β + δi ,where δi indicates deviation from the constant slope β at value zi . We focus on thesimple case of a random walk of order 1 (RW1) prior on θ = (θ1 , . . . , θn )T , condi-P
2
tionally on precision τ , the joint density is specified as π(θ|τ ) ∝ τ (n−1)/2 exp − τ2 n−1i=1 (θi+1 − θi ) .
The RW1 is a first-order intrinsic Gaussian Markov Random Field (Rue and Held(2005) ch. 3) and describes deviations from an arbitrary overall level. In the context of a VCM it is natural to interpret the latter as the constant slope β. In thissense, the RW1 characterizes a varying coefficient in a very intuitive way: it is aprior that shrinks towards a natural base model given by f (zi ) = β, ∀i = 1, ..., n,with τ controlling the amount of shrinkage. When τ −1 = 0 we have f (zi ) = β,214PC prior for VCM

which implies the linear regression model, ηi = α + βxi . For τ −1 > 0, f (zi ) incorporates higher degree of complexity w.r.t. the constant slope, leading to theflexible VCM. For a generic Gaussian Random effect conditional on τ , the PC
prior for τ is the Gumbel type 2 with density (Simpson et al. 2016),
π(τ ) = 0.5λτ −1.5 exp(−λτ −0.5 );(4)

The parameter λ in Eq. (4) can be selected through a user-defined scaling approach. The user can encode the information available (at prior) on the degree of
flexibility of the VCM model with respect to the base model. Simpson et al (2015)
suggest eliciting the probability of a tail event regarding the marginal standard√
deviation, i.e. P r(1/ τ > U ) = a, which yields λ = − log(a)/U .

2.2

An illustrative example

We use a well known example from the seminal paper by Hastie and Tibshirani(1993). The data set includes 88 measurements from ethanol-fuelled engines on
nitric oxide and nitrogen dioxide concentration (N Ox ), equivalence ratio (E) andcompression ratio (C). An initial exploratory analysis suggests that the effect of
E on N Ox is highly non-linear and that the effect of C on N Ox may depend on
E. Hastie and Tibshirani (1993) proposed the following model:
N Ox = f1 (E) + f2 (E)C + ,
where both the intercept and slope are continuous functions of E, with  a normal
error with noise variance σ 2 . Both f1 and f2 are modelled with a RW1 priorconditional on precisions τ1 and τ2 , respectively. To avoid scaling issues inherent
in RW models, the precision matrix of the RW1 has been rescaled as describedin Sørbye and Rue (2013). We assign the PC prior in Eq. (4) to each τi , withthe λi selected following the practical rule of thumb suggested by Simpson et al.
(2016): given the scale of the varying intercept and varying slope are expected
to be different, we scale the PC prior for τ1 such that P r(U1 > 3) = 0.01 (i.e.
λ1 ≈ 1.5), and the one for τ2 such that P r(U2 > 0.3) = 0.01 (i.e. λ2 ≈ 15). Notethat the higher λi , the greater the penalty for deviating from the base model.
PC priors can be implemented in INLA (Rue et al., 2009). We further assumeπ(σ 2 ) = InverseGamma(a, b), with a = 1 and b = 5 · 10−5 , since the posterior forthe noise variance is usually robust to the prior.
Fig. (1) displays the fitted varying intercept and slope, with clear indication ofvariation over the range of the effect modifier E. The different scale of f1 and f2
justifies the use of different λ’s for the two components. From Table (1) we seethat the posterior π(τ2 |y) is robust for different parametrization of the PC prior.
Similar results, not shown here, hold for π(τ1 |y).

3

Concluding remarks

Elicitation of a prior for the hyper-parameters τ is a crucial aspect for practitioners who wish to specify a Bayesian VCM model, as τ regulates the degreeof flexibility of the VCM w.r.t. the base model. Regardless the chosen model forVentrucci et al.

215

0
−2

−1

f1(E)

1

2

(a)

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.0

1.1

1.2

E

0.0
−0.4

−0.2

f2(E)

0.2

0.4

(b)

0.6

0.7

0.8

0.9

E
FIGURE 1. Fitted varying intercept (a) and slope (b).

TABLE 1. Posterior summaries for the precision of f2 (E), τ2 , for different PC
prior parameters (first column).

PC prior parameters;
U2 = 0.3 ; a = 0.01 (λ2 = 15)
U2 = 0.2 ; a = 0.01 (λ2 = 23)
U2 = 0.1 ; a = 0.01 (λ2 = 46)π(τ2 |y):0.025quantmean

0.975quant

16.76
16.69
16.61

69.25
69.1
68.94

189.22188.8
188.52the varying coefficient, a suitable PC prior shrinking to a sensible base model
can always be defined through the application of predefined principles. PC priors
avoid over-fitting by construction allowing the VCM model to arise only if datarequires it. This is a desirable property as varying coefficients are typically overparametrized models needing regularization. It is worth investigating the use of

216

PC prior for VCM

PC priors in more complex models involving spatially varying coefficients.

References
Kullback, S. and Leibler, R.A. (1951). On information and sufficiency. The Annals of Mathematical Statistics, 22, 79 – 86.
Hastie, T. and Tibshirani, R. (1993). Varying-Coefficient Models. Journal of theRoyal Statistical Society, Series B, 55(4), 757 – 796.
Rue, H. and Held, L. (2005). Gaussian Markov Random Fields. Chapman & Hall/CRC.
Rue, H., Martino, S. and Chopin, N. (2009). Approximate Bayesian inference forlatent Gaussian models using Integrated Nested Laplace Approximations(with discussion). Journal of the Royal Statistical Society, Series B, 71(2),319 – 392.
Simpson, D., Rue, H., Riebler, A., Martins, T. G. and Sorbye, S. H. (2016). Penalising model component complexity: A principled, practical approach toconstructing priors (with discussion). To appear in Statistical Science.
Sørbye, S.H. and Rue, H. (2013). Scaling intrinsic Gaussian Markov random fieldpriors in spatial modelling. Spatial Statistics, 8, 39 – 51.

Confidence intervals for posterior intercepts,
with application to the PIAAC literacy
surveyJochen Einbeck1 , Elizabeth Gray12 , Nick Sofroniou3 , Antonio
Hermes Marques da Silva Junior14 , Jacob Gledhill15
1

Durham University, UK
University of Bath, UK
3
Institute for Employment Research, University of Warwick, UK
4
Universidade Federal do Rio Grande do Norte, Natal, Brazil
5
Department for Communities and Local Government, UK
2E-mail for correspondence: jochen.einbeck@durham.ac.uk
Abstract: For variance component models, it is often the posterior estimate of
the random effect (‘posterior intercept’) rather than the estimate of the fixed
effect parameters, which is of main interest. This is the case, for instance, whenranking region–wise mortality rates (where the crude, regional rates are unreliable
due to small observed counts) or for the construction of educational league tablesfrom complex sample surveys. However, in order to be able to decide whether
two cluster–level units can actually be distinguished, it is clear that one needs ameasure of variability of these posterior intercepts. We present an explorationof methods to address this issue which appears to be still undeveloped in thecontext of the model class considered.
Keywords: Nonparametric maximum likelihood; Empirical Bayes shrinkage;
Bootstrap

1

Posterior intercepts

Consider variance component models of typeµij = E(yij |zi ) = h(xTij β + zi ),(1)

where µij is the expected response for unit j in cluster i, xij are the fixed effect covariates which may depend on i, j, or both, and zi is the random effectThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

218

Confidence intervals for posterior intercepts

operating at the cluster level. If no assumption on the random effect distribution is made, then estimation can be carried out via ‘nonparametric maximum
likelihood’ (Aitkin, 1999). Briefly, the marginal likelihood is approximated by adiscrete mixture, the parameters of which are estimated alongside with the fixed
effect parameters via the EM algorithm, yielding estimates β̂, ẑ1 , . . . , ẑk with
masses π̂1 , . . . , π̂K . Denote by θ̂ the collection of these estimates, and by yi = (yi· )the set of response values for cluster i. Aitkin (1999) suggested to estimate themean of the posterior distribution zi |yi via ‘Empirical Bayes Predictions’
z̃i =

K
X

wik ẑk ,

(2)

k=1

where wik = P̂ (k|θ̂, yi ) are the posterior probabilities (‘responsibilities’) that observation i stems from component k, which can be computed via Bayes’ theorem
from the parameter estimates θ̂ of the last M step. The quantity of interest arethese posterior intercepts, z̃i .

2

PIAAC data

The PIAAC survey of adult skills was carried out from 01/08/2011 to 31/03/2012
by the OECD in 24 countries (or sub–country entities), and was designed to assess the proficiency of adults in the key competencies of literacy, numeracy, andproblem–solving in technology–rich environments. We focus here on the ‘literacy’ output variable, with six possible outcomes for an assessed individual. We
dichotomized this variable as ‘people reaching level 3 or above’, with ‘level 2 andbelow’ being considered as low–skilled, which corresponds to the key European
Commission policy marker used to demarcate poor basic skills in the complementary PISA survey carried out at 15–years of age (Eurostat, 2016). As covariates
we will use gender, as well as a factor for age (covering the intervals 16–24, 25–
34, 35–44, 45–54, and 55+), though we have also explored more complex models
using employment status and reading habits which are not reported here. This
leads to a (rescaled) logistic regression model yij ∼ Bin(nij , µij )/nij where nij
is the (effective) sample size of the jth subpopulation (defined by the covariate
combination of interest) for country i, and function h(·) in (1) is the logistic function. Data were extracted using the PIAAC explorer. A model with age*genderinteraction and K = 5 turned out to capture the upper-level heterogeneity well
(Table 1 right).

3

Uncertainty of posterior intercepts

3.1

Analytic approximation

We initially approach the problem analytically, considering the weights wik in (2)as constants (which, strictly, they are not, since they depend on the parameters
estimated in the last M-step). It follows then from (2) that
Var(z̃i ) =

K
X
k=1

2
wik
Var(ẑk ) +X
j6=kwij wik Cov(ẑj , ẑk )(3)

Einbeck et al.

219

where the variances and covariances are available from the fitted model according
to standard GLM theory. Clearly, the covariance terms cannot be naively
omittedP
since the positions of the ẑk0 s are strongly correlated. However, as K
k=1 wik = 1
for all i, it is clear that 0 ≤ wij wik ≤ 1/4 for all pairs j 6= k. In addition, it
is often (but not always) the case that after EM convergence observations areclassified to one of the components with probability equal or close to 1, in whichcase wij wik ≈ 0. [To exemplify this point, Table 1 gives an excerpt of the matrix
W = (wik ) for the gender*age model with K = 5 components.] In either case, it
is clear that the product wij wik will be very small for all (or almost all) i, j, kwith j 6= k, so that the ‘naive’ approximation
Var(z̃i ) ≈

K
X

2
wik
Var(ẑk )

(4)

k=1

will usually be a good one. Confidence intervals
p for the posterior intercepts arethen obtained from either (3) or (4) via z̃i ± q Var(z̃i ) where q is an appropriatequantile for which we use the 97.5% Gaussian quantile, 1.96.

3.2

NPML–Bootstrap

In order to assess the variability in a potentially more realistic way, we also
developed a bootstrap routine which proceeds in two layers. Specifically, for i =1, . . . , n,(i) from the set of mass points ẑ1 , . . . , ẑk draw a masspoint ži with probability
wik ;(ii) generate new y̌ij ∼ Bin(nij , µ̌ij )/nij , where µ̌ij is defined in the natural
way via (1), using β̂ and ži .
Having y̌ij , we refit the model, yielding a new set of n posterior intercepts. Repeating these steps M times we have a bootstrap sample of estimates for posterior
intercepts. Therefore, by taking the standard deviation of these we have an estimate for their variability.

3.3

Results

Figure 1 (left) gives the z̃i along with ‘full’ confidence intervals (3) and bootstrapped confidence intervals (using N = 9999). The analytic and simulation–
based intervals are very similar, with minor differences only recognizable for asmall subset of countries. The naive intervals (4) cannot be visually distinguishedfrom the full intervals. Therefore, we provide in Figure 1 (right) the ratio of
the widths of the naive and full intervals, as well as the bootstrapped and fullintervals. All ratios are very close to 1, with slightly larger deviations for thebootstrapped intervals. We also see that five groups of countries can be robustly
distinguished (since the corresponding intervals do not overlap), with Japan being
the sole best–performing country.

220

Confidence intervals for posterior intercepts

3

4

5

0
0.0023000.9977
1

1
0
0

0
0
0

0
0

0
0

0
0

0
1

1
0

-0.490

0.011

0.273

0.622

1.307

ẑk

4000

2

0
0
0

−2 log L

1

Australia
Austria
Canada...
Japan
Netherlands
...

3000

k

5000

TABLE 1. Left: Excerpt of matrix W = (wik ) (4.d.p.) for age*gender modelwith K = 5; right: −2 log L as a function of K.

2

4

6

8

K

posterior intercepts

1.3
Naive/Full

1.2

Boot/Full

1.1

Australia
Austria
CanadaCzech Republic
Denmark
England
Estonia
Finland
Flanders (Belgium)
FranceGermany
Ireland
Italy
Japan
Netherlands
Northern Ireland
NorwayPolandRepublic of Korea
Russian Federation
Slovak Republic
Spain
SwedenUnited States

Australia
Austria
CanadaCzech Republic
Denmark
England
Estonia
Finland
Flanders (Belgium)
FranceGermany
Ireland
Italy
Japan
Netherlands
Northern Ireland
NorwayPolandRepublic of Korea
Russian Federation
Slovak Republic
Spain
SwedenUnited States

1.0

FIGURE 1. Left: Posterior intercepts [black dots] with analytic (‘full’) [innerinterval; red in the online version] and bootstrapped intervals [outer; turquoise];right: relative width of intervals.

4

Uncertainty of posterior probabilities

4.1

Sampling from posterior likelihood

A potential issue with the methodology discussed so far is that by plugging theML parameter estimates θ̂ into the expression for wik , the uncertainty in thoseestimates is ignored. Hence, the ‘certainty’ of mass point allocation when taking
the wik at face value can be considered as overstated. To address this problem,
Aitkin et al. (2014) suggested the following procedure based on the concept ofposterior likelihood (Aitkin, 2010):a) Assuming flat priors for θ, the posterior distribution p(θ|y1 , . . . , yn ) is proportional to the likelihood, L(θ). Hence, one can take M random drawsθ̂[m] , m = 1, . . . , M , from L(θ).

Einbeck et al.

221

[m]

b) Compute wik = P (k|θ̂[m] , yi ), m = 1, . . . , M .
For our purposes, one would then proceed further,[m]c) Apply step (i) in the algorithm in subsection 3.2 using wik instead of wik
in the m–th bootstrap repetition.
However, the implementation is non–straightforward and will require a computationally expensive MCMC analysis, involving Gibbs samplers for each k, alternate
draws between different types of model parameters, and ad–hoc solutions to thestarting value and the label switching problems. Also, having already carried out
a full EM procedure, a full–blown MCMC analysis only for the sake of analyzing
the posterior intercepts feels rather out of scale. Hence, a simpler alternative idea
is considered below.

4.2

Sensitivity assessment via EM process trail

As stated above, an EM algorithm has already been executed. As such, in thisprocess, a series of ‘draws’ from the full likelihood L(θ) has been obtained. Assume
that, in EM iteration s = 1, . . . , S, we have obtained parameter estimates θ̂[s]with associated weight matrices W [s] and likelihoods L[s] ≡ L(θ̂[s] ). Hence, we
possess S draws from L(θ), including the final iteration, which corresponds to
the MLE θ̂[s] ≡ θ̂. While it is clear that these S draws in no way represent thecorrect shape of L(θ), the matrices W [s] can still be used to assess the sensitivityof the NPML–Bootstrap to imprecision in the wik 0 s, especially as some of theestimates along the EM process trail correspond to really ‘bad’ likelihoods (thatis, estimates which have a likelihood of effectively 0 to be sampled in part a) of
the Aitkin routine).

4.3

Results

For the data and model at hand, the number of required EM iterations turned
out to be S = 6, and Figure 2 (left) shows L(θ̂[s] ) as a function of s. Figure 2(right) shows the interval length of the NPML–bootstrapped confidence intervals[s][6]
when using wik , s = 1, . . . , 5 relative to that using wik ≡ wik . It is clear that for
all s corresponding to appreciable likelihoods the difference is less than 10%, andeven for posterior weights corresponding to really poor likelihoods the increase isgenerally not more than 50%, indicating robust upper bounds for the uncertainty
in this process.
The present paper has demonstrated the utility of bootstrap methods to characterize the sometimes substantial uncertainty in cluster–level estimates which
commonly arises in league-table comparisons. While no claim is made that therelative magnitudes of the different intervals will in general behave in the mannerof this particular case study, the tools proposed to arrive at this judgement are
applicable for arbitrary two–level problems.

References
Aitkin, M. (1999). A General Maximum Likelihood Analysis of Variance Components in Generalized Linear Models. Biometrics, 55, 117 – 128.

222

Confidence intervals for posterior intercepts
1.8
2

3

4

5

4000

1.5

1.2

2000

L(θ)

6000

1

United States

Spain

SwedenSlovak Republic

Russian Federation

PolandRepublic of Korea

NorwayNorthern Ireland

Italy

Japan

Netherlands

IrelandFranceGermany

Flanders (Belgium)

Estonia

FinlandEnglandDenmark

Czech Republic

6

Austria

4

Canada2

number of EM iterations

Australia

00.9

FIGURE 2. Left: Maximum Likelihood L(θ̂[s] ) versus EM iteration s; right:[s][6]interval length using wik , s = 1, . . . , 5 relative to using wik ≡ wik . The value s
is given in the legend.

Aitkin, M. (2010). Statistical Inference — an Integrated Bayesian/Likelihood Approach. Chapman & Hall/CRC:Boca Raton.
Aitkin, M., Duy, V. and Francis, B. (2014). Statistical Modelling of the groupstructure for social networks. Social Networks, 38, 74–87.
Eurostat (2016). Smarter, greener, more inclusive? Indicators to support the Europe 2020 strategy. Publications Office of the European Union, European
Commission, Luxembourg.

Dealing with response styles in finite mixture
modelsRosaria Simone1 , Gerhard Tutz2
1
2

Department of Political Sciences, University of Naples Federico II, Italy
Ludwig–Maximilians- Universität München, Germany

E-mail for correspondence: rosaria.simone@unina.it
Abstract: The objective of the present work is to propose a mixture modelfor ordinal data that is able to properly identify response styles with a tendency
to choose middle or extreme categories, while accounting for uncertainty. The
proposal is grounded on CUB models’ paradigm and it is tested on a survey
collected at the University of Naples Federico II, in which sports preferences areinvestigated.
Keywords: Uncertainty; Response Styles; Finite Mixture Models

1

Introduction and preliminaries

The data generating process yielding respondents to produce an ordinal evaluation over m categories out of their latent perception can be effectively modeled
as the combination of two unobserved components: the actual feeling towards
the item, and the uncertainty accounting to the inherent indeterminacy of every
decision process, see Iannario and Piccolo (2010). Then, the response variable Riof the i-th subject has probability distribution given by:
P r(Ri = r|xi ) = πP r(Yi = r|xi ) + (1 − π)P r(Ui = r),

r = 1, . . . , m.

(1)

where P r(Yi = r|xi ) gives the preference part of the model, and P r(Ui = r)corresponds to the uncertainty - see Tutz et al. (2016). In particular, CUB models (D’Elia and Piccolo, 2005) structure the response variable Ri as a mixture
between a shifted Binomial with parameter ξ for the preference part-br (ξ)- anda uniform distribution over the support {1, . . . , m}:
P r(Ri = r|ξi , π) = πbr (ξi ) + (1 − π)1,mr = 1, . . . , m, m > 3.

(2)

This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

224

Uncertainty and response styles

The starting point of the extension proposed here are the findings in Gottard etal.(2016), where the authors introduce varying uncertainty in CUB
models, experimenting with alternative choices to the uniform distribution. In
this context, a dedicated analysis to response-styles is pursued, by presenting
an extension of CUB models where the uncertainty component itself is designed
to deal with symmetrical response styles, either with reference to extreme or to
middle-scale categories. Similar considerations are explored in Tutz and Schneider (2016) with the Beta-Binomial distribution. Our proposal here is to adjust
the uncertainty distribution by making use of the transformation:
P r(Ui = r) =P r(Si = r) + c,
1 + mc(3)where P r(Si = r) is a suitable probability distribution specifying the responsestyles attitude, and c 6= 0 is a chosen constant. A flexible tool to work on this taskturns out to be a combination of the uniform distribution-needed to maintain thefocus on heterogeneity and indecision due to the item- with the discretized Beta
distribution.
Let X ∼ Beta(α, β) be a Beta distributed random variable, with density:
Z x
1
fX (x) =tα−1 (1 − t)β−1 dt,
x ∈ [0, 1],B(α, β) 0
and the Euler Beta Function defined by:
Z 1
Γ(α)Γ(β).
B(α, β) =tα−1 (1 − t)β−1 dt =Γ(α + β)0Then we say that a discrete random variable D = D(α, β) over the support{1, . . . , m} has the Discretized Beta distribution (of parameters α, β) if:


r−1
r
P r(D = r|α, β) = P r≤ X ≤ |α, β , r = 1, . . . , m,mmthat is, if D is obtained from X ∼ Beta(α, β) by dividing its range into m equal
sub-intervals and then by integrating over them. When α = β, the distribution
is symmetric; in this case, when 0 < α < 1, the distribution is convex with twomodes located at the extreme categories, whereas the distribution is concave andconcentrated around the middle categories if α > 1. When α = β = 1, D collapses
to the discrete uniform distribution on the given support.
1
By choosing c = min (3), and D as model for the (symmetric) response stylecomponent, we propose to adjust the uncertainty specification in CUB models
by using a C ombination of a Lifted U niform and a shifted B inomial (CLUB, forshort) distribution:1 
P r(D = r|α) + m
P r(R = r|ξ, π, α) = πbr (ξ)+(1−π),
r = 1, . . . , m. (4)2Figure 1 shows, for fixed π and ξ parameters and m = 7, the effect on the
CUB probability distribution (black solid lines) of including the response stylecomponent as determined by CLUB models, both for α < 1 (top panels) and forα > 1 (bottom panels).

Simone and Tutz
pai= 0.2 , csi= 0.7

0.35

0.35

pai= 0.6 , csi= 0.5
alpha = 0.1

alpha = 0.1

alpha = 0.3

alpha = 0.3

alpha = 0.5

alpha = 0.5

0.30

0.30

225

alpha = 0.7

0.25
0.20
0.15

0.15

0.20

0.25

alpha = 0.9

alpha = 0.7
alpha = 0.9

α = 0.1

α = 0.3
α = 0.5
α = 0.7
α = 0.9

0.10

α = 0.3

0.00

0.00

0.05

α = 0.5
α = 0.7
α = 0.9

0.05

0.10

α = 0.1

1

2

3

4

5

6

7

1

2

alpha = 3alpha = 5alpha = 5

0.30

0.30

4

5

6

7

6

7

alpha = 1.5

alpha = 3alpha = 10

alpha = 10

0.20

0.25

alpha = 15

0.15
0.10

0.10

0.15

0.20

0.25

alpha = 15

α = 1.5α=3510αα==150.05

α = 1.5

0.00

α=3
5
10
αα==15

0.00

0.05

3

pai= 0.1 , csi= 0.5

0.35

0.35

pai= 0.4 , csi= 0.2
alpha = 1.5

1

2

3

4

5

6

7

1

2

3

4

5

FIGURE 1. Extension from CUB to CLUB
Estimation of CLUB models is performed via Maximum Likelihood methods(ML) and requires a specific implementation of the EM algorithm -McLachlan
and Krishnan (1997). Since CUB models are nested into CLUB (for α = 1), theLikelihood Ratio Test is performed to check if the improvement of the fit is significant. Also, the symmetrized Kullback-Leibler divergence is computed to assessthe discrepancy between observed and fitted distributions. Analytical derivationof the variance-covariance matrix of parameter estimates can be cumbersome dueto the unavailability of a closed form for the discretized Beta distribution. Thus,
standard errors are obtained either from numerical procedures or by samplingtechniques like the bootstrap. The whole analysis is run in the R environment:
the code is available upon request from the Authors, and makes use of Package
CUB -Iannario et al.(2016)- for CUB models estimation.

2

An application to Sport preferences

The performances of CLUB models are discussed on a survey collected at University of Naples Federico II in May 2016. The questionnaire has been filled by
n = 647 respondents, asked, among others questions, to rank their preferences
for 8 sports: Football, Jogging, Volleyball, Tennis, Boxing, Swimming, Cycling
and Basketball. Thus, marginally, each of these variables corresponds to rating
measurements on a m = 8 points scale, of the type: Rate your preference for the
given sport, with 1 = absolutely preferred, and 8 = not at all preferred. The frequency distributions of the ordinal variables Football and Jogging are U -shaped,
thus giving evidence for an extreme category response styles, for which CLUB226

Uncertainty and response styles

models provide an impressive fit. For all items but Boxing and Swimming, there
is empirical evidence that supports the inclusion of a middle categories responsestyles. Figure 2 displays, for each sport variable, the barplots of the observed frequency distributions, over-plotted against both the estimated CLUB distribution
and the estimated CUB distribution.

FIGURE 2. Frequency distributions of sport variables, with CLUB and CUB
estimated probabilities
Table 1 reports parameter estimates (standard errors in parentheses) for both
CLUB and the nested CUB models fitted to the sport variables, the corresponding
log-likelihoods and likelihood ratio test statistics for the null H0 : α = 1 against
the alternative H1 : α 6= 1, which will be asymptotically χ21 distributed. Nonsignificant values at level 0.05 are highlighted in bold.,Sport
FootballJogging
Tennis

CLUB
ξˆ
0.928

0.019

(0.032)(0.026)

0.118

BasketCycling
VolleyballSwimming
Boxingπ̂
0.098

Loglik CLUB(0.050)α̂
0.100(0.040)-1195.342

0.010

0.638

-1334.546(0.084)0.259(0.097)(0.046)(0.056)(0.291)2.087-1317.944

0.181

0.351

0.543

-1303.905(0.079)(0.061)(2.352)

0.208

0.114

0.713

-1314.433( 0.033)(0.020)(3.490)

0.227

0.833

2.920

-1313.173(0.038)(0.0267)(0.593)

0.199

0.847

1.180

-1327.975

1.440-1331.424(0.036)0.165(0.037)(0.029)0.092(0.040)(0.200)(0.307)

CUB
π̂
ξˆ
0.217
0.99

Loglik CUB(0.024)(0.009)

0.057

0.004

(0.022)0.275

Loglik saturated

LRT-1266.47-1192.766

142.25-1339.187-1332.782

9.282(0.049)(0.033)0.392-1328.925-1304.701

21.96

0.414

(0.048)0.430(0.021)-1309.782-1299.535

11.75

0.267

(0.049)0.322(0.040)-1329.76-1311.405

30.65

0.319

0.685

-1320.619-1310.847

14.89

0.830

-1328.448-1327.271

0.946

0.157

-1332.456-1322.767

2.065(0.046)0.202(0.037)0.177(0.040)(0.028)(0.026)(0.036)TABLE 1. Parameter estimates for both CLUB and CUB models
Table 2 reports the (normalized) dissimilarity index between the observed (relative) frequency distribution of each of the sport variables and the estimated probability distributions both for CLUB and CUB models. Similarly, the symmetrized
Kullback-Leibler divergence is computed. Both these indexes are strongly reduced
when switching from CUB to CLUB. The better performances of the new proposal
are supported also by the striking closeness between the maximized log-likelihood
and the saturated one (see Table 1).

Simone and Tutz
Sport
FootballJogging
TennisBasketCycling
VolleyballSwimming
BoxingDissimilarity
CLUB CUB
0.028 0.166
0.026 0.050
0.074 0.095
0.045 0.068
0.042 0.100
0.039 0.065
0.021 0.029
0.071 0.082

227

Symm. Kullback-Leibler
CLUB
CUB
0.004
0.107
0.003
0.010
0.020
0.038
0.007
0.016
0.005
0.029
0.004
0.015
0.001
0.002
0.013
0.015

TABLE 2. Dissimilarity and Symmetrized Kullback-Leibler divergence between
estimated probabilities and observed (relative) frequencies

2.1

CLUB models with covariates

CLUB model estimation can be enhanced by the inclusion of subjects’ characteristics zi to explain the occurrence and the effect of the response styles, to belinked to parameter α by means of a log link:0log(αi ) = ν0 + ν zi ,i = 1, . . . , n.

For illustrative purposes, let Z be a dichotomous factor with level Z = 0 if
respondent prefers individual sports and Z = 1 if he/she prefers team sports.
Then, for Volleyball preferences:log(αi ) = 1.315 −0.933 Zi ,i = 1, . . . , n,(0.242) (0.352)

implying that the tendency to place Volleyball in the middle of the ranking isstrongly associated with preferences for individual sports: indeed, α decreases
from α(0) = 3.723 when Z = 0 to α(1) = 1.46 when Z = 1 (parameter estimates are significant according to the Wald test, and the log-likelihood attains
an estimated maximum of −1310.07; see Table 1 for comparisons). For Jogging
preferences, instead, one has:log(αi ) = −0.764 + 0.735 Zi ,(0.196)i = 1, . . . , n,(0.261)

implying that the tendency to place Jogging at the extreme ranks is strongly
associated with preferences for individual sports: indeed, since α increases fromα(0) = 0.466 when Z = 0 to α(1) = 0.971 when Z = 1 (again, parameter estimates are significant according to the Wald test, and the log-likelihood attains
an estimated maximum of −1330.07; see Table 1 for comparisons).

3

Conclusions and future developments

The case study here discussed, as well as a simulation study run to validate theproposal, confirms the adequacy of CLUB models to properly identify responsestyle effects that are otherwise hidden by the uncertainty measure. Mis-specification
yields biased estimates of the feeling parameter. The framework here explored todeal with response styles can be combined in a more general setting, currently
under scrutiny, to analyze repeated measurements while modeling subjective heterogeneity.

228

Uncertainty and response styles

References
D’Elia A. and Piccolo D. (2005). A mixture model for preference data analysis.
Comput. Stat. Data An., 49, 917 – 934.
Gottard, A., Iannario, M. and Piccolo, D. (2016). Varying uncertainty in CUB
models. Advances in Data Analysis and Classification, 10(2), 225 – 244.
Iannario, M. and Piccolo, D. (2010). A New Statistical Model for the Analysis
of Customer Satisfaction. Quality Technology & Quantitative Management,
7(2), 149 – 168.
Iannario, M., Piccolo, D. and Simone, R. (2016). CUB: A Class of Mixture Models for Ordinal Data. (R package version 1.0). http://CRAN.R-project.org/package=CUB.
McLachlan, G. and Krishnan, T. (1997). The EM algorithm and extensions. New
York: J.Wiley & Sons.
Tutz, G. and Schneider, M. (2016) Mixture Models for Ordinal Responses witha Flexible Uncertainty Component. Technical Report.
Tutz, G., Schneider, M., Iannario, M. and Piccolo, D. (2016). Mixture models forordinal responses to account for uncertainty of choice. Advances in Data
Analysis and Classification, DOI:10.1007/s11634-016-0247-9.

Extending the inferential capability of ageneralised partial credit model using
Bayesian computation: An application to aninternational disability survey developed by
WHO and the World Bank
Sujit Sahu1 , Mark Bass1 , Carla Sabariego2 , Alarcos Cieza3 ,Carolina Fellinghauer4 , Somnath Chatterji51Southampton Statistical Sciences Research Institute, University of Southampton, UK
2
Department of Medical Informatics, Biometry and EpidemiologyIBE, LudwigMaximilians-University (LMU), Munich, Germany
3
Department for Management of Noncommunicable Diseases, Disability, Violence and Injury Prevention, World Health Organization, Geneva 1211, Switzerland
4
Swiss Paraplegic Research, Nottwil 6207, Switzerland
5
Department of Information, Evidence and Research, World Health Organization, Geneva 1211, Switzerland
E-mail for correspondence: S.K.Sahu@soton.ac.uk
Abstract: Generalised partial credit models are ubiquitous in many applicationsin the health and medical sciences that use item response theory. Such polytomous item response models have a great many uses ranging from assessing andpredicting an individual’s ability to ordering the items to test the effectiveness ofthe test instrumentation. By implementing these models in a full Bayesian framework, computed through the use of Markov chain Monte Carlo methods, this article extends their inferential capability in three distinct ways. First, the models
are extended to include covariate effects thus allowing simultaneous estimation ofregression and item parameters. Secondly, full Bayesian methods for ranking theitems using the Fisher information criterion (FIC) are developed. This allows us
to fully propagate and ascertain uncertainty in the inferences by calculating theitem specific FIC which facilitates the item ordering. Lastly, we propose a Monte
Carlo method for predicting the ability score of a new individual by approximating the relevant Bayesian predictive distribution. Data from a Model Disability
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

230

Extending the inferential capability of a generalised partial credit model

Survey carried out in Sri Lanka by the World Health Organization (WHO) andthe World Bank are used to illustrate the methods. The proposed approaches areshown to provide simultaneous model based inference for all aspects of disability
which can be explained by environmental and socio-economic factors.
Keywords: Bayesian Methods, Education Testing, Hierarchical Modelling, Item
ranking, Item Response Theory

1

Extended abstract

This paper sets out to achieve three extended inferential tasks when the GPCMs
are employed. The first extension enables estimation of the item parameters adjusted for the covariate effects. Inference for the covariate effects has also been
illustrated with the WHO and the World Bank Model Disability Survey data.
A Bayesian approach based on a single model, in contrast to a stage-wise procedural estimation method, allows us to accurately assess the uncertainties not
only for the item parameters but also for the regression parameters. This paper
illustrates the methods with the main effects model only, but the methods canalso be applied to higher order interaction models.
The second inferential extension task ranks the items so that a brief version of
the disability survey with fewer items can be prepared. Using an expected FIC
we have developed a method for ranking the items. It is up to the practitioner todecide how many items can be afforded in the reduced survey and we acknowledge
that there may be other practical considerations which may influence the final
item choice. The proposed method will guide item selection based on a desired
percentage of information that must be present in the reduced survey.
The third inferential extension is the MCMC based methodology to predict theability scores of new individuals whose data are observed after model fitting
has already been performed. The methodology uses all the relevant covariate
information of each new individual so that the best possible Bayesian estimates
are obtained. The proposed prediction methodology has been empirically verified
by re-estimating the ability scores of a large number (1000) of new individuals
by fitting the model to the full data set. Close agreement between the predictedscores and the estimated scores based on all the data shows the effectiveness ofthe new methodology.
These three methodological extensions allow us to extract a lot more informationfrom the data than what has been possible before. Using a unified model it isconcluded that the main effects of the three covariates: gender, age and income
are significant in the presence of, hence accounting for, the latent ability, item
discriminatory and item difficulty parameters. In addition, the main advantageof the unified model also lies in its ability to make coherent inference on item
ordering and ability score prediction for new individuals. By eliminating a stagewise procedure for the three different inferential tasks, the developed Bayesian
methodology proposes a rigorous and coherent inference framework wherever
GPCM models are to be used in practice. This framework ensures coherency
by having the correct and mutually consistent levels of uncertainty in the threedifferent inferential tasks.

Response Styles in the Partial Credit Model
Gunther Schauberger1 , Gerhard Tutz1 , Moritz Berger2
1
2

LMU Munich, Germany
University of Bonn, Germany

E-mail for correspondence: gunther@stat.uni-muenchen.de
Abstract: Ignoring the response style in the modelling of ordinal responses in
psychological measurement can lead to poor and considerably biased estimatesof item parameters. This work focuses on the modelling of a tendency to extremeor middle categories in item response data with Likert scales. An extension of thePartial Credit Model is proposed that explicitly accounts for this specific responsestyle. The proposed model contains person-specific response style parameterswhich are estimated using the framework of generalized mixed linear models.
The method is applied to a real data example to illustrate the effect of ignoring
response styles.
Keywords: Partial credit model; Likert-type scales; Rating scales; Responsestyles.

1

The Partial Credit Model

Let the categories 0, . . . , k represent graded agree-disagree attitudes with a natural symmetry like strongly disagree, . . . , strongly agree where Ypi ∈ {0, 1, . . . , k},
p = 1, . . . , P , i = 1, . . . , I denotes the ordinal response of person p on item i.
Then the partial credit model (PCM) as proposed by Masters (1982) is denotedby
P (Ypi = r)
= θp − δir , r = 1, . . . , klog
P (Ypi = r − 1)
where θp is the person parameter and (δi1 , . . . , δik ) are the item parameters of
item i. It can be seen that the model is locally (given response categories r − 1,r) a binary Rasch model with person parameter θp and item difficulty δir .
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

232

2

PCMRS

The Partial Credit Model with Response Style

Response styles that account for a tendency to extreme categories or a tendency
to middle categories are modeled by modifying the parameter δir . An additional
person parameter γp is introduced that describes the response style of the person.
The resulting partial credit model with response style (PCMRS) has the form
P (Ypi = r)log
= θp + (m − r + 0.5)γp − δir , r = 1, . . . , k
P (Ypi = r − 1)for an odd number of categories (k is even) and
P (Ypi = r)
= θp + (m − r)γp − δir ,log
P (Ypi = r − 1)

r = 1, . . . , k

for an even number of categories. Here, m represents the middle category whereis m = k/2 for an odd number of categories and m = [k/2] + 1 for an evennumber of categories. The parameter γp can be seen as a shifting of thresholds.
If γp is positive , the thresholds δir shift away from each other and, therefore,
result in a higher tendency for person p to middle categories. The extreme case
γp → ∞ yields P (Ypi = m) → 1. If γp is negative one has the reverse effect;
the person has a tendency to the extreme categories. For illustration we show
Four response categories (k=3)γp = 0γp = −1.5~~ ~δ 2δ 3 δ 1−3 −2 −1

0 1
θp

2

3

0.0

0.4

P( Ypi =r)

0.8

~
δ3

0.0

0.4

P( Ypi =r)

0.8

~ ~
δ1 δ2

0.8

~
δ3

~
δ2

0.0

P( Ypi =r)

~
δ1

0.4

γp = 1.5−3 −2 −1

0 1
θp

2

3

−3 −2 −1

0 1
θp

2

3

FIGURE 1. Probabilities P (Ypi = r) against θp for positive, negative γp and
γp = 0; in the upper panels the number of categories is three, in the lower panels
it is four.
in Figure 1 the probabilities of response categories as functions of the person
abilities for varying response style parameter γp in the case k = 3 (four response
categories). It is immediately seen that for γp = −1.5 the extreme categories havemuch higher probabilities than for γp = 0. The inverse is seen for γp = 1.5. It isnoteworthy that the probabilities of adjacent categories are now equal, that is,
P (Ypi = r) = P (Ypi = r − 1), if θp = δ̃ir where δ̃ir = δir − (m − r + 0.5)γp is thenew (person-specific) threshold parameter. The basic concept, explicit modellingof a tendency to middle or extreme categories, has been used before by Tutz andBerger (2016) who considered just one item and modelled the effect of covariates.
The model is estimated using marginal likelihood estimation. For the personparameters, a two-dimensional normal distribution N (0, Σ) is assumed. The diagonals of the matrix Σ contain the variance of the response style parameters σγ2
and the variance of the person effects, σθ2 , the off diagonals are the covariances
between response style and location effects, covγθ . Therefore, one also allows for acorrelation between both person parameters. In the marginal likelihood approach,Schauberger et al.

233

the common density of the person parameters is integrated out numerically by
Gauss-Hermite integration.

3

Illustrative example

In our applications we use data from the standardiziation sample of the Freiburg
Complaint Checklist (FCC) (Fahrenberg, 2010). The FCC is a questionnaire thatis used to assess physical complaints of adults, we will focus on the scale tenseness.
The data set contains 2070 participants (2032 complete cases). Each of the 9 items
from the scale tenseness is measured on a 5-point response scale that refers to
the frequency of the complaint: “never”, “about 2 times a year”, “about 2 times
a month”, “approximately 3 times a week” or “almost every day”.
Figure 2 shows the estimates for the item parameters. The solid lines represent
estimates for the PCMRS and the dashed lines represent estimates for the PCM.
The estimates for the PCMRS are more stable across the categories. Figure 3

●

3

4

●
●

1

2

●

3

4

●
●
●
●

●
●
●

1

2

●

3

4

●
●
●

●

●
●
●
●

1

2

3

ThresholdThresholdThresholdThresholdRestless hands

Restless feet

Twitching eyes

Twitching mouth

●
●
●

●

●
●

1

●

2

3

Threshold

4

●
●

●

●
●

●

●
●

1

2

3

Threshold

4

●
●
●
●

●
●

●

●

1

2

3

Threshold4−3.0 −2.5 −2.0 −1.5 −1.0 −0.5

●
●

2

●

Wavering hands−3.0 −2.5 −2.0 −1.5 −1.0 −0.5●●

Clumsiness−3.0 −2.5 −2.0 −1.5 −1.0 −0.5●●●

−3.0 −2.5 −2.0 −1.5 −1.0 −0.5●

−3.0 −2.5 −2.0 −1.5 −1.0 −0.5●●

1−3.0 −2.5 −2.0 −1.5 −1.0 −0.5

Sudden attacks of sweating●●−3.0 −2.5 −2.0 −1.5 −1.0 −0.5

−3.0 −2.5 −2.0 −1.5 −1.0 −0.5

Clammy hands

●
●
●

4

●
●

●
●
●

1

2

3

4

Threshold

FIGURE 2. Estimates of item parameters for items on tenseness (FCC), separately for each item. Solid line represents estimates for PCMRS, dashed line
represents estimates for PCM.
shows the response curves, exemplary for the item “Clammy hands”, for threedifferent response style parameters (columns) along the person parameter θp .
Overall, the last category has very high probabilities, whereas the first category
has very low probabilities. In particular, for a person with a rather high tendency
to the extremes (γp = −σ̂γ ), the probability for the last category is above 0.6
throughout the whole range of person parameters.
In a regular PCM, the estimated variance of the person parameters σ̂ 2 = 0.546
while the estimated covariance matrix between person and response style parameters for the PCMRS is
 2
 

σ̂θ
cov
ˆ γθ
0.449 0.263
Σ̂ =
=
.
cov
ˆ γθ
σ̂γ2
0.263 1.172

PCMRS

0.5

P( Xpi = 5 )

0.8

−0.5

0.0
θp

0.5

−0.5

0.0
θp

0.5

0.0

0.0

0.4

P( Xpi = 5 )

0.8
0.4

0.8
0.0

0.0
θp

0.5

0.8

P( Xpi = 3 )

−0.5

0.0

0.4

0.5

0.0
θp

0.4

0.5

0.0
θp

P( Xpi = 3 )

0.0
θp

−0.5

−0.5

0.0

−0.5

0.5

0.8

0.5

0.0
θp

0.8

0.0
θp

−0.5

0.4

−0.5

0.4

P( Xpi = 1 )

0.8
0.0

0.4

P( Xpi = 1 )

0.8
0.4

0.5

0.8

0.0
θp

0.0

P( Xpi = 3 )

−0.5

0.0

P( Xpi = 5 )γp = −1 σγ

γp =0

0.0

P( Xpi = 1 )γp = 1 σ γ

0.4

234

FIGURE 3. Response curves for item “Clammy hands” (FCC) along person parameter θ for categories 1, 3 and 5 and different response styles. Solid lines represent estimates for PCMRS, dashed lines represent estimates for PCM.
The variance (i.e. magnitude) of the response style effects is quite high. Therefore,
ignoring the response style in such a case would lead to severely biased item
estimates. The covariance of 0.263 between both person parameters refers to anoteworthy correlation of 0.36.

4Concluding Remarks

The proposed PCMRS model has several advantages, in particular in comparison
to mixture models as for example proposed by Eid and Rauber (2000). Mixture
models assume that respondents come from different latent classes. Different item
response models are fitted within different classes, some may represent the substantive trait, some may represent response style behaviour. One of the problems
with mixture models is always the number of classes, which is unknown. Typically
one gets quite different models if one fits, for example, two or three classes, since
all the parameters change when considering one more class. If one has chosen anumber of classes it is still difficult to interpret the difference between classes andexplain what feature is represented by a class, it might be a response style or someother dimension that is involved when responding to items. Since the classes are
not pre-specified, for example, by explicitly modelling response style behaviour,
there is much uncertainty involved and the interpretation of the model withinclasses often tends to be vague. In contrast, in the PCMRS model the explicit
modelling of the response style allows to decide if it is present, and if, how strong
it is.

References
Eid, M. and M. Rauber (2000). Detecting measurement invariance in organizational surveys. European Journal of Psychological Assessment, 16(1), 20 –30.

Schauberger et al.

235

Fahrenberg, J. (2010). Freiburg Complaint Checklist [Freiburger Beschwerdenliste (FBL)]. Göttingen, Hogrefe.
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika,
47(2), 149 – 174.
Tutz, G. and M. Berger (2016). Response styles in rating scales - simultaneousmodelling of content-related effects and the tendency to middle or extreme
categories. Journal of Educational and Behavioral Statistics, 41(3), 239 –268.

Semi-parametric ordinal regression models
for continuous scales
Maurizio Manuguerra1 , Gillian Heller1 , Jun Ma1
1

Statistics Department, Macquarie University, Sydney, Australia

E-mail for correspondence: maurizio.manuguerra@mq.edu.au
Abstract: We extend the ordinal regression model for continuous scales byintroducing non-parametric terms in the linear predictor. The model parameters
are estimated using constrained optimisation of the penalised likelihood and thepenalty parameters are automatically selected via maximisation of their marginal
likelihood. An application of the model in a study on alternative chemotherapy
treatments for advanced breast cancer is shown. The outcome of interest, thequality of life of patients, is modelled with fixed effects, smoothing terms andindividual random effects and its association with the treatment arm over thechemotherapy cycle is estimated. The methodology has been implemented in an
R package, ordinalCont, available on CRAN.
Keywords: Ordinal regression; Continuous scale; VAS.

1Introduction

Continuous self-rating scales are commonly used to evaluate the intensity of outcomes which are intangible and difficult to measure, such as pain and quality oflife. Two examples of such scales are the Visual Analogue Scale (VAS) and theLinear Analog Self-Assessment (LASA) scale, used in the pain literature and inquality of life (QoL) studies respectively. Subjects are typically given a linear
scale of 100 mm and asked to put a mark where they perceive themselves. The
relevance of continuous self-rating scales in the pain literature has been describedin Heller et al. (2016), where the frequent use of suboptimal methods in the analysis of VAS and the superior power of ordinal regression analysis (Manuguerra
and Heller, 2010) are discussed. In this paper we extend the formulation of theordinal regression model for continuous scales, including fixed effects, random
effects and smoothing terms, explain how to compute the constrained maximum
penalised likelihood (MPL) estimates of the regression coefficients and show how
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Manuguerra et al.

237

to automatically select the optimal smoothing parameters. We also show an example in which the new formulation of the model allows for a more insightful
analysis of the data.

2

Methods

2.1

Ordinal regression for continuous scores

In the ordinal regression model for continuous scales, the covariates are modelled
on a latent scale W . This is linked to the observed ordinal scale V by an increasing
function g (the “g function”) such that W = g(V ). We assume a stochastic modelfor W of the form of W (η) = h? (η) + , where h? (η) is the deterministic partof the model that depends on the covariates η, and  is a random error term.
In the following, we will assume the standard logistic distribution for , butother distributions, such as the normal, can be used. The cumulative distributionfunction for the score V can then be written as:γ(v|η) = P (W ≤ g(v)|η) = P ( ≤ g(v) − h? (η)) =where h(v, η) = g(v)−h? (η) and g(v) =Peh(v,η)1 + eh(v,η)(1)θu Ψu (v) is modelled with monotonic I-usplines (Ramsey, 1988). This guarantees an increasing g function when θu ≥ 0, ∀u.
Assuming p fixed effects, r random effects and s smoothing terms, the predictor
can be written as:
h? (η) = Xβ +r
X

Z j bj +j=1sXΦl (yl )ϑl =

l=1

T
X

Am ηm

(2)

m=1


where X T = X1T , . . . , XpT is a p × n design matrix for the fixed effects, Zj isthe design matrix for the j th random effect and Φl is the matrix containing thebasis functions (B-splines)
evaluated at 
the values of the covariate yl . Here η is
TT
the set of parameters η1T , . . . , ηtT= β T , bT , ϑT , with β T = β1 , . . . , βp ,
 T
 T
T
T
T
T
b = b1 , . . . , br , ϑ = ϑ1 , . . . , ϑs , t = p + r + s and Am is the mth elementin X1 , . . . , Xp , Z1 , . . . Zr , Φ1 , . . . Φs .

2.2

Penalised likelihood estimation

The log-likelihood for subject i is obtained by differentiating equation (1):`0i = log

m
X




θu Ψ0u (v) + h(v, η) − 2 log 1 + eh(v,η)

(3)

u=1


T
The model parameters we want to estimate are η = η1T , . . . , ηtT
and θ =
T
θ1 , . . . , θm . Their constrained MPL estimates are defined by
n
t
n
o
X
X
(η̂, θ̂) = arg max `p =`0i − λg Jg (θ) −λm Jm (ηm )
η,θ

i=1

m=1

(4)

238

Semi-parametric continuous ordinal models

where the J’s are penalty functions, such as roughness penalties, the λ’s aresmoothing parameters and the first p terms in the last sum account for penalty
terms relative to the fixed effects, and are then zero. The roughness penalties
can be written in general asRJ(θ) = θT Rθ where the R terms are square matrices
with j, k elements given by Ψ00j (y)Ψ00k (y)dy. For the random effect terms, R is aunit matrix and the penalty term is J(b) = bT Rb b = bT b, which implies normally
distributed random effects b with variance σb2 = 2λ1 b .

2.3

Estimation of model parameters

The estimation procedure iterates through two steps repeated until convergence.
First, given the current values of the λ’s, η and θ are estimated, then given
the current value of θ and η the λ’s are estimated. We comment that this is aconstrained optimisation as θ ≥ 0.
The Karush-Kuhn-Tucker (KKT) necessary conditions for the constrained MPL
estimation of θ and η are:∂`p
= 0 for any m
∂ηm
∂`p
∂`p
= 0 if θu > 0 and< 0 if θu = 0
∂θu
∂θu

(5)

Equations (5) are solved iteratively, with the unconstrained parameters η estimated using a Newton method and the positively constrained parameters θestimated using the multiplicative iterative (MI) algorithm (Ma, 2010).

η: Newton step Given the estimated values of η and θ at iteration k, thevalues of each ηm ∈ η at iteration k + 1 are obtained with the Newton algorithm:


−1 
(k+1)
(k)
ηm
= ηm
+ ω (k) ∇2ηm `p (η (k) , θ(k) )∇ηm `p (η (k) , θ(k) )(6)where ω (k) ∈ (0, 1] is a line search step size that guarantees that `p (η (k+1) , θ(k) ) ≥`p (η (k) , θ(k) ).
The first and the second derivative of the penalised log-likelihood (4) with respectto η are:∇ηm `p = ATm 1n − 2γ(v) − 2λm Rm ηm

∇2ηm ηn `p = ATm DAm − 2δm,n λm Rm
h

(7)

e
where D is the diagonal matrix with diag(D) = −2 (1+eh )2 and δm,n = 1 if m = n,
and 0 otherwise.

Manuguerra et al.

239

θ: MI algorithm Given the estimated values of η at iteration k + 1 and θ atiteration k, the values of θ at iteration k + 1 are obtained with the MI algorithm.
When θ > 0, by the second KKT condition (5):∇θ `p = ΨT 1n − 2γ(v) + Ψ0T E − 2λθ Rθ θ = 0(8)where E is a diagonal matrix with diag(E) = g01(v) and 0 otherwise.,
Following Ma (2010), it is easy to derive the updating algorithm:(k)θ(k+1) = θ(k) + ωθ s(k) ∇θ `p (η (k+1) , θ(k) )

(9)

(k)

θ
and a line search of size ω (k) ∈ (0, 1] is introduced
where s(k) = 2ΨT γ(v)+2λ[R θ]+
θθ

to guarantee that `p (η (k+1) , θ(k+1) ) ≥ `p (η (k+1) , θ(k) ).

2.4

Automatic smoothing parameter estimation

Fixing the values of η to the most current estimate, the optimal smoothing parameters are the roots of the partial derivatives with respect to each λm of themarginal posterior computed integrating out θ and η from (4). The log-posterior
can be written as:

Φ=

n
X

`0i −

i=1

p

θ

2

log(σθ2 ) +t
1 T  X  pm
2
T
θ
Rθ
−
log(σm) + λ m ηm
Am ηm (10)
2
2σθ
2
m=1

and the marginal log-posterior is obtained integrating out θ and η:ZΦmarg = log
exp Φ(θ, η; σθ2 , ση2 )dθdη

(11)

Exact solution to this interval is infeasible but it can be approximated following
Laplace:Φmarg = −

t
X
pθ
pm
1
2
log(σθ2 )−log(σm)+`p (θ̂, η̂; σθ2 , ση2 )− log −Q(θ̂, η̂; σθ2 , ση2 )
2
2
2
m=1
(12)

where Q(θ, η; σθ2 , ση2 ) = ∇2θη `p (θ, η; σθ2 , ση2 ). Solving∂Φmarg=0∂σθ2∂Φmarg=0∂ση2mfor σθ2 and ση2m we obtain the desired smoothing parameters estimates:

(13)
(14)

240

3

Semi-parametric continuous ordinal models

σθ2 =


θT Rθ θPpθ − tr (G − Qθ − Qm )−1 Qθ

(15)

2
σm
=


T
ηm
Rm ηm

P
pm − tr (G − Qθ − Qm )−1 Qm(16)

ExampleMetastatic breast cancer is the most common cause of cancer death among Australian women. The ANZ0001 trial is a randomized trial with three chemotherapy

Better QoL

2
−1

0

1

Worse QoL

Effect of treatments over chemotherapy cycles
Intermittent Capecitabine
Continuous CapecitabineStandard CMF

0

10

20

30

40

50

60

Chemotherapy cycle

FIGURE 1. Effects of treatments over chemotherapy cycles
treatment arms (n = 292 patients with complete quality of life measurements)
concluded in 2005 (Stockler et al., 2011). QoL is assessed at each chemotherapy
treatment cycle on a LASA scale. The treatments Intermittent Capecitabine (IC)and Continuous Capecitabine (CC) are compared with the standard combination
treatment CMF, each with its own protocol. There is no maximum duration oftreatment, but it is interrupted on disease progression, or when patient intolerance or unacceptable toxicity are recorded. In this analysis we aim to verify which
treatment has a better impact on QoL, and in particular how this impact changes
over chemotherapy cycle. The research questions can be answered by modellingthe overall QoL vij for patient i at chemotherapy cycle j as:
log

γij= g(vij ) + xi β + s(j|ti )1 − γij

where xi is the age of patient i, s(j|ti ) is a smooth term that depends on cycle
number j and the treatment arm ti . In Figure 1, the effects of the three treatments
over chemotherapy cycles are shown. The CC treatment has no clear advantage
over the CMF treatment in terms of QoL, but can be sustained for longer periods,

Manuguerra et al.

241

while the IC treatment has a positive effect on the QoL, with a peak after about
30 cycles (higher values mean higher quality of life).
All the analyses have been performed in the R Statistical Software (R Core Team,
2016) using the package ordinalCont (Manuguerra and Heller, 2016). The data
set used in the analysis is included in the package.

References
Heller, G.Z., Manuguerra, M., and Chow, R. (2016). How to analyze the Visual
Analogue Scale: Myths, truths and clinical relevance. Scandinavian Journalof Pain, 13, 67 – 75.
Ma, J. (2010) Positively constrained multiplicative iterative algorithm for maximum penalized likelihood tomographic reconstruction. IEEE trans. on
Nucl. Sci., 57(1), 181 – 192.
Manuguerra, M. and Heller, G.Z. (2010). Ordinal Regression Models for Continuous Scales. The International Journal of Biostatistics, 6(1).
Manuguerra, M., and Heller, G.Z. (2016). ordinalCont: Ordinal Regression Analysis for Continuous Scales. R package version 1.0.2, URL https://cran.rproject.org/web/packages/ordinalCont.
R Core Team (2016). R: A language and environment for statistical computing.
R Foundation for Statistical Computing, Vienna, Austria. URL https://www.Rproject.org/.
Ramsey, J.O. (1988) Monotone Regression Splines in Action. Statistical Science,
3(4), 425 – 441.
Stockler, Martin R et al. (2011). Capecitabine versus classical cyclophosphamide,
methotrexate, and fluorouracil as first-line chemotherapy for advanced breast
cancer. J of Clinical Oncology,29, 34, 4498 – 4504.

Modeling agreement on continuousrecordings in the presence of a binary scale
Sophie Vanbelle1 , Emmanuel Lesaffre2
1

Department of Methodology and Statistics, CAPHRI, Maastricht University,
The Netherlands
2
Leuven Biostatistics and Statistical Bioinformatics Centre, Belgium
E-mail for correspondence: sophie.vanbelle@maastrichtuniversity.nl
Abstract: Recording variables continuously over time is increasingly frequent
in medical and behavioral sciences. Indeed, the use of video and portable devices
permits to study individuals in their natural environment and provide real-timeassessments. The measurement instruments developed for ambulatory use andthe coding schemes used to analyze video recordings have to show reliability andvalidity before daily practice implementation. Studies using continuous records
are usually characterized by a large amount of data presenting serial correlation
and a small sample of subjects. We developed a method to derive appropriate
reliability and agreement indexes to analyse this type of data in the presence ofa covariate structure (including time as component) when records are made on abinary scale. The method is illustrated on a validation study for a new single-unit
activity monitor (CAM) in patients with chronic organ failure.
Keywords: Bayesian methods, transient event, time-event sequential data, reliability, e-Health.

1

Introduction

Continuous recordings, defined as second by second or even closer records intime, are frequent in medical, behavioral and social sciences. Indeed, complex
behaviors can be quantified by observers on video recordings or data can becollected with the use of handled portable electronic data-entry and storage devices (e.g. e-Health). In both cases, the reliability and the validity of the coding
scheme/measurement instrument have to be assessed before using the continuousrecordings in daily practice.
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Vanbelle and Lesaffre

243

For example, the following study was designed to validate a new single-unit activity monitor (CAM) in patients with chronic organ failure (Annegarn et al.,2011). This new monitor was developed to implement daily physical activity as
an outcome measure of cardiopulmonary rehabilitation. In the study, the activity (non-weight bearing posture (NWBP), weight-bearing posture (WBP) or
dynamic activity (DA)) of 10 patients in rehabilitation was recorded during one
hour of daily routine by the CAM. In the present paper, we restrict attention
on the distinction between non-weighted bearing postures (NWBP) and the twoother activities (WPB+DA). The CAM was wore at two different places of the
body simultaneously for comparative purposes, namely on the leg and on thetrunk. The patients were also videotaped by a researcher during the one hour
time period. The video, considered as criterion standard, was then analyzed second by second by a researcher blinded to the values obtained by the CAM. The
aim of the study was to determine (1) the agreement level between the video
assessments and the CAM, wore on the leg or on the trunk and (2) the bodyplace providing the highest agreement level with the video.
The existing population-based and unit-specific models developed to determine
agreement in the presence of a covariate structure and repeated measurements
have to be ruled out to analyze the present study. Population-based models are
not practicable. The large amount of repeated measurements (3600 in the CAM
study) and the small number of subjects involved (10 in the CAM study) do not
permit to obtain stable parameter estimates. In the same way, the only unitspecific approach permitting to directly model agreement according to predictors
in the presence of repeated measurements (Vanbelle and Lesaffre, 2015) is not
manageable because of the large amount of observations. We therefore developed
a partial-Bayesian approach, based on this latter work, to model agreement obtained on continuous recordings over time in the presence of a covariate structure.

2

MethodInstead of considering each time point separately, we group them into small timeintervals. The cumulative distribution of the outcome over one interval (i.e. number of time points under NWBP) is binomial given three assumptions: (1) theintervals have fixed length, (2) the probability of being engaged in NWBP is constant within each time interval and (3) the observations within a time interval
are independent. The first assumption holds by definition of the intervals. We
assume that the second assumption holds because the time intervals are small.
The third assumption does not hold as observations close in time are very likely
to be correlated. To account for this correlation, we assumed that the cumulative
distribution of the outcome within each interval follows a beta-binomial distribution. We make the same assumption for the cumulative distribution of agreement
between the CAM and video assessments within each interval.
Then, based earlier work, we consider the likelihood formed by the cumulative
distribution of the probability to observe NWBP with the CAM and the video
and the cumulative distribution of the agreement. Models relating these cumulative distributions to covariates (namely time, body place) are then fitted jointlywith a partial-Bayesian approach using Markov chain Monte Carlo (MCMC).
The MCMC calculations were performed using JAGS. Vague independent priors

244

Modeling agreement on continuous recordings in the presence of a binary scale

were considered for all model parameters. The dependency between the probabilities over the one hour observation period was taken into account using random
effects. Marginal posterior probability distributions were obtained by averaging
over these random effects. The length of the time intervals was fixed to 30 seconds.

3

Results

The probability to be engaged in a NWBP is displayed in Figure 1 over the
one hour observation period. It can be remarked that the probability of being in
NWBP varies over time and differs markedly between CAM-trunk on one hand
and CAM-Leg and video on the other hand.

FIGURE 1. CAM study. Probability of being in a NWBP in the sample (lightgray). Data were summarized every 5 seconds for the clarity of the graph. Posterior marginal probability of being in a NWBP using a binomial (gray) andbeta-binomial (black) model. The plain line corresponds to the video, the dottedline to the CAM-leg and the dashed line to the CAM-trunk
The evolution of the agreement level between the CAM and the video, quantified
through Cohen’s kappa coefficient, is depicted in Figure 2 over the one hourobservation period. Two important results have to be noted. First, the agreementlevel between the CAM and the video decreases over time. This is an indicationof a possible inappropriate observation period. The observation period was either
too short to obtain stable agreement estimates or too long for the researcher toavoid fatigue effects when rating the video recordings. Secondly, the agreementlevel is higher between the CAM-leg and the video than between the CAM-trunk
and the video. The new CAM device should therefore best be placed on the leg
than on the trunk to detect non-weight bearing postures.

4

Discussion

We developed a method to model the agreement in the presence of continuous recordings and a covariate structure. To this end, we summarized the data
into small time intervals. The difference between the results obtained with thebinomial and the beta-binomial models supports the need to take the dependency between observations within time intervals into account. Taking the average probability over a time interval corresponds to temporal aggregation, a kindof smoothing technique used in time series analysis. We should be careful about

Vanbelle and Lesaffre

245

FIGURE 2. CAM study. Agreement (Cohen’s kappa coefficient) on NWBP in
the sample (light gray). Data were summarized every 5 seconds for the clarity of
the graph. Posterior marginal mean for the agreement level between CAM-Legand video (dashed line) and between CAM-trunk and video (plain line) using abinomial (gray) and beta-binomial (black) models.

the choice of the interval length not to hide important patterns in the evolutionof the outcome and the agreement over time. The dependency between time intervals is currently taken into account using a random intercept in the models.
However, the use of autoregressive models is currently under study.

Acknowledgments: This research is part of project 451-13-002 funded by the
Netherlands Organisation for Scientific Research.

References
Annegarn, J., Spruit, M.A., Uszko-Lencer, N.H.M.K., Vanbelle, S., Savelberg,
H.H.C.M., Schols, A.M.W.J., Wouters, E.F.M. and Meijer, K. (2011). Objective Physical Activity Assessment in Patients With Chronic Organ Failure: A Validation Study of a New Single-Unit Activity Monitor. Archivesof Physical Medicine and Rehabilitation, 92, 1852 – 1857.e1.
Vanbelle, S. and Lesaffre, E. (2015). Modeling agreement on categorical scales
in the presence of random scorers. Biostatistics, 17, 79 – 93.

Bayesian Inference for Mixed Effects
Multinomial Logit Models
Helga Wagner1 , Sylvia Frühwirth-Schnatter2
1
2

Johannes Kepler University Linz, Austria
Vienna University of Economics and Business, Austria

E-mail for correspondence: helga.wagner@jku.at
Abstract: We consider Bayesian inference for very flexible multinomial logit
models with choice and unit specific covariates where we allow for global as wellas category specific fixed and random effects. To achieve model identification we
use a sum-to-zero constraint on the category specific fixed and random coefficients. In contrast to the conventional identification strategy where one of theoutcome categories is chosen as baseline and the corresponding category specificparameters are fixed to zero, the sum-to-zero constraint allows for a symmetricprior of the category specific effects of all outcome categories. By augmenting themodel with latent utilities posterior inference using MCMC methods is straightforward. In our empirical analysis we use panel data from EU-SILC in Austria
2004 – 2007 to analyse labor supply of households. Inference is particularly challenging as due to the fact that data are collected using a rotating panel design,
panels are very short with a maximum of 4 observations per household.
Keywords: random utility model, sum-to-zero identification, discrete choicemodel, longitudinal data, EU-SILC

1

Mixed Effects Multinomial Logit Model

We consider modelling of longitudinal discrete choice data using covariate information. Covariates can be either choice or subject specific and we allow for
choice-specific effects of subject-specific covariates. Additionally, to model bothheterogeneity between subjects as well as dependence within subjects we include
a subject-specific random intercept and allow for random effects of choice specificcovariates. We assume that each subject i, i = 1, . . . , N chooses one of m + 1 unordered alternatives {0, . . . , m} at occasion t = 1, . . . , Ti and denote this choice
by Yit . The probability that alternative k is chosen by subject i at occasion t is
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Wagner and Frühwirth-Schnatter247

then modelled by a multinomial logit model asexp(xfitk α + xit βk + xritk ζi + ξik )P(yit = k) = Pmexp(xfitl α + xit βl + xritl ζi + ξil )k = 0, . . . , m

(1)

l=0

Here xit denotes the vector of subject specific covariates with choice specificeffects βk ; α is the vector of fixed effects of the choice specific covariates xfitk andζi the vector of random effects of the choice specific covariates xritk . Finally ξik
is the category specific random intercept of subject i.
As the same choice probabilities would result for choice specific effects βk + β̃˜
and subject specific random intercepts (ξik + ξ),˜
exp(xfitk α + xit (βk + β̃) + xritk ζi + (ξik + ξ))P(yit = k) = P,m˜
exp(xfitl α + xit (βl + β̃) + xritl ζi + (ξil + ξ))
l=0

the model is not identified. To achieve identification the usual strategy is tochoose one of the categories, e.g. category 0 as reference category and set thecorresponding category specific effects to zero, i.e.
β0 = 0andξi0 = 0.

However this identification strategy causes problems in the interpretation of theestimated subject- and category-specific random intercepts ξik . Though no random intercept is explicitely included for the reference category 0, this is implicitely
the case as the probability that subject i chooses category 0 depends on all othercategory-specific random intercepts ξik , k = 1, . . . , m due to the fact thatmX

P(yit = k) = 1.

k=0

An alternative identification strategy, which is symmetric with respect to all
categories, is to impose a sum-to zero constraint on the category-specific effects(Burgette and Hahn, 2013),mX

βk = 0andmX

ξik = 0k=0

k=0

In a Bayesian approach this constraint can be imposed by specifying a reduced
rank Normal prior distribution on the corresponding effects, where the sum tozero constraint holds a priori with probability 1.

2

Prior Distributions

To specify the prior on the fixed effects, we define the vector of the category
specific regression coeffcients for each covariate j as β j = (β0,j , . . . , βm,j )T and
let em+1 = (1, 1 . . . , 1)T . We assume that the vectors β j are a priori independentfor j = 1, . . . , d. The sum-to-zero constraint on β j given asmX
k=0

βk,j = eTm+1 β j = 0

(2)

248

Mixed Effects Multinomial Logit Models

is imposed by specifying a reduced rank Normal prior
β j ∼ N (bj , δjβ Cm ),where the mean vector bj is restricted to sum to zeroeTm+1 bj = 0.
δjβ is a scale factor and the matrix Cm has a compound symmetry structurem+1
1
Im+1 − em+1 eTm+1 =
m
m


1
1
1
−m
... −m
 1

1 
−
1
... −m
 m


=
 ..
..
..  .
.
.
 .

.
.
.


1
1
−m −m ...
1

Cm =With this specification the sum to zero constraint (2) holds a priori with probability 1, as E(eTm+1 · β j ) = eTm+1 · bj = 0 andV(eTm+1 · β j ) = δjβ eT
m+1 Cm em+1 = O.
For the subject-specific vectors of random intercept ξi = (ξi0 , ξi1 , . . . , ξim )T we
assume prior independence across subjects and specify the prior hierarchically as
ξi ∼ N (γ, δ ξ Cm )γ ∼ N (0, δ γ Cm )δ ξ ∼ G −1 (ν, Q)
Note, that the specification of this reduced rank prior yields a sparse parameterisation of the covariance matrix of the category specific random intercepts where
only one parameter, the scale parameter δ ξ has to be estimated from the data.
Finally, for the global fixed and random effects we specify standard multivariate
Normal priors.

3

Posterior Inference

Posterior inference is performed by MCMC methods, where we use the representation of the model in terms of random latent utilities, first introduced by
McFadden (1974). Let uitk denote the latent utility of category k for subject i attimepoint t
uitk = xfitk α + xit βk + xritk ζi + ξik + itk ,k = 0, . . . , mwhere itk are i.i.d extreme value distributed random variables. If yit is definedas the category with maximum utility
yit = k ⇔ uitk = max uitllthe marginal distribution of yit is given by the multinomial logit model in equation(1). MCMC iterates between sampling the latent utilities and the effects in themultinomial logit model, for which we use the data augmented independence MH
sampler with a multivariate proposal density described in Fussl et al. (2013).

Wagner and Frühwirth-Schnatter

4249

Application

The goal of our empirical analysis is to provide a microsimulation model for
labour supply of couples with children in Austria, which allows to evaluate theconsequences of interventions like changes in tax system, social security contributions or social transfers (e.g. parental leave benefits) on labour supply of thehouseholds. We use a unitary model with discrete choices, i.e. we assume thatchoices on labour supply are made on the unit level of the household.

1200
1000
800
600
0

200

400

600
0

200

400

Frequency

800

1000

1200

For the analysis we use data from EU-SILC (European Survey on Income andLiving Conditions) from 2003 – 2006. EU-SILC has a four year rotating panel
design and provides longitudinal microdata on income, poverty, housing, laborand living conditions. The data for our analysis comprises 4218 observations of2289 households which are observed in at least one but not more than four years.
Figure 1 shows the labour supply (in working hours) of men and women.

0

20

40

60

Working hours (males)

80

100

0

10

20

30

40

50

Working hours (females)

FIGURE 1. Working hours of males (left) and females (right) in couple househoulds with children (in the first panel year).
We exclude non-working men from the analysis and – as the vast majority of men
is working full-time – define three choices with respect to the labour supply of
women as not working (N), working part-time (P), and working full time (F). A
challenging feature of the data is their structure as an unbalanced panel with 1079
households observed only at one time point and only 167 households observedin all four panel time points. Additionally only one choice is observed for the
majority of households (1936 of 2289), whereas all three possible labour supply
choices were observed in only 12 households. For the three possible labour supply
choices we fit a multinomial logit model where we include household income andhousehold leisure time, both measured on the log-scale, as choice-specific covariates with fixed effects. Additionally we include the following global covariates:
indicators for age of the youngest child (categorized; baseline: 0-3; 3-5, 6-10; over10) and the number of further children in each of these age groups and model
their effects as category-specific. To account for heterogeneity of households withrespect to their labour supply choice we include a household specific random
intercept.

4
2

ξi,2
0
−2
−4

−4

−2

0

ξi,1

2

4

6

Mixed Effects Multinomial Logit Models

6

250

−4

−2

0

2

4

6

−4

−2

ξi,0

0

2

4

6

ξi,1

FIGURE 2. Random effects for groups defined by observed working status of
mothers (not working (N)/working part-time (P)/ working full-time (F); blue:
N; green: P; red: F; magenta:N/P; yellow: N/F; cyan: P/F; black: N/P/F)

As expected, the global effect of household income is positive, thus increasing
the utility of part-time as well as full time work , and the effect of leisure timeis negative. The utilities for both part-time as well as full-time work of mothers
increase with the age of the youngest child and decrease with the number ofchildren. Heterogeneity among mothers is considerable with an estimated random
intercept variance of 4.8. Figure 2 shows bivariate plots of the random intercepts,
in different colors for clusters of households defined by the observed labour supply,
e.g. for households observed only with the mother non-working (shown in blue)
the mean random intercept is positive for not working (N), roughly zero for parttime working (P) and negative for full-time working (F).
Predicted household choices, which are determined as the category with the highest mean posterior probability, are compared to the observed choices in the sample
in Table 1. Results indicate good insample predictive performance of the modelwith 92.2% choices predicted correctly.
TABLE 1. Comparison of predicted to observed choices in the sample

predicted choiceactual choicenon-working part-timefull-timeSum

non-working

1711

106

46

1863

part-time

71

1436

66

1573

full-time

12

31

739

781

1794

1573

851

4218

Sum

Wagner and Frühwirth-Schnatter

251

References
Burgette, L. and Hahn, P. R. (2013). A symmetric prior for multinomial probit
models. Technical Report.
Fussl, A. et al. (2013). Efficient MCMC for Binomial Logit Models. ACM Transactions on Modeling and Computer Simulation. 23, Article 3, 121.
Frühwirth-Schnatter S. and Frühwirth R. (2010). Data augmentation and MCMC
for binary and multinomial logit models. In: Kneib T. and Tutz G. (Eds..),Statistical Modelling and Regression Structures Festschrift in Honour of
Ludwig Fahrmeir. Heidelberg: Physica-Verlag, 111-132.
Mc Fadden, D. (1981). Econometric models of probabilistic choice. In: C. F.
Manski and D. McFadden (Eds.), Structural Analysis of Discrete Datawith Econometric Applications, Cambridge MA: MIT Press, 198 – 272.

Simple Effect Measures for Interpreting
Models for Ordinal Categorical Data
Alan Agresti1 , Claudia Tarantola2
1
2

University of Florida, USA
University of Pavia, Italy

E-mail for correspondence: aa@stat.ufl.edu
Abstract: We survey effect measures for models for ordinal categorical datathat can be simpler to interpret that the model parameters. For describing theeffect of an explanatory variable while adjusting for other explanatory variables,
we present probability-based measures, including a measure of relative size andeffect measures based on an instantaneous rate of change. We also survey summary measures of predictive power that are analogs of R-squared and multiple
correlation measures for quantitative response variables. We suggest new measures of effect and of predictive power. In a longer companion paper availableat www.stat.ufl.edu/~aa/articles/agresti_tarantola.pdf, we illustrate themeasures for an example and provide R code for calculating them.
Keywords: Cumulative link models; Cumulative logits; Marginal effects; Proportional odds; R-squared

1

Introduction

Popular models for ordinal categorical response variables are generalized linear
models that employ non-linear link functions. As a consequence, the model effectparameters relate to measures, such as odds ratios and probits, that are not easily
understood by non-statisticians. This article surveys simpler ways to interpret theeffects of explanatory variables and to summarize the model’s predictive power.
Section 2 presents alternative ways to summmarize the effect of an explanatory
variable. These include simple comparisons of the probability of extreme-response
outcomes at extreme values of an explanatory variable, measures of instantaneous
effect on the extreme-response probabilities, and measures for comparing groups
that result directly from latent variable models that induce the standard ordinalmodels. Section 3 surveys measures of predictive power, including measures that
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Agresti and Tarantola

253

resemble those for ordinary linear models, possibly estimated for a latent variable model. We also propose a multiple correlation measure that generalizes theSpearman correlation. A companion paper (Agresti and Tarantola 2017) availableat www.stat.ufl.edu/~aa/articles/agresti_tarantola.pdf illustrates existing and newly-proposed measures with an example, gives further details andreferences, and provides R code for the analyses.

2

Ordinal Effect Measures for Individual Explanatory
Variables

For an ordinal response variable y with c categories, we consider models in which
the explanatory variables may be a mixture of quantitative and categorical variables. We denote explanatory variable values by x = (x1 , . . . , xp )T . To discuss
ways of summarizing effects for a categorical explanatory variable, we refer also
to a separate indicator variable z that distinguishes between two groups. The
most popular models are special cases of the cumulative link model
link[P (y ≤ j)] = αj − βz − β1 x1 − · · · − βp xp , j = 1, . . . , c − 1,(1)for link functions such as the logit and probit. The nonlinear link function naturally produces effects on the link scale. For example, for cumulative logit models
with this proportional odds structure, β is the difference between logits of cumulative probabilities for the two groups, and −β1 is the change in the cumulative
logit per each 1-unit increase in x1 , adjusting for the other explanatory variables.
This leads to odds ratios as natural effect measures. For instance, exp(−β1 ) isa multiplicative effect of each 1-unit increase in x1 on the cumulative odds of
response ≤ j vs. > j, for each j.
Such effect measures are not easy to interpret by scientists who need to understand the effects in more real-world terms. In addition, with nonlinear linkfunctions, effects often behave in a way that is counterintuitive to those who aremainly familiar with ordinary linear models. In logistic regression models, for example, if an explanatory variable is added to the model that is uncorrelated with
x1 , the partial effect of x1 is typically different than in the model without theother explanatory variable, whereas it would be identical in an ordinary linear
model fitted by least squares. Probability-based effects summarized in this paper
are easier to understand and are typically more stable.

2.1

Extreme-category range-based probability summaries

In practice with ordinal responses, the highest and lowest response categories
often represent a noteworthy state, such as the best or worst outcome. It is
informative to report how P (y = 1) and P (y = c) change as explanatory variables change. As any explanatory variable increases, cumulative link models that
contain solely main effects imply monotonicity in these probabilities. A simple
summary of the effect of an explanatory variable reports the way that estimatesof P (y = 1) and P (y = c) change over the range of values of that explanatory
variable, when other explanatory variables are set at particular values such as
their means.

254

Effect Measures for Ordinal Categorical Data

2.2

Marginal effect measures

A second type of simple summary for ordinal responses considers the rate ofchange in P (y = 1) and P (y = c), as a function of the explanatory variable. We
express the model (1) as
F −1 [P (y ≤ j)] = αj − βz − xT β, j = 1, . . . , c − 1,

(2)

−1

where Fis the inverse of a standard cdf, x is a column vector of explanatory
variable values (excluding z), and β is a column vector of parameters for x. Let
f (y) = ∂F (y)/∂y.
For a quantitative explanatory variable xk , the rate of change in P (y = 1) ata particular value of xk , when other explanatory variables are fixed at certain
values x∗ , is ∂P (y = 1|x = x∗ )/∂xk . Some books, such as Long and Freese (2014),
refer to such an instantaneous effect as a marginal effect. For the cumulative linkmodel, the marginal effect of xk is −f (αj − βz − xT β)βk on P (y = 1) andf (αj − βz − xT β)βk on P (y = c). For the n sample observations, we can find themarginal effect of xk at each of the n observed values of the explanatory variables,
and then average them. This is called an average marginal effect (AME). For
categorical explanatory variables, we can find the difference between P (y = 1)
(and/or P (y = c)) when z = 1 and when z = 0, for the n sample observations on
x, and average the obtained values.
For binary data, Mood (2010) pointed out that such measures have behavior
reminiscent of effects in ordinary linear models. For example, they are roughly
stable when we add an explanatory variable to the model that is uncorrelatedwith the variable for which we are describing the effect.

2.3

A probability summary for ordered comparison of groups

We next present an alternative way to summarize the effect of a categorical explanatory variable on an ordinal response y, suggested by Agresti and Kateri(2017). We discuss this in the context of comparing two groups (z = 0 andz = 1). We regard the ordinal response as crude measurement of an underlying
continuous latent variable y ∗ that is the response variable in an ordinary linear
model. Anderson and Philips (1981) showed that the cumulative link model (2)is implied by a model in which a latent response has conditional distribution
with standard cdf given by the inverse of the link function. Let y1∗ and y2∗ denoteindependent underlying latent variables for y, representing the underlying distributions when z = 1 and when z = 0 respectively. At a particular setting x forother explanatory variables, P (y1∗ > y2∗ ; x) is a summary measure of relative size.
This measure is most meaningful when the groups are stochastically ordered.
The normal latent variable model with y ∗ ∼ N (βz + β1 x1 + · · · + βp xp , 1) implies
model (2) with probit link. For this model,
 ∗



(y1 − y2∗ ) − ββ−β√P (y1∗ > y2∗ ; x) = P> √=Φ √ .
(3)
2
2
2
This is true regardless of the x value, so we denote it by P (y1∗ > y2∗ ). For thelogit link, Agresti and Kateri (2017) showed that√exp(β/ 2)∗∗√ ,P (y1 > y2 ) ≈(4)[1 + exp(β/ 2)]

Agresti and Tarantola

255

for the β coefficient of z in the cumulative logit model. For a log-log link, whichis relevant when we expect underlying latent variables to have extreme-value
distributions, Agresti and Kateri noted that
P (y1∗ > y2∗ ) =exp(β),[1 + exp(β)]

for the β coefficient of z in the cumulative link model with log-log link. Ordinary
confidence intervals for the β model parameter induce confidence intervals for thestochastic superiority measure.
Agresti and Kateri suggested that practitioners can more easily interpret P (y1∗ >y2∗ ) than parameters such as odds ratios and differences in probits that naturally
result in cumulative link models. They also proposed related measures for theobserved y scale that need not relate to latent variables.

3

Summary Measures of Predictive Power

Next we discuss ways to summarize how well we can predict y using the fit of
the chosen ordinal model. Measures of predictive power can be useful for comparing different models, such as to see whether it helps substantively to add aninteraction term.

3.1

Concordance index

Consider all pairs of observations that have different outcomes on y. The concordance index estimates the probability that the predictions and the outcomesare concordant, that is, that the observation with the larger y-value also has astochastically higher set of model-fitted probabilities. Appealing features of theconcordance index are its simple structure and its generality of potential application. Because it utilizes ranking information only, however, it cannot distinguishbetween different link functions or linear predictors that yield the same stochastic
orderings.

3.2

R-squared type measures

An alternative approach to summarizing predictive power adapts standard measures for quantitative response variables. A way to construct such a measure
without assigning arbitrary scores to the categories of y is to estimate R2 for the
linear model for an underlying latent response variable. McKelvey and Zavoina(1975) suggested this measure for the cumulative probit model, for which theunderlying latent variable model is the ordinary normal linear model. Let yi∗ denote the value of the latent variable for subject i. The R2 measure for the latentvariable model has the usual proportional reduction in variation form
P ∗
P
P ∗
∗ 2
ȳ ∗ )2 − i (yi∗ − ŷi∗ )2i (yi −Pi (ŷi − ȳ )
P
R2 =
=
,
∗
∗
∗ 2
∗ 2
i (yi − ȳ )i (yi − ȳ )the estimated variance of ŷ ∗ divided by the estimated variance of y ∗ . After fitting
a cumulative link model we can estimate the variance of ŷ ∗ by the variance of256

Effect Measures for Ordinal Categorical Datathe linear predictor ŷ ∗ = β̂z + β̂1 x1 + · · · + β̂p xp without the intercept. We
cannot observe the latent variable or its sample variance, but we can estimatethat variance by the estimated variance of ŷ ∗ plus the variance of the latentvariable distribution, which is 1 for the probit link and π 2 /3 = 3.29 for the logit
link (i.e., standard logistic distribution).
An alternative proportional-reduction-in-variability approach uses alikelihood-based measure. Denote the maximized log-likelihood values by LM forthe working model fit and L0 for the null model (i.e., containing only intercept
terms). The pseudo R-squared measure is
1−

LM
.
L0

A weakness of such a measure is that the scale is not the same as for y. Interpreting the numerical value is difficult, other than in a comparative sense for different
models.

3.3

Multiple correlation measures

Some statisticians prefer correlation measures over related R2 measures, becauseof the appeal of working on the original scale and its proportionality to the effectsize. For example, for the ordinary linear model, for fixed marginal standard
deviations, doubling the slope also doubles the correlation.
We could estimate the multiple correlation for an underlying latent variable
model, such as by taking the square root of the McKelvey and Zavoina (1975)
R2 measure. Another way to form a multiple correlation measure that connectswith models for cumulative probabilities uses as scores the average cumulative
proportions for the marginal distribution of y. For sample proportions {pj } inthat marginal distribution, the average cumulative proportion in category j is
vj =

j−1
X
k=1

pk +

 
1
pj ,
2

j = 1, 2, . . . , c.

Such scores, which are linearly related to the midranks, are sometimes referred
to as ridits. With such created scores, one could construct the correlation for
the n sample observations between the score for the observed outcome category
for a subject and the estimated mean score generated by the model-fitted probability values for the subject. With ridit scores, this is a multiple correlation
version of the Spearman correlation. In future research, it would be of interest
to study properties of such a measure, including bias reduction in estimating the
population analog.

References
Agresti, A., and Kateri, K. (2017). Ordinal probability effect measures for groupcomparisons in multinomial cumulative link models. Biometrics, to appear.
Agresti, A., and Tarantola, C. (2017). Simple effect measures for interpretingmodels for ordinal categorical data. Submitted for publication.

Agresti and Tarantola

257

Anderson, J. A., and Philips, P. R. (1981) Regression, discrimination, and measurement models for ordered categorical variables. Applied Statistics, 30,
22–31.
Long, J.S. and Freese, J. (2014). Regression Models for Categorical Dependent
Variables Using Stata, 3rd ed. College Station, TX: Stata Press.
McKelvey, R.D. and Zavoina, W. (1975). A statistical model for the analysis ofordinal level dependent variables. J. Math. Sociol. 4: 103–120.
Mood, C. (2010). Logistic regression: Why we cannot do what we think we cando, and what we can do about it. Eur. Sociol. Rev. 26: 67–82.

The truth about the effective dimension
Paul H. C. Eilers1
1

Erasmus MC, Rotterdam, the Netherlands

E-mail for correspondence: p.eilers@erasmusmc.nl
Abstract: The hat matrix of a model is valuable. Its trace gives the effective
dimension. In mixed models partial effective dimensions can be defined that makevariance estimation easy and reliable. Efron (2004) casts doubt on the trace of thematrix and advocates an alternative definition of the effective dimension, based ona covariance formula. Unfortunately he uses the robust lowess smoother, thereby
blurring the issue.
Keywords: Hat matrix; Penalty; Mixed model; Robustness

1

Introduction

In a linear model we can write ŷ = Hy, where y is an observed vector and H is theso-called hat matrix (Ye, 1998). It depends only on the explanatory variables andpenalties,
P if present. The diagonal of H provides the effective (model) dimension:
ED = i hii . The situation seems clear cut, but one can find several publications
that cast doubt on this definition, e.g Efron (2004) and Janson et al. (2015). HereI want to show that H deserves its place. Also I show that in mixed models we
can define partial hat matrices and partial effective dimensions. The latter playa central role in the estimation of variances, as already pointed out by Harville(1972).
Efron uses the robust lowess smoother, which has a redescending influence function, effectively making the model strongly non-linear and generating worrying
results. A linear smoother, like locfit or P-splines eliminates all issues.
The effective dimension is a property of a model, when fitted to data, not of thedata themselves. I illustrate that with the Whittaker smoother, varying the orderof the differences in the penalty.
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Eilers

2

Theory and examples

2.1

The hat matrix

259

Consider a mixed model with fixed and random components: y = Xβ + Zc + e,
with c ∼ N (0, G) and e ∼ N (0, R). It is well known that one finds estimates for
β and c by solving the system  0 −1 
 0 −1X̂ R yXR X
X 0 R−1 Z
β̂
.
(1)
=
0 −1
0 −1
−1
Z R X Z R Z +GĉẐ 0 R−1 y
In many applications one can assume that R = σ 2 I1 and G = τ 2 I2 , with unknown
σ and τ , and proper sizes of the identity matrices I1 and I2 . Then one can write
 0



  0   0 
XX
X 0Z
X̂ y
X
β̂
β̂
=S
y,
(2)
=
=
Z 0 X Z 0 Z + λI
Z0ĉĉẐ 0 ywith λ = σ 2 /τ 2 . Because ŷ = [X Z][β̂ 0 ĉ]0 , the hat matrix is
 0 


X
.
H = X Z S −1
Z0
Another useful matrix is

K11
K=
K21

K12
K22


=S

−1



X 0X
Z0X

X 0Z
Z0Z

(3)


.

(4)

It is easy to show that tr(H) = tr(K) = tr(K11 ) + tr(K22 ). Hence the effective
dimension ω = tr(H) can be written as the sum of two components, one for the
fixed and one for the random part.
Generalization is straightforward. If we have two random components: y = Xβ +Zc+ Z̆c̆, with c̆ ∼ N (0, τ̆ 2 I), we have the blocks K11 , K22 and K33 on the diagonalof K. Their traces add up to the effective dimension.
Harville (1977) presents an algorithm for variance estimation that uses traces ofsub-matrices of a matrix I −T . It turns out that I −T is equal to the sub-matrix ofK that corresponds to the random components. The algorithm uses the fact thatc0 c/tr(K22 ) = τ 2 and c̆0 c̆/tr(K33 ) = τ̆ 2 . Also σ 2 = (y − ŷ)0 (y − ŷ)/(n−ED), where
n is the number of observations. These relations hold for the true values of σ 2 , τ 2
and τ̃ 2 , but it immediately suggests an iterative algorithm. Practice has shownthat iterating with these expressions, updating the variances, re-estimating thecoefficients and traces, is reliable and relatively fast. Harville’s algorithm predates
Schall’s (1991) work. It has been used successfully by Rodriguez-Alvarez et al.
(2016) for modeling agricultural field trials with up to ten unknown variances. As
reported by Velazco and others (2017), it never failed on a large set of agriculturalexperiments.
Harville’s formula for T is quite complicated, because he starts by explicitly
eliminating the fixed part of the mixed model. Everybody else does it too. Thisis an intriguing element of the mixed model folklore, as if the REML paradigm,
correcting for fixed effects, should be explicitly visible to be believed. It is acomplication we can easily avoid, using the matrix K.

260

The truth about the effective dimension
Data and fit

Estimate of hat diagonal

● ●

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+
+
+●+
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+
+
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+
+
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+●+●●●+ ●+●++ ● ●+●+●+● +

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+●
+●

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0.1

Total kidney function
−2
0
2

●

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●

70

80

90

20

30

40

50 60
Age

70

80

90

FIGURE 1. Smoothing of kidney data with lowess. Left: data and lowess fit.
Right: estimates of ∂ ŷi /∂yi , obtained by data perturbation (red dots) and bybootstrapping (blue crosses).

2.2

The covariance alternativeYe (1998) not only used the diagonal of the hat matrix as the effective dimension,
but also an equivalence: ∂ ŷi /∂yi = cov(ŷi , yi )/σ 2 . Remarkably, it seems to be
more popular than the trace of H. Efron (2004) casts doubt on the hat matrixby showing a scatterplot of data on kidney function and applying the lowesssmoother. He also presents a plot of ∂ ŷi /∂yi . It shows a rather wild pattern andeven a number of negative values. Efron shows how to get alternative estimates,
using the above equivalence and bootstrapping. The covariance in the formulacannot be computed from the data directly, but by simulating pseudo-data vectors
with mean ŷ and independent errors with standard deviation σ̂ and smoothing
each of them.
In Figure 1 I have reproduced the data, the lowess fit, and the corresponding
estimates of ∂ ŷi /∂yi . In the paper Efron does not mention how he obtained
them, but the example also appears in the recent book (Efron and Hastie, 2016),
where numerical differentiation is mentioned. Following this approach, I made asmall change (0.01) to each observation in turn and re-computed the fit.
These surprising results for ∂ ŷi /∂yi are caused by the non-linearity of the smoother.
Figure 2 influence function of lowess and its derivative. The latter is negative formoderately large residuals. Their influence is reduced so aggressively that forthe more extreme observations an increase of the size of the residual moves thesmooth curve in the opposite direction.
If Efron would have used locfit instead of lowess, he would have obtained noncontroversial estimates for the hat diagonal. P-splines would have been even more
attractive, because the hat matrix (or the matrix K) is easily computed. Writing
P-splines with a second order difference penalty as a mixed model (Currie andEilers261

Smoothing with lowess

Influence function and its derivative

1.0

0.4

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−5
0
5
Standardized residual

10

−2

−1

0
1
Residuals●●

2

FIGURE 2. Left panel: the lowess smoother has a re-descending influence function, with a derivative that is negative for moderately large residuals. For standardized residuals with absolute values between 2.7 and 5, a fitted value will
move towards the smooth trend when the observation moves away from it. Rightpanel: estimates of the hat diagonal vs the residuals for lowess smoothing of thekidney data.

Durban, 2002) and using the simplified Harville algorithm to estimate variances,
we find that very strong smoothing is indicated. This means that effectively we
get a straight line fit. The kidney data turn out to be no good candidate forsmoothing at all: simple linear regression is sufficient.

2.3

The dimension of the model or the data?

The effective dimension is not a property of the data, but of the model, once
penalty parameters have been set. Figure 3 shows simulated data (a sine wave
plus independent errors) and the (optimized) fits obtained with the Whittakersmoother: ŷ = (I + λDd0 Dd )−1 y, with Dd the matrix that forms differences oforder d. For d = 1, the effective dimension is much larger than for d = 2 or d = 3,
while the fits look very similar. The reason is that fist order differences do notallow much flexibility. A further increase of λ would give a smoother fit, but at
the price of a larger bias.
These results have practical consequences. It is attractive to specify a smoother interms of the effective dimension of the fitted model, because a penalty parameter
like λ has no intuitive appeal. But a chosen effective dimension might lead tobias if the penalty is not flexible enough.
In the limit, when λ is very large, the Whittaker smootherP fits a polynomial of
degree d−1. The second order penalty can be written as λ i (zi −2φzi−1 +zi−2 )2 ,with φ = 1. If we change that to φ = cos(2πf /n), the limit is a (co)sine with f

262

The truth about the effective dimension

periods on the domain from 0 to 1, if covered by n observations. Amplitude andphase are automatically det to values that give the least squares fit to the data.
If the data indeed are a cosine wave plus noise, a very large λ would be indicated,with an effective dimension equal to 2 (or equal to 3, if the mean is not zero, andan extra parameter is introduced.) This shows that a good penalty should matchthe data, a largely unexplored aspect of smoothing.
Optimal smooth●●●●●● ●● ● ●
●● ●
● ●●
●

1.0

Fixed ED = 6

d = 1 ED: 18.4
d = 2 ED: 10.7
d = 3 ED: 7.1

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FIGURE 3. Whittaker smoother. Left: optimal smooth for various orders of thedifferences. Right: fits for a fixed effective dimension.

Eilers

263

References
I D Currie, I.D. and Durban, M. (2002) Flexible smoothing with P-splines: a unified approach. Statistical Modelling, 2, 333–349.
Efron, B. (2004). The Estimation of Prediction Error. Journal of the American
Statistical Association, 99, 619–632.
Efron, B. and Hastie, T (2016) Computer Age Statistical Inference. Cambridge
University Press.
Harville, D. A. (1977). Maximum likelihood approaches to variance component
estimation and to related problems. Journal of the American Statistical
Association, 72, 320-338.
Janson L., Fithian W. and Hastie T.J. (2015) Effective degrees of freedom: a flawed
metaphor Biometrika 102, 479–485.
Rodrı́guez-Álvarez, M. X., Boer, M. P., van Eeuwijk, F. A., and Eilers, P. H. C.
(2016). Spatial Models for Field Trials. http://arxiv.org/abs/1607.08255.
Schall, R. (1991). Estimation in generalized linear models with random effects.
Biometrika, 78, 719–727.
Velazco, J.G., Rodrguez-lvarez, M.X., Boer, M.P. et al. (2017) Modelling spatial
trends in sorghum breeding field trials using a two-dimensional P-spline
mixed model. Theoretical and Applied Genetics, doi:10.1007/s00122-0172894-4.
Ye, J. (1998). On Measuring and Correcting the Effects of Data Mining andModel Selection. Journal of the American Statistical Association, 93, 120–
131.

Score inference in LASSO regression
Giovanna Cilluffo1,2 , Gianluca Sottile1 , Stefania La Grutta2 ,Vito M.R. Muggeo1
1

Department of Economical, Business and Statistical Sciences, University ofPalermo (Italy),2
Institute of Biomedicine and Molecular Immunology A Monroy (IBIM) - National Research Council (CNR) of Palermo (Italy)
E-mail for correspondence: giovanna.cilluffo@unipa.it
Abstract: We discuss the score statistic to carry out inference on the regression
coefficients in LASSO regression. The proposed approach relies on the Induced
Smoothing framework and leads to results exhibiting good performance in different settings, including the high dimensional one n < p. We focus on interval estimation where few proposals have been discussed in literature with unsatisfactory
results in some settings. We present results from some simulation experiments
and an analysis of the well known prostate cancer dataset.
Keywords: Score Statistics; Induced Smoothing; LASSO; Confidence Intervals.

1

Introduction

The Least Absolutes Shrinkage and Selection Operator (LASSO) represents avery elegant and relatively widespread solution to carry out variable selection andparameter estimation simultaneously (Tibshirani 1996). While point estimation
can be performed quite straightforwardly, possible current limitations are computation of standard errors and consequently inferences, namely reliable hypothesis
testing and confidence intervals. Cilluffo et al. (2016) present a smooth approximation for the LASSO regression, based on the recent idea of induced smoothing(IS). The IS-LASSO allows to gain reliable standard errors which satisfactorily
quantify the estimator variance, even for zero coefficient. In hypothesis testing
problems, simulations show good performance of the IS-LASSO Wald statistic
when compared with the two competitors covTest (Lockart et al. 2016) and postSel (Lee et al. 2014). However, the regression coefficient estimator is still biased
for nonzero coefficient, that prevents the Wald statistic to be used for intervalThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Cilluffo et al.

265

estimation. In this paper we discuss an approach based on Score statistic. The
paper is structured as follows: Section 2 describes the basics of the approach andSection 3 reports some simulation evidence. Section 4 is devoted to a real-data
analysis, while conclusions are reported in the last section.

2

Methods

For the penalized objective 12 ||y − Xβ||22 + λ||β||1 , the IS score ise (β) = X T (y − Xβ) + λ{2Φ(β/v 1/2 ) − 1p },U
where Φ(·) is the cumulative distribution function of the standard Normal, v
is the main diagonal of the covariance matrix of estimates, V , and a/b means
the element-wise ratio of two vectors a and b. V is computed via the sandwiche −1 I H
e −1 where H
e = ∂ Ue is the hessian, and I is theformula, namely V = H∂βInformation matrix.
ej , He jk and Ijk
Let β1 and β2 be the interest and nuisance subsets of β, and U(j, k = 1, 2) the corresponding blocks of the Score vector, and of the hessian andInformation matrices. It is well know that score inference on β1 relies on the
profiled score−1 e
e1|2 = U
e1 − He 12 H
e 22
U
U2 ,

(1)

which can be expressed in matrix notation via
e1|2 = AUe = [I, −b][Ue1T , Ue2T ]TUe 12 He −1 . The variance is easily obtained
where I is the identity matrix, and b = H
22
ase1|2 ) = AV(Ue )AT with V(Ue ) = I = σ 2 (X T X).
V(UUnlike the usual inferential contexts where the regularity conditions are met,e ] 6= 0 (at the true parameter value β 0 6= 0), and thus the studentized Score
E[Ustatistic to be used for inference takes the formde1|2 − E[Ue1|2 ]]T V(Ue1|2 )−1 [Ue1|2 − E[Ue1|2 ]] →Se1|2 = [U
χ2p1 ,(2)

where p1 is the dimension of the interest parameter β1 . We propose to use thestudentized Score statistic (2) to carry out hypothesis testing and interval estimation for the regression coefficients in LASSO regression. In particular a (1 − α)confidence interval for β1 is given by
CI1−α = {β̄1 : Se1|2 (β̄1 ) ≤ χ2p1 ;1−α }.
To illustrate, Figure 1 portrays an example of profile score for two coefficients ina toy dataset.

Score inference in LASSO

0

U1|2
−30

−10

−20

U1|2

−10

10

0

266

−3

−2

−1

0

1

2

3

β1

−3

−2

−1

0

1

2

3

β6

FIGURE 1. Illustrating the profile score with corresponding point estimate and95% confidence intervals in a simulated dataset. The left and right panels referto a non-zero and zero coefficient respectively. The functions have been shiftedto guarantee U1|2 (β̂) = 0 (thus the dashed horizontal lines do not correspond toquantiles z.025 and z.975 ).

3

Simulation Evidence

The proposed Score statistic can be employed for hypothesis testing and interval estimation, but we here focus on interval estimation only. At the bestof our knowledge, the only approach currently available to build confidence
intervals is via selective inference (pSel) due to Lee et al. (2014). Thus wecompare Score-based confidence intervals with respect to those obtained via
the pSel. We consider scenarios fulfilling the theoretical condition for guarantee consistency (Zhao and You 2006) of the LASSO, namely: irrepresentablecondition; number of nonzero coefficients β1 , . . . , βk √(out of p covariates) suchthat k ≤ n/(2 log p) having magnitudes at least cσ 2 log p for some unspecified numerical constant c (Wainwright 2009). We generate data from a linear
regression model y = Xβ +  with X ∼ Np (0, Ip ) and  ∼ N (0, 1) for fivedifferent scenarios given by some combinations of sample sizes n = 50, 100,and number of covariates p = 20, 40, 60, 80, 120 with only 5 non-null coefficients
β = (3, −3.1, 4, 3.5, −5, 0, . . . , 0)T . Standard theory of pSel provides inference fora fixed λ, but because it is unknown in practice, we use cross-validation for choosing λ at each simulation run. Table 1 shows results based on 500 and 100 runs,
respectively in low and high dimensional settings, wherein the tuning parameterλ has been selected via 5-fold cross validation at each replicate.
Coverage levels referring to the first 10 coefficients are comparable for the nonzero
coefficients at the ‘simpler’ scenario of n = 50 and p = 20, but otherwise pSel
exhibits coverage levels lower than the nominal one, 0.95. Interestingly when
the sample size increases to n = 100 (keeping fixed the ratio p/n) the coveragelevels do not improve. The proposed score-based confidence intervals exhibit good
performance in all scenarios.
We also consider scenarios in which the theoretical conditions are not met, namely
with same combinations of n and p, but σ = 3. Table 2 reports results. Coverage

Cilluffo et al.

267

TABLE 1. Coverage levels of 95% CIs from Score and pSel for 10 selected parameters. Results are based on 500 and 100 replicates in low and high dimensionalsetting, σ is equal to 1 and the optimal λ has been obtained via cross validation.

Score
pSel

3
0.95
0.94

-3.1
0.95
0.95

4
0.95
0.94

coefficents true value
3.5
-5
0
0
0.95
0.95
0.95
0.96
0.95
0.95
1.00
1.00

0
0.95
1.00

0
0.96
1.00

0
0.94
1.00

(50,40)

Score
pSel

0.96
0.63

0.96
0.63

0.96
0.62

0.97
0.65

0.98
0.64

0.95
0.87

0.98
0.95

0.97
0.96

0.98
0.94

0.96
0.94

(100,80)

Score
pSel

0.99
0.39

0.98
0.39

0.99
0.39

0.98
0.39

0.97
0.39

0.99
0.91

0.98
0.89

0.98
0.89

0.99
0.86

0.99
0.90

(50,60)

Score
pSel

0.98
0.76

0.94
0.73

0.99
0.77

0.94
0.75

0.96
0.75

0.95
1.00

0.95
0.99

0.97
1.00

0.93
1.00

0.95
1.00

(100,120)

Score
pSel

0.98
0.88

0.92
0.85

0.97
0.88

0.96
0.87

0.99
0.89

0.96
1.00

0.97
0.99

0.96
0.96

0.95
0.99

0.96
1.00

(n, p)(50,20)levels from pSel are again lower than the nominal level, even in the simple scenario
of n = 50 and p = 20. In high dimensional settings, pSel performs quite bad.
TABLE 2. Coverage levels of 95% CIs from Score and pSel for 10 selected parameters. Results are based on 500 and 100 replicates in low and high dimensionalsetting, σ is equal to and the optimal λ has been obtained via cross validation.

Score
pSel

3
0.96
0.64

-3.1
0.97
0.64

4
0.95
0.65

3.5
0.97
0.65

true value
-5
0
0.97
0.96
0.64
0.94

0
0.97
0.90

0
0.97
0.92

0
0.97
0.90

0
0.94
0.91

(50,40)

Score
pSel

0.96
0.49

0.97
0.48

0.99
0.49

0.98
0.49

0.99
0.48

0.98
0.74

0.98
0.86

0.98
0.86

0.98
0.88

0.98
0.87

(100,80)

Score
pSel

0.99
0.33

0.98
0.34

0.98
0.34

0.97
0.34

0.98
0.35

0.97
0.86

0.98
0.84

0.98
0.88

0.99
0.83

0.97
0.84

(50,60)

Score
pSel

0.99
0.25

0.99
0.24

0.99
0.22

0.99
0.23

0.99
0.25

1.00
0.86

0.97
0.82

0.98
0.86

0.98
0.80

0.99
0.83

(100,120)

Score
pSel

1.00
0.34

0.99
0.32

1.00
0.33

1.00
0.33

1.00
0.33

1.00
0.92

1.00
0.93

0.99
0.90

1.00
0.90

1.00
0.94

(n, p)(50,20)

4

Application and conclusion

We apply the IS-Lasso to the well-known Prostate Cancer dataset analyzed in
Tibshirani (1996). Data refer to n = 97 subjects, p = 8 covariates and the
response variable is the log of prostate specific antigen. Table 3 reports the estimates from ordinary LASSO along with p-values and the 95% confidence intervals
returned by pSel and the proposed Score.
Findings are substantially similar between Score and pSel for all but one coefficient. In fact for the variable svi, pSel returns a somewhat ‘large’ p-value andconfidence interval including the zero, while the Score inference provides evidence
for a significant effect of such covariate. However for the remaining coefficientswith the same findings in terms of significance, the Score based confidence intervals are generally narrower. Results for pSel have been obtained via the package

268

Score inference in LASSOTABLE 3. Estimates, p-values and 95% confidence intervals for the regression
coefficients in the Prostate Cancer dataset. The widths are reported below each
CI. Estimates come from ordinary LASSO, p-values and confidence interval comefrom Score statistic and pSel.

95% confidence interval
Score
pSel
0.394, 0.726
0.395, 0.717
0.332
0.322
lweight
0.349 0.008 0.038
0.122,0.820 -0.057, 0.783
0.698
0.840
age
-0.001 0.118 0.203-0.039, 0.004 -0.039, 0.026
0.044
0.065
lbph
0.053 0.078 0.099-0.012, 0.228 -0.066, 0.222
0.240
0.288
svi
0.570 0.003 0.077
0.223, 1.141 -0.290, 1.069
0.918
1.359
lcp
0.000 0.265*-0.283, 0.079
*
0.363
*
gleason0.000 0.812*-0.275, 0.346
*
0.622
*
pgg45
0.002 0.424 0.166-0.005, 0.012 -0.004, 0.010
0.170
0.014
*lcp and gleason variable are not selected by pSel
covariatelcavolEst
0.517

p-value
Score pSel
0.000 0.000

selectiveInference version 1.2.2 (Tibshirani et al., 2017). An R package toperform Score-based inference in LASSO regression will be distributed in due
time.

5

Conclusions

The proposed Score statistic seems to be a good inferential tool to build confidence intervals in LASSO regression. The coverage levels of the interval estimators
are pretty close to the nominal level in different scenarios, even when the theoretical conditions are not met. While current results are quite promising, further
research is necessary to compare our approach with others not yet considered,such as the method proposed by Javanmard and Montanari (2014), which use adesparsifying LASSO to build CI.

References
Cilluffo, G., Fasola, S., La Grutta, S. and Muggeo, V.M.R. (2016) The induced
smoothed LASSO. In: Proceedings of the 31st International Workshop on
Statistical Modelling, Rennes, (France), 1, 69–74.

Cilluffo et al.

269Javanmard, A., Montanari, A. (2014) Confidence intervals and hypothesis testing for high-dimensional regression. Journal of Machine Learning Research
15, 2869–2909.
Lee, J.D., Sun, D.L., Sun, Y., and Taylor, J. E. (2016) Exact post-selection inference, with application to the lasso. Annals of Statistics, 44, 907–927.
Lockhart, R., Taylor, J., Tibshirani, R. J., and Tibshirani, R. (2014) A significancetest for the lasso. Annals of statistics, 42, 413.
Tibshirani, R. (1996) Regression shrinkage and selection via the lasso. Journalof Royal Statistical Society B, 58, 267–288.
Tibshirani, R., Tibshirani, R., Taylor, J., Loftus., J., and Reid, J., (2017) selectiveInference: Tools for Post-Selection Inference. R package version 1.2.2,
https://CRAN.R-project.org/package=selectiveInference
Wainwright, M.J. (2009) Sharp thresholds for high-dimensional and noisy sparsity recovery using-constrained quadratic programming (lasso). IEEE Trans.
Inform. Theory, 55, 2183–2202.
Zhao, P., and Yu, B. (2006) On model selection consistency of Lasso. Journal of
Machine learning research, 7, 2541–2563.

Using chain graph models for structuralinference with an application to linguistic
data
Craig Alexander1 , Jane Stuart-Smith2 , Tereza Neocleous1 ,Ludger Evers 1
12School of Mathematics and Statistics, University of Glasgow, UK
School of Critical Studies, University of Glasgow, UK

E-mail for correspondence: c.alexander.1@research.gla.ac.uk
Abstract: Graphical models provide a visualisation of the conditional dependence structure between variables, making them an attractive inference tool.
The improved readability makes this an appealing approach to represent complex
model output to non-statisticians. In this paper, we introduce a novel approach
using graphical models to visualise the output of a mixed effects model withmultivariate response with an application to linguistic data.
Keywords: Hierarchical models; Graphical models; Linguistic change.

1

Introduction

In this work, we discuss the use of a chain graph model structure to represent
the output from a hierarchical regression model with multivariate response. The
chain graph model is inferred in three parts. The dependency structure of thecovariates is modelled independently using standard structural inference methods
in graphical models. The relationship between the response and explanatory variables and the dependence structure between response variables is jointly inferredusing a multivariate Bayesian hierarchical model, where the precision estimates of
the residuals and random effects are assumed to conform to an undirected graphical model. We report an application of this model to linguistic data obtained
from the Sounds of the City corpus, consisting of Glaswegian speech recordings
from the 1970’s to the 2000’s. From this data, we look to recover the underlying
chain graph model detailing which factors affect vowel change.
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Alexander et al.

2271

Chain graph model structure

Implementing a chain graph (Lauritzen & Wermuth, 1989) structure allows theuse of directed and undirected edges. Nodes are partitioned into blocks, one for
explanatory variables and one for response variables. Edges within blocks are
undirected and edges between blocks are directed.
The directed edges between blocks are modelled using a Bayesian hierarchicallmodel. The response is defined as yij, which is the j th measurement from the iththlgroup on the l response. β is the corresponding vector of regression coefficients.
The random effects coefficients are denoted by bl .
In vector-matrix notation, the likelihood is defined as:p(y | β, b, Ω , X) = N y | Xβ + Ub, Ω−1
 ⊗I



.

(1)

As conjugate priors have been specified for each parameter where possible, aGibbs sampler can be used for parameter inference.
The presence of a directed edge in the graphical model corresponds to the valueof the respective βij coefficient:• βij 6= 0 → an edge is present between variable i and response j.
• βij = 0 → no edge is present between variable i and response j.
If any interactions are in the model, a factor graph structure is used to represent
these, with the interacting explanatory variables connecting to a factor node,
which is connected to the relevant response variable.
At each step of the sampler, a new candidate model is proposed by either adding
or removing an explanatory variable. By integrating out β, model evidences arethen computed to determine whether we accept or reject the candidate model.
For random effect parameters b a Gaussian prior is specified of the form:.
b | Ωb ∼ N 0, Ω−1
b ⊗I

(2)

To maintain conjugacy, the random effect and model error precision matrices
have G-Wishart (Dobra et al, 2011) hyperpriors placed on them:Ωb , Ω |G ∼ WG (νb , Sb ).

(3)

where G is a defined graph.
The precision estimates from the hierarchical model are taken as input to a zeromean Gaussian graphical model, defined as:
MG = N (0, Ω−1 ).

(4)where ωi,j = 0 corresponds to a missing edge between response i and j.
The normalising constant, IG (ν, S), for chordal graphs can be obtained via a
closed form solution by factorising it into a product of density functions:
Qd
i=1 ITi (ν, STi ,Ti )IG (ν, S) = Qd−1.
(5)
j=1 ISi (ν, SSi ,Si )where Ti are cliques and Si the separators of G. In the case of a non-chordal graph,model selection can be performed by using trans-dimensional MCMC methods
such as those discussed in Mohammadi & Wit (2015).

272

Chain graph models for structural inference

FIGURE 1. Spectrogram of the word GOAT spoken by a male Glaswegianspeaker. The three coloured lines show the first three formants of the vowel /o/.

3

Application: vowel change in Glasgow

We apply the methodology to data from the Sounds of the City (Stuart-Smith,2017) project, which is a study modelling vowel change in the Glaswegian dialect
over the 20th century. Recordings of speakers over various decades are used andvowel measurements are taken along with phonetic quantities of interest andsocial and biological factors.
Acoustically, a vowel can be characterised by its main resonances, known asformants (Hz). This is illustrated in Figure 1, with the /o/ vowel formants represented by the three coloured lines. Vowel change is studied in terms of how such
frequencies alter over time.
Figure 2 shows the best posterior model selected for the vowel /o/, found inwords like GOAT, HOPE, etc, selected with posterior probability 0.414.

Alexander et al.

273

Gender

F1.meanDecadeAge

F2.meanPrecedingF3.meanFollowing

Posterior probability of model: 21.7%

FIGURE 2. Best posterior model selected for the GOAT vowel.

From the graphical model, we observe the following significant predictors:• Age for F1; younger speakers are leading the change in vowel quality. Shifts
in F1 relate to raising of the vowel quality, so /o/ becoming more like /u/.
• Shifts in F2 relate to fronting/backing of the vowel quality according to
the nature of following consonant.
• Decade and Gender for F3, evidence of vowel change over time and additional differences due to speaker gender.

4

Conclusions & future work

In this work, we have extended beyond previous modelling of sociolinguistic data,by considering multiple formants within the one model, accounting for the highcorrelation between formants. This approach can also be extended beyond linguistic data, with several academic trials using datasets with similar multivariate
and nested features.
In order to promote the usability of our method, we aim to turn it into an online
application, allowing users to input their own data.

274

Chain graph models for structural inference

Acknowledgments: CA is grateful to the Lord Kelvin Adam Smith scholarship for his research studentship.

References
Lauritzen, S.L. and Wermuth, N. (1989). Graphical models for associations between variables, some of which are qualitative and some quantitative. The
Annals of Statistics, 17, 31-57.
Dobra, A., et al (2011). Bayesian inference for general Gaussian graphical models with application to multivariate lattice data. JASA , 106.
Stuart-Smith, J., et al (2017). Changing sounds in a changing city. Language
and a sense of place, E. Moore & C. Montgomery, CUP, 38-64.
Mohammadi, A. and Wit, E.C. (2015) Bayesian Structure Learning in Sparse
Gaussian Graphical Models. Bayesian Analysis, 10, 109-138.

Sample quantiles corresponding to mid
p-values for zero–modification tests
Paul Wilson1 , Jochen Einbeck2
1

School of Mathematics and Computer Science/Statistical Cybermetrics Research Group, University of Wolverhampton, WV1 1LY, United Kingdom
2
Department of Mathematical Sciences, Durham University, DH1 3LE, United
KingdomE-mail for correspondence: pauljwilson@wlv.ac.uk
Abstract: Wilson and Einbeck (2015, 2016) propose a test for zero-modification
relative to a stated model. The basis of the test is that the number of observed
zeros follows a Poisson-binomial distribution. The decision to reject, or otherwise,
the non zero–modified model is made by either (i) computing the mid p-value
corresponding to the number of observed zeros, or (ii) comparing the number ofobserved zeros to the relevant “traditional” quantile of the appropriate Poisson–
binomial distribution. In general either approach will result in the same decision,but occasionally discrepancies may occur. In this paper we investigate the use of
mid-distribution quantiles in approach (ii) above, and show that this reduces thepossibility of discrepancies.
Keywords: zero-modification, mid p-values, quantiles

1

Introduction

Wilson and Einbeck (2015) proposed a new and intuitive test for zero-modificationthat uses the observed number of zeros, n0 , in a given sample y = y1 , y2 , . . . , ynfrom count variables Yi and a set of covariates xi , i = 1, 2, . . . , n to establish
whether the distributional assumption Yi |xi ∼ G(yi |µi ) where µi is a pre-specified
parametric function of the xi is consistent with N0 , the distribution of the number of zeros under G. This is achieved by referencing the value of n0 to theappropriate Poisson-Binomial distribution (Chen and Liu, 1997).
To illustrate, consider the case where G is a Poisson model, and thus pi =p(0|µi ) = e−µi and let Ti be a random variable which takes the value 1 if yi = 0
and 0 otherwise. Clearly Ti is a Bernoulli random variable with parameter pi andThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

276

Sample quantiles for zero–modification tests

thus N0 may be formulated as the sum over independent Bernoulli experiments
T1 , T2 , . . . , Tn .
Based on this simple observation, consider the special case that there are nocovariates, that is µ1 = µ2 = · · · = µn = µ. In this case, the pi ’s are equal also,
and so the distribution of N0 is a binomial distribution Bin(n, p), where p = e−µ ,and thus has mean np and variance np(1 − p). Based on this distribution, one canimmediately compute quantiles corresponding to a given significance level, anduse these as critical values for the test; alternatively one may determine the pvalue corresponding to n0 , and reject or otherwise the Poisson model based uponthis. If the µi do depend on covariates, N0 is the sum of n independent Bernoulli
random variables T1 , T2 , . . . , Tn , and hence is a Poisson-Binomial distribution
with parameters p1 , p2 , . . . , pn and one proceeds by computing quantiles or pvalues relative to this distribution, using, for example, the R package poibin(Hong, 2013).
Wilson and Einbeck (2016) proposed the use of mid p-valuesα̂T,0.5 (t) = P0 [T > t] + 0.5P0 [T = t] = 0.5 (P0 [T ≥ t] + P0 [T ≥ t + 1])
which Franck (1986) argues are more appropriate when the test statistic is discrete. Note that if T were continuous, then P0 [T = t] = 0 and the mid p-value isequivalent to the “traditional” p-value. It may be shown that the attainment rateof the proposed test when mid p-values are employed is superior to that when
traditional p-values are used.
Wilson and Einbeck (2015) utilise the “traditional” quantile Q(p) = inf{t |F (t) ≥ p} where F (x) = P (X ≤ x) is the cumulative distribution function ofa random variable X. This may lead to discrepancies. An example, based upon
the one-sided version of the test (i.e. we are testing for zero-inflation only), is thefollowing:

1.1

Trajan Data

The data are the number of roots produced by n = 270 micropropagated shootsof the columnar apple cultivar Trajan. During the rooting period, all shoots weremaintained under identical conditions, but the shoots themselves were cultured
on media containing different concentrations of the cytokinin BAP, in growthcabinets with an 8 or 16 hour photoperiod. Full details of the experiment are to
be found in Marin (1993). A striking feature of the data is that although almost
all shoots produced under the 8 hour photoperiod rooted, only about half of those
produced under the 16 hour photoperiod did. Overall n0 = 64 shoots produced
zero roots, of which only 2 were from the shorter photoperiod.
These data were analysed by Ridout and Demétrio (1992) and Ridout et al.
(1998). If the model of the null hypothesis is a negative binomial (type–II) model,
where both the mean and the size parameter are modelled by photoperiod, thena (mid) p-value of 0.0871 for the test of Wilson and Einbeck (2015, 2016) isreturned, indicating non-rejection of the negative binomial model at α = 0.05.
The traditional 5th and 95th quantiles of the distribution of N0 are 47 and 66;
the interval [47, 66] is referred to as a 90% fluctuation interval. As n0 = 64 isinterior to this interval we conclude that n0 is consistent with such a model (andWilson and Einbeck277

inconsistent with the zero-inflated model) at a level of significance of α = 0.05. An
80% fluctuation interval however is [49, 64], and thus based upon this fluctuationinterval we would fail to reject the negative-binomial model in favour of thestrictly zero-inflated model at a level of significance of 0.10, but we would do sounder the “p–value criterion”.

2

Quantiles based on mid-distribution functions

Let X be a discrete random variable with distinct values v1 < v2 < · · · < vd , let
P (X = vi ) = pi . Ma et al. (2011) recommend the following quantile function for
discrete distributions:−1

Q(p) = Fmid (p) =Where πk =

2.1


v1








v

 k

if p < p1 /2

λvk + (1 − λ)vk+1










vd

if p = λπk + (1 − λ)πk+1
0 < λ < 1, k = 1, . . . , d − 1

if p = πk , k = 1, . . . , dif p > πdPk−1

i=1 pi + pk /2, that is, πk is a lower-tailed mid-p-value.

Example: Mid Quantiles for a Binomial Distribution

Let X ∼ Bin(7, 0.35), and thus X has pmf and cdf:x
P (X = x)
P (X ≤ x)

0
0.049
0.049

1
0.185
0.234

2
3
4
0.298 0.268 0.144
0.532 0.800 0.944
thand hence the “traditional” 90 quantile of X is 4.

5
0.047
0.991

6
0.008
0.999

7
0.001
1.000

We determine the “mid–quantile” as follows:
v5 = 4, v6 = 5, p5 = 0.144, p6 = 0.047.
Hence π4 = 0.800 + 0.144/2 = 0.8720, π5 = 0.944 + 0.047/2 = 0.9675.
Note that 0.9 = 0.707π4 + (1 − 0.707)π5 , hence:−1
Q(0.9) = Fmid(0.9) = 0.707v4 + (1 − 0.707)v5 = 3.213

2.2

Example: Simulated Poisson Data

The 25 data of Table 1 are a random draw from a random variable W that isbelieved to follow a Poisson distribution. It is wished to test this belief.
It is estimated, using the adaptive mixture estimator of Wilson and Einbeck(2016), that the mean of W is µ = 1.171, and hence under the null (Poisson)model P (W = 0) = exp(−1.171) = 0.310. Hence the observed number of zeros in278

Sample quantiles for zero–modification tests

TABLE 1.

0
16

1
4

2
4

3
1

random samples of size 25 drawn from W will be Bin(25, 0.310) distributed. The“traditional” 2.5th and 97.5th quantiles of such a distribution are 7 and 16 respectively, and hence a 95% fluctuation interval for the number of observed zeros
under the Poisson distribution is [7, 16] indicating non-rejection of the Poissonmodel at a level of significance of α = 0.05, consistent with the traditional p-valueof 0.064, but inconsistent with the mid p-value of 0.045. The 95% fluctuation interval based upon the mid quantiles is however [6.52, 15.57], consistent with themid p-value. These results are summarised in Table 2.2.
TABLE 2. n = 25, H0 :Poisson

n0 = 16
traditional
mid

2.3

p-value
0.064
0.045

95%F I[7, 16][6.52, 15.57]

Example: Trajan Data Revisited

Here we re-compute the 80% fluctuation interval for the negative binomial model
fitted to the Trajan data of Section 1.1 using the mid-distribution quantiles defined above. (Recall, here we are testing for strict zero-inflation, and thus the upper bound of the fluctuation interval serves as a test statistic for a one-sided test).
We find that π47 = 0.073 and π48 = 0.101, thus 0.1 = 0.069π47 +(1−0.069)π48 andhence Q(0.1) = (0.069 × 47) + ((1 − 0.069) × 48) = 47.931. Similarly π63 = 0.882
and π64 = 0.913, thus 0.9 = 0.419π63 + (1 − 0.419)π64 and hence Q(0.9) =(0.419 × 63) + ((1 − 0.419) × 64) = 63.581. Thus, using the mid-distribution quantile we obtain a 80% fluctuation interval of (47.931, 63.581), and hence n0 = 64
is exterior to the confidence interval, and we reject the negative-binomial modelin favour of the zero-inflated negative binomial model under both criteria. These
results are summarised in Table 2.3.
TABLE 3.

n0 = 64
traditionalmid

3

p-value
0.10100.0871

90% FI[47, 66][46.902, 65.679]

80% FI[49, 64][47.931, 63.581]

Conclusion

Decisions based upon mid-distribution quantiles as defined above will agree withthose based upon mid p-values unless p < p1 /2 or p > πd . With respect to the testWilson and Einbeck279

proposed in Wilson and Einbeck (2015, 2016) these exceptions correspond to theobserved data either containing no zeros, or consisting entirely of zeros, and hence
the adoption of quantiles based upon mid-distribution functions results in fluctuation intervals that nearly entirely removes discrepancies that may sometimes
occur between decisions based upon fluctuation intervals and mid p–values. Given
that the power and attainment rates of the test when based upon mid p-values are
excellent, such alignment is desirable. The adoption of such quantiles is straightforward. In this paper we only discuss the use of mid-distribution quantiles inrelation to the test of Wilson and Einbeck (2015, 2016), but their application toother tests with discrete test statistics is worthy of investigation.

References
Chen, S.X. and Liu, J.S. (1997). Statistical applications of the Poisson-binomial
and conditional Bernoulli distributions. Statistica Sinica 7, 875–892.
Hong, Y. (2013). poibin: The Poisson Binomial Distribution. R package version
1.2. https://CRAN.R-project.org/package=poibin
Ma, Y., Genton M. and Parzen, E. (2011) Asymptotic properties of sample quantiles of discrete distributions. Annals of the Institute of Statistical Mathematics 63, 227 – 243.
Ridout and Demétrio (1992) Generalized Linear Models for Positive Count Data.
Revista de matematica e estatistica 10, 139 – 148.
Ridout, M.S., Demétrio C.B. and Hinde, J. (1998) Models for count data withmany zeros. In Proceedings of the XIXth international biometric conference
19, 179 – 192).
Wilson, P. and Einbeck, J. (2015). A simple and intuitive test for number–inflationor number–deflation. In: Wagner, H. and Friedl, H. (Eds). Proc’s of the 30th
IWSM, Linz, Austria, Vol 2, pages 299–302.
Wilson, P. and Einbeck, J. (2016). On statistical testing and mean parameter
estimation for zero–modification in count data regression. In: Dupuy, J.
and Josse, J. (Eds). Proc’s of the 31st IWSM, Rennes, France, Vol 1, pages
325 – 330.

Two wrongs make a right: addressing
underreporting in binary data from multiple
sources
Betsabe Blas1 , Scott J. Cook2 , Raymond J. Carroll3 , SamiranSinha41

Department of Statistics, Federal University of Pernambuco, Brazil
Department of Political Science, Texas A&M University, US
3
Departmentof
Statistics,
Texas
A&M
University
School of Mathematical Sciences, University of Technology Sydney
4
Department of Statistics, Texas A&M University2

andE-mail for correspondence: betsabe.bg@de.ufpe.br
Abstract: Media-based event data i.e., data comprised from reporting by media outlets are widely used in research in political science. However, events ofinterest (e.g., strikes, protests, conflict, etc.) are often underreported by these primary and secondary sources, producing incomplete data that risks inconsistency
and bias in subsequent analysis. While general strategies exist to help ameliorate
this bias, these methods do not make full use of the information often available
to researchers. Specifically, much of the event data used in the social sciences
is drawn from multiple, overlapping news sources (e.g., Agence France-Presse,
Reuters, etc.). Therefore, we propose a novel maximum likelihood estimator that
corrects for misclassification in data arising from multiple sources. In the mostgeneral formulation of our estimator, researchers can specify separate sets ofpredictors for the true-event model and each of the misclassification models characterizing whether a source fails to report on an event. As such, researchers areable to accurately test theories on both the causes of and reporting on an event ofinterest. Simulations evidence that our technique regularly outperforms current
strategies that either neglect misclassification, the unique features of the datagenerating process, or both. We also illustrate the utility of this method with amodel of repression using the Social Conflict in Africa Database.
Keywords: missclasification; binary model; multi-source.
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Blas 1 et al.

1

281

Misclassification

Misclassification is likely to occur with media-based event data, where primaryor secondary-source reports fail to include the occurrence of an actual event. To
introduce our method, consider two news outlets, 1 and 2, providing reports, Y1
and Y2 , on the event of interest YT . However, we possess two incomplete reports
Y1 6= YT and Y2 6= YT , explained by pr(Y1 = 1|X, Z1 ) and pr(Y2 = 1|X, Z2 ),where Z1 and Z2 are predictors of the (mis-)reporting of an event (e.g., distance toreporting office) by that source, which are otherwise unrelated to YT . Following
convention in the applied literature, we aggregate these sources to reduce theindividual missingness by Ysum = 1(Y1 + Y2 ≥ 1). If Ysum = YT , the data arecomplete and we find thatpr(YT = 1|X, Z1 , Z2 ) = pr(YT = 1|X) = F (β t X),(1)

where F (·) is specified up to the parameter β. We are interested in Eq. 1. However, where Ysum 6= YT , we are unable to simplify as in 1. This means thatwhen observed outcomes are misclassified, fitting Equation 1 will result in biased
estimates of X on YT . We make the following assumptions:(a) Y1 and Y2 are independent given (YT , X, Z1 , Z2 ). (b) Ysum = 1 implies YT = 1 with probability
1. (c) YT = 0 implies that Ysum = 0 = Y1 = Y2 with probability 1.
If we treat Ysum as the response variable, the problem is related to one studied
by Carroll and Pederson (1993), and Hausman et al. (1998). The misclassification probabilities are pr(Ysum = 0|YT = 1, X, Z1 , Z2 ) = γ(X, Z1 , Z2 ), andpr(Ysum = 1|YT = 0, X, Z1 , Z2 ) = 0, which follows from Assumption (b). In
Hausman et al. (1998) these misclassification probabilities do not depend of covariates, and are instead simply an unknown constant to be estimated. Therefore, we generalize Hausman et al. (1998)’s estimator to allow for misclassification
probabilities that are dependent upon the covariates. We have thatpr(Ysum = 0|X, Z1 , Z2 )={1 − γ(X, Z1 , Z2 )}{1 − F (X, β)} + γ(X, Z1 , Z2 );pr(Ysum = 1|X, Z1 , Z2 )={1 − γ(X, Z1 , Z2 )}F (X, β).

However, the data are not (Ysum , X, Z1 , Z2 ), but (Y1 , Y2 , X, Z1 , Z2 ), that is, wehave multiple sources of data. As such, there may be different misclassification
rates, which is the fundamental difference between our estimator and existing
approaches. Rather than neglect this information, thereby failing to use all of thedata, we define
α1 (X, Z1 )=pr(Y1 = 0|YT = 1, X, Z1 );α2 (X, Z2 )=pr(Y2 = 0|YT = 1, X, Z2 ).

Here by Assumption (b) we have that pr(Y1 = 1|YT = 0, X, Z1 ) = pr(Y2 =1|YT = 0, X, Z2 ) = 0.
Under Assumption (a-c), we havepr(Y1 = 0, Z2 = 0|X, Z1 , Z2 )=1 − F (X, β) + α1 (X, Z1 )α2 (X, Z2 )F (X, β);pr(Y1 = 0, Z2 = 1|X, Z1 , Z2 )=α1 (X, Z1 ){1 − α2 (X, Z2 )}F (X, β);pr(Y1 = 1, Z2 = 0|X, Z1 , Z2 )={1 − α1 (X, Z1 )}α2 (X, Z2 )F (X, β);pr(Y1 = 1, Z2 = 1|X, Z1 , Z2 )={1 − α1 (X, Z1 )}{1 − α2 (X, Z2 )}F (X, β).

Thus, we can construct the likelihood function.

282

Binary model from multiple sources

TABLE 1. Simulation Study Results
Naive
Probit

2

Bias
Std
SE
MSE
CP(%)

0.074
0.051
0.055
0.008
77.4

Bias
Std
SE
MSE
CP(%)

0.397
0.055
0.057
0.161
0.0

Hausman Hausman
Const Pr
w/ Covβ0 = −1−0.663−0.023
0.227
0.106
0.175
0.096
0.490
0.012
3.7
95.8
β1 = 1−0.379−0.035
0.258
0.158
0.218
0.132
0.210
0.026
56.9
94.0

Multi-Source
Const Pr

Multi-Source
w/ Cov−0.048
0.060
0.067
0.006
91.3

0.003
0.063
0.063
0.004
95.8

0.287
0.092
0.079
0.091
11.1

−0.010
0.094
0.095
0.009
96.3

Simulations

We consider the following five methods: Naı̈ve Probit: pr(Ysum = 1|X) =Φ(β0 + β1 X). Hausman, Constant Probabilities: the approach outlined in
Hausman et al. (1998), which fits pr(Ysum = 1|X) = {1 − pr(Y = 1|YT =1, X)}Φ(β0 + β1 X). Hausman with Covariates: our generalization of Hausman et al. (1998), pr(Ysum = 0|YT = 1, X, Z1 , Z2 ) = γ(X, Z1 , Z2 ) = Φ(η00 +η01 X + η02 Z1 + η03 Z2 ), giving event probabilities pr(Ysum = 1|X, Z1 , Z2 ) ={1 − γ(X, Z1 , Z2 )}Φ(β0 + β1 X). Multi-Source, Constant Probabilities: ourmulti-source method, pr(Y1 = 0|YT = 1, X, Z1 ) = Φ(η10 ) and pr(Y2 = 0|YT =1, X, Z2 ) = Φ(η20 ). Multi-Source with Covariates: our general multi-source
method, pr(Y1 = 0|YT = 1, X, Z1 ) = Φ(η10 + η11 X + η12 Z1 ) and pr(Y2 = 0|YT =1, X, Z2 ) = Φ(η20 + η21 X + η22 Z2 ). We generated N = 1000 Monte Carlo trials, with sample size n = 1000. X, Z1 and Z2 from N (0, 1) distribution. YTfrom Bernoulli{F (X, β) = Φ(β0 + Xβ1 )}, where Φ denotes the CDF of the standard normal distribution. Next, generate misclassification probabilities by using
α1 = Φ(η10 + η11 Xi + η12 Zi,1 ) and α2 = Φ(η20 + η21 Xi + η22 Zi,2 ), then generate
Yi,1 = Yi,T (1 − Bernoulli(α1 )) and Yi,2 = Yi,T (1 − Bernoulli(α2 )). Given Y1 andY2 , generate Ysum using Ysum = 1(Yi,1 + Yi,2 ≥ 1). We set β0 = −1 and β1 = 1,
(η10 , η11 , η12 , η20 , η21 , η22 ) = (−0.7, 1, 1, −1.4, 1, 1).

3

Application

we estimate a model of repression in Africa using the four methods. Our outcome
data is taken from SCAD (Salehyan et al. 2012), which generates event data on
forty-seven African countries using key word searches of Associated Press (AP)and Agence France-Presse (AFP) news wires.

Blas 1 et al.

283

TABLE 2. Model of Repression in Africa

N

Naive
Probit0.020(0.062)0.407(0.053)−0.655(0.151)−8.011(1.000)1092

4Conclusion

Model
GDP pct−1
P opt−1
Demot−1
Constant

HausmanConst Pr
0.020
(0.062)0.407(0.053)−0.655(0.151)−8.011(0.994)1092

Hausmanw/ Cov−0.164(0.125)0.314(0.085)−0.739(0.261)−4.679(1.624)
1092

Multi-Source
Const Pr
0.022
(0.072)0.458(0.063)−0.757(0.172)−8.568(1.161)
1092

Multi-Source
w/ Cov−0.292(0.145)0.330(0.095)−0.819(0.315)−3.857(2.063)1092

Traditionally researchers devote less attention to measurement error in the outcome, however, here we have highlighted the severity of the bias induced by differential misclassification in binary outcomes. Our simulations show that misclassification can produce substantial bias when researchers employ either: i) strategies
which assume no misclassification or ii) strategies which assume non-differential
misclassification. We show how researchers possessing more than one source ofdata-generating information can achieve this desired result. Specifically, we derive an estimator for applications in which researchers have at least two sourcesof potentially misclassified data on a single outcome of interest. Under few assumptions, our estimator returns unbiased estimates of the risk probability andallows for source-specific misclassification estimates. We illustrated the utility ofthis method in a model of state repression in Africa, observing that predictor
effects change dramatically when misclassification is ignored.

Acknowledgments: This work was supported by a grant from the National
Cancer Institute [U01-CA057030 to R.J.C.]; and a post-doctoral fellowship from
Conselho Nacional de Desenvolvimento Cientı́fico e Tecnológico [201192/2015-2
to B.B.].

References
Carroll, R. J., Pederson, S. (1993). On robustness in the logistic regression
model. Journal of the Royal Statistical Society. Series B (Methodological),693 – 706.
Hausman, J. A., Abrevaya, J., Scott-Morton, F. M. (1998). Misclassification of
the dependent variable in a discrete-response setting. Journal of Econometrics, 87(2), 239 – 269.
Salehyan, I., Hendrix, C. S., Hamner, J., Case, C., Linebarger, C., Stull, E.,
Williams, J. (2012). Social conflict in Africa: A new database. International
Interactions, 38(4), 503 – 511.

Estimation of multivariate distributions forrecurrent event data
Luı́s Meira-Machado1 , Beatriz Sampaio1
1

Centre of Mathematics & Department of Mathematics and Applications, University of Minho, Campus de Azurem, 4800-058 Guimarães, Portugal.

E-mail for correspondence: lmachado@math.uminho.pt
Abstract: In many longitudinal studies information is collected on the times ofdifferent kinds of events. Some of these studies involve repeated events, where asubject or sample unit may experience a well-defined event several times along
his history. Such events are called recurrent events. In this work we consider
the estimation of the marginal and joint distribution functions of two gap times
under univariate random right censoring. We also consider the estimation of thebivariate survival function.
Keywords: Censoring; Kaplan-Meier; Nonparametric estimation; Recurrent events;
Survival Analysis.

1

Introduction

In many longitudinal studies, subjects can experience recurrent events. This typeof data has been frequently observed in medical research, engineering, economy
and sociology. In medical research, the recurrent events could be multiple occurrences of hospitalization from a group of patients, multiple recurrence episodes
in cancer studies, repeated heart attacks or multiple relapses from remission forleukemia patients. In this work we consider the estimation of the marginal andjoint distribution / survival functions of the gap times under univariate random
right censoring. These issues have received much attention recently. Among others they were investigated by Lin, Sun and Ying (1999), de Uña-Álvarez andMeira-Machado (2008) or de Uña-Álvarez and Amorim (2011).
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Meira-Machado and Sampaio

2285

Nonparametric estimators

In the context of recurrent event data, each individual may go through a welldefined event several times along his history. Assume that each study subject
can potentially experience K consecutive events at times T1 < T2 < ... < TK ,which are measured from the start of the follow-up. In this work we are primarily
interested in the gap times Y1 := T1 , Y2 := T2 − T1 , ..., Yk := Tk − Tk−1 ,k = 2, ..., K. For simplicity we assume K = 2.
Then, (Y1 , Y2 ) is a vector of gap times of successive events, which we assume tobe observed subjected to (univariate) random right-censoring. Let C be the rightcensoring variable, assumed to be independent of (Y1 , Y2 ). Because of this, theobserved data consists of (Ye1i , Ye2i , ∆1i , ∆2i ), 1 ≤ i ≤ n, which are n independent
replications of (Ye1 , Ye2 , ∆1 , ∆2 ), where Ye1 = Y1 ∧C, ∆1 = I(Y1 ≤ C), Ye2 = Y2 ∧C2 ,∆2 = I(Y2 ≤ C2 ) with C2 = (C − Y1 )I(Y1 ≤ C) the censoring variable of thesecond gap time. Here and thereafter, a ∧ b = min(a, b) and I(·) is the indicatorfunction.
Let Fk , k = 1, 2 denote the distribution function of the k-th event time Tk .
Since Tk and C are independent, the Kaplan-Meier product-limit estimator (Kaplan and Meier, 1958) based on the pairs (Teki , ∆ki )’s, consistently estimates the
distribution of the time to the k-th event. Because Y2 and C2 will be in general dependent, the estimation of the marginal distribution of the second gap
time is not a simple issue. The same applies to the joint distribution function
F12 (t1 , t2 ) = P (Y1 ≤ t1 , Y2 ≤ t2 ) and the joint survival function S12 (t1 , t2 ) =P (Y1 > t1 , Y2 > t2 ). Some estimators for these quantities will be presented below.
Below we present several different approaches for estimating the bivariate distribution function of (Y1 , Y2 ). An estimator based on Inverse Probability of Censoring Weights was first introduced by Lin, Sun and Ying (1999):
nn
1 X I(Ye1i ≤ t1 , Ye2i > t2 )1 X I(Ye1i ≤ t1 )∆1i
IPCW
−
.
Fb12
(t1 , t2 ) =b 1 (Ye1i )b Ye1i + t2 )
n i=1
n i=1
G
G(

e 1 and Ge stand for the Kaplan-Meier estimator (of the censoring distriwhere G
bution) based on the (Ye1i , 1 − ∆1i )’s and (Te2i , 1 − ∆2i )’s, respectively.
A simple estimator based on the Kaplan-Meier weights was later introduced byde Uña-Álvarez and Meira-Machado (2008). The idea behind their estimator is toweight the data by the Kaplan-Meier weights (Wi ) pertaining to the distributionof the total time (in this case, T2 ) of the process:

KMW
Fb12
(t1 , t2 ) =nX

Wi I(Ye1i ≤ t1 , Ye2i ≤ t2 ).

i=1
PKMW
A related estimator based on presmoothing (Fb12) was later proposed by deUña-Álvarez and Amorim (2011).

Given that P (Y1 ≤ t1 , Y2 ≤ t2 ) = P (Y2 ≤ t2 | Y1 ≤ t1 )P (Y1 ≤ t1 ) we also
consider the landmark estimator (LDM) for which to estimate P (Y2 ≤ t2 | Y1 ≤

286

multivariate distributions for recurrent event datat1 ) the analysis is restricted to the individuals with an observed first event timeless or equal than t1 . This is known as the landmark approach (van Houwelingen
et al. 2007). The corresponding estimator (LDM) is given by
LDM
Fb12(t1 , t2 ) =

n
X

(t ]

Wi 1 I(Ye2i ≤ t2 ) × Fe1KM (t1 )i=1where F1KM is the Kaplan-Meier estimator of the distribution of the first time(t ]and Wi 1 denote the Kaplan-Meierweightsof the distribution of T2 computednoefrom the given sub sample i : Y1 ≤ t1 .
In this work we also introduce new estimators which are constructed using the
cumulative hazard of the total time given a first time but where each observation
has been weighted using the information of the first duration. The proposed
WCH
estimator (WCH - weighted cumulative hazard) is given by Fb12
(t1 , t2 ) = Pb(Y1 ≤bbt1 )(1 − P (Y2 > t2 | Y1 ≤ t1 )) where P (Y1 ≤ t1 ) is estimated by the KaplanQ
Meier estimator of the first event time and Pb(Y2 > t2 | Y1 ≤ t1 ) = v≤t2 (1 −Λ̂Y2 |Y1 ≤t1 (dv)),
wherePn
b Yb1i + v)
I(Ye1i ≤ t1 , Ye2i = v, ∆2i = 1)/G(.
Λ̂Y2 |Y1 ≤t1 (dv) = Pi=1
n
e
e
b b
i=1 I(Y1i ≤ t1 , Y2i ≥ v, ∆1i = 1)/G(Y1i + v)
Finally we compare the aforementioned methods with the estimator of the bivariate distribution which is obtained using Nearest Neighbor Estimation (NNE).
Now, we consider the estimation of the bivariate survival function S(t1 , t2 ) =P (Y1 > t1 , Y2 > t2 ). For this quantity, the estimator constructed using the
Kaplan-Meier weights was built assuming the following equality S(t1 , t2 ) = 1 −P (Y1 ≤ t1 ) − P (Y1 > t1 , Y2 ≤ t2 ) where the first probability on the right handside is estimated using the Kaplan-Meier estimator of the first event and thesecond probability is estimated using Kaplan-Meier weights pertaining to the
distribution of the total time (i.e., T2 ) in a similar way as introduced above. The
weighted cumulative hazard estimator of the bivariate survival function is given
WCH
by Sb12(t1 , t2 ) = Pb(Y2 > t2 | Y1 > t1 )(1 − Pb(Y1 ≤ t1 )) where Pb(Y2 > t2 | Y1 > t1 )is obtained using the same ideas given above. This is the Wang and Wells (1998)
estimator.
Finally, landmark-based estimators can be introduced to estimate the bivariate
survival function. Given that P (Y1 > t1 , Y2 > t2 ) = 1 − P (Y2 ≤ t2 | Y1 >t1 )(1 − P (Y1 ≤ t1 )) the idea is to estimate P (Y2 ≤ t2 | Y1 > t1 ) by restricting the analysis to the individuals with an observed first event time greater or
LDM
equal than t1 . The corresponding estimator (LDM) is given by Sb12
(t1 , t2 ) =
Pn
[t1 )
[t1 )
KM
e
e
1 − i=1 Wi I(Y2i ≤ t2 ) × (1 − F1 (t1 )) where Wi denote theKaplan-Meiernoweights of the distribution of T2 computed from the sub sample i : Ye1 > t1 .

3

Example of Application

Our methodology is motivated by the re-analysis of the German breast cancer
data. In this study, patients were followed from the date of breast cancer diagnosis

Meira-Machado and Sampaio

287

until censoring or dying from breast cancer. From the total of 686 women, 299
developed a recurrence and 171 died. These data can be viewed as arising froma model with two consecutive events: ‘Alive with Recurrence’ and ‘Dead’. In
this section, we present plots for the proposed methods to estimate the bivariate
distribution function and bivariate survival function of the two gap times, Y1 =“Time to recurrence” and Y2 = “Time from recurrence to death”.

500

1000

1500

2000

Time

2500

0.4

S[x=730,y]

0.00
0

0.5

0.6

WCH
KMW
LDM

0.3

0.15
0.10

LIN
WCH
KMW
PKMW
LDM

0.05

F12[x=730,y]

0.20

0.25

0.7

Figure 1 reports estimated probabilities for a fixed value of t1 = 365 (days), alongtime. Plot shown in the left hand side (bivariate d.f) show that all proposed
methods behave quite similar something that is not true with regard to theestimation of the bivariate survival function (right hand side).

0

500

1000

1500

2000

2500

Time

FIGURE 1. Estimates of the bivariate d.f. and bivariate s.f. using the proposed
methods. Breast cancer data.

4

Simulation Studies

In this section, we investigate the performance of the proposed estimators throughsimulations. To simulate the data we consider the bivariate exponential distribution with marginal exponentials with rate parameter 1. This corresponds tothe so-called FarlieGumbelMorgenstern copula, where the single parameter controlling for the amount of dependence between the gap times. An independent
uniform censoring time C was generated, according to models U nif orm(0, 4) andU nif orm(0, 3). For each simulated setting we derive the analytic expression of
F12 (t1 , t2 ) and S12 (t1 , t2 ) for several (t1 , t2 ) pairs, corresponding to combinationsof the percentiles 20%, 40%, 60%, and 80% of the marginal distributions of the
gap times (i.e., 0.2231, 0.5108, 0.9163, 1.6094). Sample sizes n = 100, n = 250,
and n = 500 were considered.
Results reveal that the all proposed methods for estimating the bivariate distribution function perform quite well, though the performance of all methods ispoorer at the right tail (i.e., larger values of t1 and t2 ) where the censoring effects
are stronger. At these points the standard deviation (SD) is in most cases larger.
The SD decreases with an increase in the sample size and with a decrease of thecensoring percentage. All methods proposed in this work obtain in all settings anegligible bias.
Attained results for the bivariate survival function reveal that the weighted cumulative hazard estimator (WCH) is the recommended approach. This is illustrated

288

multivariate distributions for recurrent event data

0.1

0.5

0.2

0.6

0.3

0.4

0.7

0.5

0.8

0.6

in Figure 2 in which we show the boxplots of the estimates for the bivariate
distribution function.

KMW LDM CHW KMW LDM WCH KMW LDM WCH
n = 100n = 250n = 500

0.0

0.1

0.1

0.2

0.3

0.2

0.4

0.3

0.5

KMW LDM WCH KMW LDM WCH KMW LDM WCH
n = 100n = 250n = 500

KMW LDM WCH KMW LDM WCH KMW LDM WCH
n = 100n = 250n = 500

KMW LDM WCH KMW LDM WCH KMW LDM WCH
n = 100n = 250n = 500

FIGURE 2. Boxplot with estimated probabilities S12 (t1 , t2 ). On the top results
for the pair (0.2231, 0.2231) (left) and (0.2231,0.9163) (right); on the bottom
results for the pair (0.9163, 0.5108) (left) and (1.6094,1.6094) (right).

Acknowledgments: This research was financed by Portuguese Funds throughFCT - “Fundação para a Ciência e a Tecnologia”, within Project
UID/MAT/00013/2013.
Referencesde Uña-Álvarez J. and Meira-Machado L. (2008). A simple estimator of the bivariate distribution function for censored gap times. Statistics and Probability Letters, 78, 2440 – 2445.
de Uña-Álvarez J and Amorim, A.P. (2011). A semiparametric estimator of thebivariate distribution function for censored gap times. Biometrical Journal,
53, 113 – 127.
Kaplan, E. and Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53(282), 457 –481.

Meira-Machado and Sampaio

289

Lin, D., Sun, W. and Ying, Z. (1999). Nonparametric estimation of the time distributions for serial events with censored data. Biometrika, 86(1), 59 – 70.
van Houwelingen, H.C. (2007). Dynamic prediction by landmarking in event history analysis. Scandinavian Journal of Statistics, 34, 70 – 85.
Wang, M.C. and Wells, M.T. (1998). Nonparametric Estimation of successive duration times under dependent censoring. Biometrika, 85, 561 – 572.

Multi-Parameter Regression and Frailty
Kevin Burke1 and Gilbert MacKenzie1,2
12

University of Limerick, Limerick, Ireland
CREST, Ensai, Rennes, France

E-mail for correspondence: kevin.burke@ul.ie
Abstract: We introduce a class of survival models which account for the presenceof frailty and in which both scale and shape parameters of the hazard function
depend on covariates (we call this “Multi-Parameter Regression”). This generalformulation allows us to consider potential origins of time-dependent covariate
effects. We apply the models to lung cancer data using the mpr package in R.
Keywords: frailty; multi-parameter regression; survival; time-dependent effects

1

Introduction

Consider the following basic hazard modelh(t) = λ h0 (t ; γ)(1)where λ > 0 controls the overall size of h(t), and h0 (t ; γ) is a non-negative
function with shape parameter γ > 0 characterising its time-evolution. Survival
data may exhibit heterogeneity beyond that of (1). Thus, we introduce a latentvariable, U ∈ [0, ∞), representing this heterogeneity and consider the conditional
hazard functionh(t | U ) = U λ h0 (t ; γ).
Furthermore, we assume that U follows a one-parameter gamma distribution
such that E(U ) = 1 and V ar(U ) = ν, i.e., the Gamma Frailty (GF) model(cf. Duchateau and Janssen (2007)). Since U is unobservable, so too is the conditional hazard; in fact, we observe the marginal hazard,
hm (t) =where H0 (t) =Rt0

λ h0 (t ; γ)
1 + ν λ H0 (t ; γ)(2)h0 (u)du. Note: when ν = 0, we have hm (t) = h(t).

This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Burke and MacKenzie

2291

Multi-Parameter Regression Modelling

Naturally, we wish to consider the effect of covariates on the hazard function.
Although it is standard practice to allow one distributional parameter to dependon covariates (the “interest” parameter), more flexibility is achieved by allowing multiple parameters to depend on covariates simultaneously – we refer tothis as Multi-Parameter Regression (MPR) (Burke and MacKenzie, 2016) – and,
therefore, we suggest the following:log λ = xT β,

log γ = xT α,

where x = (1, x1 , . . . , xp )T is a vector of covariates, and β = (β0 , β1 , . . . , βp )T and
α = (α0 , α1 , . . . , αp )T are the corresponding vectors of scale and shape regression
coefficients. We could go even further and let the frailty variance, ν, depend on
covariates, but this is beyond the scope of the current paper. This regression
specification leads to the marginal hazard functionThm (t | x) =T

ex β h0 (t ; ex α )1 + ν exT β H0 (t ; exT α )(3)which is quite general as it covers a variety of models:• PH: Proportional Hazards model when ν = 0 and xT α = α0 . If h0 (t) is
non-parametric, we have Cox’s (1972) semi-parametric model.
• PH-GF: The Gamma Frailty extension of the PH model when xT α = α0 .
See Duchateau and Janssen (2007).
• MPR: The Multi-Parameter Regression extension of the PH model when
ν = 0. See Burke and MacKenzie (2016) for details.
• MPR-GF: The GF extension of the MPR model is given by (3) without
constraining any parameters.
Let x(−j) = (1, x1 , . . . , xj−1 , 0, xj+1 , . . . , xp )T be the covariate vector with its jthelement set to zero so that we may write
xT β = xj βj + xT(−j) β,

xT α = xj αj + xT(−j) α,i.e, we explicitly express the linear predictor as a term involving xj and all otherterms. This is useful when considering the hazard ratio for xj :ψj (t) =hm (t | xj = 1)
= exp(βj ) · ηj (t) · ρj (t)hm (t | xj = 0)
| {z } | {z } | {z }
PH

where

MPR

α +xT
(−j) αηj (t) =andρj (t) =h0 (t ; e j

xT
α
(−j)

GF

)

h0 (t ; e

)

xT

xT

β

α

1 + ν e (−j) H0 (t ; e (−j) )
β +xT
β
(−j)

1+νe j

(4)

α +xT
α
(−j)

H0 (t ; e j.
)

Thus, the leading term in ψj (t) is the familiar PH constant, exp(βj ), the second
term, ηj (t), appears due to the MPR extension, and the third term, ρj (t), appears

292

Multi-Parameter Regression and Frailty

due to the GF extension. This general formulation is very useful as it allows us
to compare the various models and, in particular, to consider the nature of timedependent hazard ratios.
The presence of frailty is a standard explanation for observing time-dependent
effects in practice (Aalen, 2008, chap. 6). However, frailty on its own is notvery flexible in that context: we gain a single additional parameter, ν, whichis unlikely to capture the time-dependence arising for each covariate, xj , j =1, . . . , p. Furthermore, frailty imposes time-dependence on all hazard ratios (i.e.,ρj (t) always depends on time), and, in the absence of the MPR extension (i.e.,setting xT α = α0 ), the hazard ratios also must converge to one with time.
Hazard ratios are constant in the basic PH model since h0 (t ; γ) is common to allindividuals. Thus, compared with frailty, a more fundamental explanation for theappearance time-dependent hazard ratios is perhaps simply that hazard functions
take on different shapes for different individuals, i.e., γ depends on covariates.
In contrast to the GF extension, the MPR extension produces an additionalparameter per covariate, αj , j = 1, . . . , p, which allows flexibility in modellingthe hazard ratios for each of these covariates separately. Moreover, when αj = 0,
we see that ηj (t) = 1 and, therefore, in the absence of frailty (i.e., setting ν = 0),
the hazard ratio reduces to the usual PH constant, i.e., the MPR extension does
not impose time-dependence on all hazard ratios.
The MPR and GF extensions offer alternative explanations for non-PH effects.
However, we may contemplate the general MPR-GF model and whether or notboth extensions are required simultaneously in practice.

3

Analysis of Lung Cancer Data

We applied the above models to a lung cancer dataset collected in Northern
Ireland between October 1991 and September 1992 (see Burke and MacKenzie,2016) using the mpr package in R (Burke, 2016). We consider only single-factor
treatment models here and specialise to the case where h0 (t) = γ tγ−1 (Weibull
baseline).
A brief summary of the models appears in Table. 1 where, interestingly, the most
TABLE 1. Summary of Treatment Models.

ν̂
AIC
BIC

PH

PH-GF

MPR

MPR-GF

—–
57.7
33.9

0.91
25.3
6.3

—–
20.4
15.6

0.74
0.0
0.0

complex model (MPR-GF) minimises both AIC and BIC. Thus, it appears thatboth extensions (MPR and GF) can be supported simultaneously in practice;the improved fit is evident when we compare the fitted models to the KaplanMeier curves (not shown). The hazard ratios arising from the various models(not shown) clearly display the flexibility achieved by modelling the shape of thehazard function. Note that the estimated frailty variance, ν̂, is smaller in the case

Burke and MacKenzie

293of the MPR-GF which we may expect as more variation is explained under thismodel compared with the simpler PH-GF model.

4Discussion

The class of models considered here allows us to compare various approaches.
From a practical perspective, frailty as an explanation of time-dependent effects
is somewhat contrived, whereas the MPR approach directly extends the PH model
more fundamentally without latent assumptions. However, in practice, combining
MPR and frailty is useful: in our application, the MPR-GF model is best interms of AIC and BIC. Thus, the GF extension of the PH model fails to absorb
all unexplained variation (e.g., due to covariate-dependent hazard shape) but,equally, unexplained variation still exists beyond the MPR model.

Acknowledgments: The first author would like to thank the Irish ResearchCouncil (www.research.ie) for supporting this work.
References
Aalen, O. O. and Borgan, Ø. and Gjessing, H. K. (2008). Survival and event history: A process point of view. Springer.
Burke, K. (2016). mpr: Multi-Parameter Regression (MPR). R package version
1.0.4 (http://cran.r-project.org/package=mpr).
Burke, K. and MacKenzie, G. (2016). Multi-parameter regression survival modeling: An alternative to proportional hazards. Biometrics, DOI: 10.1111/biom.12625.
Cox, D.R. (1972). Regression models and life tables (with Discussion). Journalof the Royal Statistical Society, Series B, 34, 187 – 220.
Duchateau, L. and Janssen, P. (2007). The Frailty Model. Springer.

Comparison of Semiparametric Copula andFrailty Models for Clustered Survival Data
Il Do Ha1 , Jong-Min Kim2
12

Department of Statistics, Pukyong National University, Busan, South Korea
Division of Science and Mathematics, University of Minnesota-Morris, USA

E-mail for correspondence: idha1353@pknu.ac.kr
Abstract: In clustered survival data, dependence among individual survival
times within a cluster has been usually described using copula models and frailty
models. In this paper we compare the two modelling approaches under semiparametric setting. We also investigate relationships between copula-based likelihood and frailty-based marginal likelihood, and propose corresponding profile
likelihood-based efficient procedures. The proposed method is demonstrated using a well-known CGD recurrent infection data set with different cluster sizes.
Keywords: Copula model; Frailty model; Marginal likelihood; Profile likelihood.

1

Introduction

A dependence among individual survival times within a cluster can be modelledusing frailty or copula. Here, the frailty is an unobserved random effect acting multiplicatively on the individual hazard rate (Vaupel et al., 1979) and has
been widely used for modelling the dependence among clustered survival datain biomedical studies (Duchateau and Janssen, 2008; Ha, Jeong and Lee, 2017).
The copula is a convenient way to study the dependence between random variables; following Sklar’s (1959) theorem, a copula expresses the joint distributionof random variables as a function of marginal distributions of each variable (Joe,1997). In particular, copula is useful for modelling a dependency in finance risks(Kim and Jung, 2016).
In this paper we consider clustered survival data where the size of each cluster may
be different. Let Ti1 , . . . , Tini be survival times (time-to-events) for the jth (j =1, . . . , ni ) observation of the ith (i = 1, 2, . . . , q) cluster. The survival data may becorrelated because Tij ’s are observed on the same cluster. Here q is the numberof clusters and ni is the number of individuals in cluster i (i.e. cluster size). Let
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

Ha and Kim

295

xij = (xij1 , . . . , xijp )T be the p covariates corresponding to Tij . The observablerandom variables are Yij = min(Tij , Cij ) and δij = I(Tij ≤ Cij ), where Cij is the
censoring time. We assume the non-informative and independence for censoring.
In this paper we compare two popular models (i.e. semi-parametric marginal
copula and conditional frailty models) using profile likelihood approaches. The
proposed method is illustrated using a practical CGD example (Fleming andHarrington, 1991).

2

Likelihood-based procedure for copula models

Under the Archimedean copula family, the joint survival function of Ti1 , . . . , Tinifor cluster i given xij can be expressed as−1
S(ti1 , . . . , tini |xij ) = ϕθ [ϕ−1
θ {S1 (ti1 |xi1 )} + · · · + ϕθ {Sni (tini |xini )}],

where Sj (tij |xij ) is a marginal survival function for Tij given xij (j = 1, 2, . . . , ni ).
Here, the generator ϕθ of Archimedean copula can also be expressed as a Laplacetransform of a positive distribution function Gθ (y); e.g., for the Clayton copulaϕθ (s) = (1 + θs)−1/θ for θ ≥ 0. Thus, the joint survival function above for cluster
i can be rewritten as


Z
ni
X
(1)
ϕ−1
S(ti1 , . . . , tini |xij ) = exp −yθ {Sj (tij |xij )} dG(y).
j=1Here we assume the marginal survival function is obtained from the Cox’s proportional hazards (PH) model:
λij (t|xij ) = λ0 (t) exp(xTij β),(2)where λ0 (·) is an unspecified baseline hazard and β = (β1 , . . . , βp )T is a p × 1
vector of regression parameters corresponding to covariates xij . Since ϕθ is the
Laplace transform of Gθ , following Prenen et al. (2017) and the derivative of (1),the Clayton copula-based log-likelihood for all q clusters has a closed form:

X

ni
q  ni
X
X 
(di )
−1
δij log fij − log ϕ0θ {ϕ−1
(S
)}
+
log
ϕ
ϕ
(S
)
`c =
ij
ij
θ
θ
θ
i=1 j=1=

X

j=1

δij {log λij + θΛij } −

ij

q 
X

di −1
∗
(di + θ−1 ) log(1 + Si+
)−

i=1

P ni

X


log(1 + lθ) ,

l=0

P i
−θ
∗
where di = j=1 δij , Sij = Sj (yij |xij ), fij = fj (yij |xij ) and Si+= n
j=1 (Sij −T
T1). Here λij = log λ0 (yij )+xij β and Λij = Λ0 (yij ) exp(xij β) with the baseline cumulative hazard Λ0 (t) are from the marginal Cox model (2). For the estimation of
copula models with the the model (2), we present the two-stage semi-parametricestimation procedure (Shih and Louis, 1995). Since the functional form of λ0 (t)in (2) is unknown, we consider a step function for Λ0 (t) (Breslow, 1972). Thatis, in the first stage, β is estimated by maximizing the Breslow partial likelihood(`p (β)) and Λ0 is estimated by the Breslow’s estimator. In the second stage, θ isestimated by maximizing a profile likelihood based on the estimates (β̂ and Λ̂0 )obtained from the first stage,`c (θ) = `c |β=β̂,Λ0 =Λ̂0 .

296Comparison of Copula and Frailty Models

In this paper, we also propose a likelihood ratio test (LRT) for testing the absenceof association parameter θ, H0 : θ = 0. However, care is necessary because sucha null hypothesis is on the boundary of the parameter space (θ ≥ 0). Thus, thestandard chi-square distribution can not be applied. The null distribution for
the LRT statistic follows an asymptotic chi-square mixture distribution, i.e. a
mixture of χ20 and χ21 with equal weights of 0.5, (Stram and Lee, 1994). Thus, it
is calculated as
LRc = −2{`p (β̂) − `c (θ)},
leading that the p-value is calculated from p = 12 P (χ21 > LRc ). Here `p (β̂) is thepartial log-likelihood evaluated at β = β̂ under H0 : θ = 0.

3

Likelihood-based procedure for frailty models

Let Ui be a shared frailty (random effect) of the ith cluster. The semi-parametricfrailty model (Duchateau and Janssen, 2008) is described as follows. The conditional hazard function of Tij given xij and Ui = ui takes the form ofλij (t|xij , ui ) = λ0 (t) exp(xTij β)ui ,(3)

where λ0 (·) is an unspecified baseline hazard. Popular distributions for the frailty
Ui are gamma and log-normal; for the gamma frailty model E(Ui ) = 1 andvar(Ui ) = θ. Following Nielsen et al. (1992), the marginal joint survival function
is given by


Z
ni
X
ϕ−1
(4)
Sm (ti1 , . . . , tini |xij ) = exp −uiθ {Sjm (tij |xij )} dGθ (ui ),j=1

Rwhere Sjm (tij |xij ) = S(tij |xij , ui )dGθ (ui ) with distribution function Gθ (·) of
Ui . As mentioned by Goethals (2008) and Prenen et al. (2017), the two joint
survival functions (1) and (4) are similar in that both joint functions take theform of the same copula, but Sj (tij |xij ) 6= Sjm (tij |xij ), which gives a majordifference between both models. We find that the marginal log-likelihood for all
q clusters under frailty models (3) is obtained by by replacing fij and Sij in `cmmby the corresponding functions fij
and Sij; under gamma frailty model (3), it
has also a closed form:


X

ni
ni
n X
X
(di )
m
m
−1
m
δij log fij− log ϕ0θ {ϕ−1
`m =
(S
)}
+
log
ϕ
ϕ
(S
)
.
ij
ij
θ
θ
θ
i=1 j=1

=

X
ij

j=1


di −1
q 
X
X
F
−1
F
δij (log λij ) −(di + θ ) log(1 + θΛi+ ) −log(1 + lθ) ,
i=1

l=0

Pni

T
F
T
where λFij = log λ0 (yij ) + xij β and Λi+ =j=1 Λ0 (yij ) exp(xij β) are from thefrailty models (3). Following Ha et al. (2010), we propose the use of a simple gridsearch method to implement the likelihood method. In the inner loop, given θ,(β, Λ0 ) are obtained by solving the estimating equations ∂`m /∂(β, Λ0 ) = 0 via the
Newton-Raphson method. In the outer loop, given (β, Λ0 ), the profile likelihood`m (θ) = `m |β=β̃,Λ0 =Λ̃0 is maximized for θ. Here β̃ and Λ̃0 are the estimatesobtained from `m . We can again use the LRT for testing the absence of θ in thefrailty models, which is equivalent to test ui = 0 for all i (i.e. the absence of frailty
effect). Here, the corresponding LRT is given by LRm = −2{`p (β̂) − `m (θ)}.

Ha and Kim

297

TABLE 1. Results of fitting copula and frailty models for the CGD data.

Treatment

Association

Profile likelihoodβ̂ (SE)θ̂ (p-value)`(θ̂)−1.086(0.268)−1.138(0.345)

0.804(< .001)
1.383(< .0001)−347.86−345.84

Model

−348
−350

gPL
cPL

−352

profile likelihood−346

Clayton copula
Gamma frailty

0.0

0.5

1.0

1.5

2.0

2.5

3.0

θ

FIGURE 1. Profile likelihoods (PL) for association parameter θ in the CGD data;
gPL, `m (θ) under gamma frailty model; cPL, `c (θ) under Clayton copula model.

4

An illustrated example

Fleming and Harrington (1991) provided a clustered survival data set on a placebocontrolled randomized trial of gamma interferon (γIFN) in chronic granulomatous
disease (CGD). Here 128 patients had recurrent infections with ni = 1 ∼ 8, andwe consider a main covariate, i.e. Treatment (0 = placebo, 1 = γ-IFN). Table
1 shows both models give similar significance, but different estimates for θ (seealso Figure 1) as compared to β̂.

Acknowledgments: This research was supported by Basic Science Research
Program through the National Research Foundation of Korea (NRF) funded by
the Ministry of Education (No. NRF-2015R1D1A3A01015663).

References
Duchateau, L. and Janssen, P. (2008). The frailty model. Springer.
Ha, I. D., Noh, M. and Lee, Y. (2010). Bias reduction of likelihood estimators
in semi-parametric frailty models. Scandinavian Journal of Statistics, 37,
307–320.
Ha, I. D., Jeong, J.-H. and Lee, Y. (2017). Statistical modelling of survival datawith random effects: h-likelihood approach. Springer, in press.
Kim, J.M. and Jung, H. (2016). Linear time-varying regression with CopulaDCCGARCH models for volatility. Economics Letters, 145, 262-265.

298

Comparison of Copula and Frailty Models

Prenen, L., Braekers, R. and Duchateau, L. (2017). Extending the Archimedean
copula methodology to model multivariate survival data grouped in clustersof variable size. JRSS, B, in press.

Conditional inference survival trees for
nonstandard data
Wei Fu1 , Jeffrey S. Simonoff1
1

New York University, USA

E-mail for correspondence: jsimonof@stern.nyu.edu
Abstract: Tree methods (recursive partitioning) are a popular class of nonparametric methods for analyzing data. One extension of the basic tree methodology
is the survival tree, which applies recursive partitioning to censored survival data.
This has mainly been designed for right-censored data. We discuss application of
the conditional inference survival tree method to two important but less standard
data types, left-truncated and right-censored (LTRC) data and interval-censoreddata. Further, we show that LTRC trees can be used to analyze survival datawith time-varying covariates, essentially building a time-varying covariates survival tree. Implementation of the methods is easy, and simulations and real data
analysis results show that the proposed methods work well from both a predictive
point of view and in uncovering tree structure in the underlying survival process.
Keywords: Interval-censored data; Left-truncated and right-censored data; Survival tree.

1

Introduction

Right-censored data are often studied using a (semi-)parametric model such as
the Cox proportional hazards (PH) model or accelerated failure time model.
However, the parametric assumptions imposed by these models are often either
not met or unrealistic in practice, and more flexible nonparametric methods aredesired. One such method is the survival tree.
Various authors have proposed tree methods for right-censored data. Hothorn etal. (2006) (hereafter HHZ), in particular, implemented a survival tree using the
log-rank test as the splitting method. They embedded the survival tree algorithm
into a large framework of conditional inference trees, which has the desirable property of selecting the splitting variable in an unbiased way (an unbiased tree hasThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

300Survival trees for nonstandard data

the property that when there is no relationship between the response and any covariates all covariates have the same probability of being the split variable). Likeother proposed methods, this method is designed for the most basic setup of survival outcome – right-censored data with time-independent covariates. However,other types of survival data such as left-truncated and right-censored (LTRC)
data, interval-censored (IC) data, and survival data with time-varying covariates
arise commonly in practice.
In this paper, we discuss extension of the conditional inference survival tree
method to LTRC and IC data. The resulting methods are easy to implementin practice. Through data reformulation, the LTRC survival tree method canthen also be used to fit survival data with time-varying covariates.

2

Conditional inference trees for survival data

The conditional inference tree algorithm of HHZ is based on the idea of separating the two steps of choosing the variable for splitting and then choosing the
split point of that variable. The splitting variable is chosen based on conditional
distributions that are constructed assuming that the response and covariates are
independent. After the splitting variable is selected, the split point can be determined by any criterion. The association of Y and a covariate Xj is measured bylinear statistics of the form!
nXTwi gi (Xji )h (Yi )∈ Rpj qTj (Ln , w) = vec
i=1(equation 3.1 in HHZ), where gj : Xj → Rpj is a nonrandom transformation ofcovariate Xj and h : Y ×Y n → Rq is the influence function of the response Y . Fora univariate numeric response Y , the choice of influence function is the identity,i.e. h (Yi ) = Yi .

2.1

Log-rank score for right-censored data

HHZ discussed the construction of a conditional inference survival tree for rightcensored data. Subjects can be represented as a triple (ti , δi , xi ), i = 1, 2, ..., n,where ti is the observed event time or censored time for the ith subject, δi = 1 if
ti is the event time and δi = 0 if ti is the censored time and xi is the covariate
vector for the ith subject. We assume that censoring is noninformative given xi .
The response variable for the ith subject is Yi = (ti , δi ). The influence function
for such a bivariate response is the so-called log-rank score. The main functionof the log-rank score is to assign a univariate value Ui (scalar) to the bivariate
response Yi = (ti , δi ), so the algorithm can then execute in the same way as in
the univariate numeric response case.

2.2

Log-rank score for LTRC and IC data

Pan (1998) extended the rank invariant tests of Peto and Peto (1972) to lefttruncated and interval-censored data, with the log-rank score being
Ui =Ŝ(li ) log Ŝ(li ) − Ŝ(ri ) log Ŝ(ri )− log Ŝ(Li ).
Ŝ(li ) − Ŝ(ri )Fu and Simonoff

301

Here, Li is the left-truncation time and (li , ri ) is the interval in which the true
event time lies. The log-rank scores for LTRC and IC data can be derived fromthis score equation as special cases, which are then used in the general conditional
inference tree framework. The log-rank score for an LTRC observation (Li , Ri , δi )is
Ui = 1 + log Ŝ(Ri ) − log Ŝ(Li ) if δi = 1
and
Ui = log Ŝ(Ri ) − log Ŝ(Li ) if δi = 0.
Note that Ŝ is the nonparametric maximum likelihood estimator (NPMLE) of thesurvival function, constructed based on the product-limit estimator, i.e. KaplanMeier (KM) estimator. Details are given in Fu and Simonoff (2017a), and implementation of the method is provided in the R package LTRCtrees, which isavailable at CRAN.
A crude but common approach to IC data is to impute the midpoint or endpointof the censoring interval as the actual survival time, but it is known that this canlead to bias and incorrect inferences (Lindsey and Ryan, 1998). The appropriate
conditional inference tree, however, is easily defined using the log-rank score for
an IC observation, which is given by
Ui =Ŝ(Li ) log Ŝ(Li ) − Ŝ(Ri ) log Ŝ(Ri ),Ŝ(Li ) − Ŝ(Ri )where Li and Ri are the lower and upper boundaries of the censoring interval forthe ith observation, respectively. Once again Ŝ is the NPMLE of the survival function, here constructed using the EM-algorithm as proposed by Turnbull (1976).
If the event time is observed, so Li = Ri and the interval (Li , Ri ) reduces to apoint, the corresponding log-rank score is Ui = 1 + log Ŝ(Li ). More details are
given in Fu and Simonoff (2017b), and an R function that implements the method
is provided in the R package LTRCtrees.

3

Properties of the trees

Simulations show that when the true structure is a tree, the LTRC and IC trees
perform significantly better than the Cox PH model, regardless of sample size,
censoring distribution and left-truncation rate, as would be expected. Further,when the true structure is a complex nonlinear relationship the trees can outperform the Cox PH model, and never perform significantly worse than the Cox PH
model, demonstrating that the trees have more robust performance than does theCox PH model. The IC trees are uniformly better than building a tree based on
imputing the midpoint or endpoint of the censoring interval as the actual survival
time.

4

Real data examples

4.1

LTRC data

The assay of serum free light chain data in the R package survival is used as adata example. An analysis could use age as a covariate and time from enrollment

302

Survival trees for nonstandard datain the study as the response, but as noted by Klein and Moeschberger (2003),
age should actually be used as a left-truncation point, since the real response ofinterest should be the subject’s life length, not the time from enrollment in the
study.
Figure 1 gives conditional inference trees with the actual death/censoring time as
response. The tree that accounts for the left truncation (left panel) identifies the
top FLC decile (FLC=10) as the most important covariate of overall survival,
the same split used in the original analysis based on subject area knowledge. In
contrast, the tree that ignores the left-truncation (right panel) only uncovers theeffect of FLC for males.
1
FLC
p < 0.001

1sex
p < 0.001

≤9

>9

2
sex
p < 0.001

≤ 1.9

M

3
FLC
p < 0.001

6FLC
p < 0.001
≤7

≤5

≤2

M
3
FLC
p < 0.001

> 1.9

12
sex
p = 0.039

≤9

>9

4
FLC
p = 0.001

>7

7
FLC
p = 0.033

>5

F

11
creatinine
p < 0.001

F

F

M

≤7

>2

1

Node 2 (n = 3589)

1

Node 5 (n = 1946)

7
creatinine
p = 0.04>71

Node 6 (n = 649)

≤ 1.8
1

Node 8 (n = 271)

> 1.8
1

Node 4 (n = 1895)
Node 5 (n = 1354)
Node 8 (n = 505)
Node 9 (n = 1441)
Node 10 (n = 649)
Node 13 (n = 307)
Node 14 (n = 284)
Node 15 (n = 86)
1
1
1
1
1
1
1
1

0.8

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0.2

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0.2

0.2

0

0

0

0

0

0

0

0

0

0

0

0

0 40 80

0 40 80

0 40 80

0 40 80

0 40 80

0 40 80

0 40 80

0 40 80

0 20 40 60 80 100

0 20 40 60 80 100

0 20 40 60 80 100

Node 9 (n = 66)

0.2
00 20 40 60 80 100

0 20 40 60 80 100

FIGURE 1. Survival trees for the serum FLC data, accounting for the left-truncation (left) and ignoring it (right), respectively.

4.2

IC data

The Signal Tandmobiel R study is a longitudinal prospective oral health study
that was conducted in the Flanders region of Belgium for which data are provided
as the tandmob2 data set in the R package bayesSurv. The response variable isthe time to emergence of tooth 24, which is interval-censored since emergence isonly identified at scheduled dentist visits.
Figure 2 gives the IC tree for the emergence time of the tooth. Decay and gender
are associated with earlier emergence time. More orthodontic removal of deciduous teeth is associated with earlier emergence time, which could reflect a reverse
causality effect. There are province effects for some children, apparently due tomisclassification effects of certain examiners.

5

Adapting LTRC trees to time-varying covariate
data

Fu and Simonoff (2017a) also described how LTRC trees can be used to construct
trees for time-varying covariate data. The general strategy is to first split eachsubject into several “pseudo-subjects,” inside which covariates are constant. So,for example, if a covariate changes at time points t1 and t2 with the event time T ,the subject is split into three pseudo-subjects over the intervals (0, t1 ), [t1 , t2 ), and[t2 , T ]. This results in three created LTRC pseudo-subjects with time-independentFu and Simonoff

303

1
DMF.Scorep < 0.001≤6>6

2
Genderp < 0.001

15
Provincep = 0.016

boy

girl

3
BAD.Scorep < 0.001≤08
DMF.Scorep < 0.001

>0

≤2

>2

4
DMF.Scorep = 0.001

10
BAD.Scorep = 0.008

A, O, V

≤0

≤2

12
DMF.Scorep = 0.012

>2

≤3
1

Node 5 (n = 862)

Node 6 (n = 529)

1

L, W

>0

1

Node 7 (n = 288)

1

Node 9 (n = 938)

Node 11 (n = 471)

1

1

>3

Node 13 (n = 35)

1

Node 14 (n = 154)

Node 16 (n = 714)

1

0.8

0.8

0.8

0.8

0.8

0.8

0.8

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0.6

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0

0
0 1 2 3 4 5 6 70
0 1 2 3 4 5 6 70
0 1 2 3 4 5 6 70
0 1 2 3 4 5 6 70
0 1 2 3 4 5 6 70
0 1 2 3 4 5 6 7

0.2

0
0 1 2 3 4 5 6 7Node 17 (n = 397)

1

0.8

0
0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

FIGURE 2. IC tree for the emergence time in years after the age of 5 of tooth
24 (the permanent upper left first premolar).

covariates, and the LTRC tree algorithm is then applied on all such constructedpseudo-subjects to fit a tree. Simulations show that this constructed tree has
properties similar to those of the (underlying) LTRC tree.
The PBC data in the R package survival are used as an example. The data
were collected at entry and at yearly intervals on 45 variables. Figure 3 gives
conditional inference trees for these data. The tree in the left panel is based on
only baseline covariate values, while the one is the right panel uses the follow-up
data, in which all of the covariates except age become time-varying covariates.
The trees are quite noticeably different, with different variables appearing andthe top-level split variables being different.
1bili
p < 0.001≤ 0.6422
edema
p < 0.001> 0.6425
protimep < 0.001

1
albuminp < 0.001≤ 2.389≤ 1.037> 2.389≤03
protimep = 0.003

0 40000 40000 4000≤ 2.434

15
protimep = 0.002> 2.434

9
spidersp = 0.02

12age
p = 0.018>0≤ 2.526> 2.526≤ 56.772> 56.772

Node 4 (n = 80)
Node 5 (n = 80)
Node 6 (n = 77)
Node 10 (n = 1157)
Node 11 (n = 361)
Node 13 (n = 72)
Node 14 (n = 10)
Node 16 (n = 92)
Node 17 (n = 16)
1
1
1
1
1
1
1
1
1

Node 3 (n = 224)
Node 4 (n = 25)
Node 6 (n = 83)
Node 10 (n =Node7) 11 (n =Node20) 12 (n = Node50) 13 (n = 9)
1
1
1
1
1
1
1
0.8
0.8
0.8
0.8
0.8
0.8
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0.6
0.6
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0.2
0.2
0.2
0.2
0.2
0.2
0.2
0
0
0
0
0
0
0
0 4000>0

8bili
p < 0.001≤0≤ 44.14> 44.14

0 4000≤0≤ 2.434> 2.434≤ 1.224 > 1.224
9age
p = 0.019

0 40007
ascitesp < 0.001≤ 2.197 > 2.197≤ 1.314> 1.314
8bili
p = 0.035>0> 1.0372bili
p < 0.001

7albumin
p = 0.006

0.8

0.8

0.8

0.8

0.8

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0

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0

0 30000 30000 30000 30000 30000 30000 30000.4
0.2
00 30000 30000 4000FIGURE 3. Survival trees for PBC data based on baseline covariates (left) andtime-varying covariates (right), respectively.

304

Survival trees for nonstandard data

References
Fu, W. and Simonoff, J.S. (2017a). Survival trees for left-truncated and rightcensored data, with application to time-varying covariate data. Biostatistics, 18, 352 – 369.
Fu, W. and Simonoff, J.S. (2017b). Survival trees for interval-censored survival
data. arXiv:1702.07763, https://arxiv.org/abs/1702.07763.
Hothorn, T., Hornik, K., and Zeileis, A. (2006). Unbiased recursive partitioning:
A conditional inference framework. Journal of Computational and Graphical Statistics, 15, 651 – 674.
Klein, J.P. and Moeschberger, M.L. (2003). Survival Analysis: Techniques for Censored and Truncated Data, 2nd ed., Springer, New York.
Lindsey, J.C. and Ryan, L.M. (1998). Methods for interval-censored data. Statistics in Medicine, 17, 219 – 238.
Pan, W. (1998). Rank invariant tests with left truncated and interval censoreddata. Journal of Statistical Computation and Simulation, 61, 163 – 174.
Peto, R. and Peto, J. (1972). Asymptotically efficient rank invariant test procedures. Journal of the Royal Statistical Society, Series A, 135, 185 – 207.
Turnbull, B.W. (1976). The empirical distribution function with arbitrarily grouped,censored and truncated data. Journal of the Royal Statistical Society, Series B, 38, 290 – 295.

Bayesian Joint Modelling of Distributional
Regression
Elisabeth Waldmann1 , Nadja Klein2 , David Taylor-Robinson3
1

Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Georg-August-Universität Göttingen, Germany
3
University of Liverpool, England
2

E-mail for correspondence: elisabeth.waldmann@fau.de
Abstract: When modelling repeated measurement and time to event data simultaneously, analysis is often based on combining random effects regression on themean with survival analysis in a so called joint model. The assumptions made bythis model, however, are not necessarily met, when analysing real life data sets.
Especially the assumption of equidispersion is too strict in many cases. In simple
longitudinal setups this issue is often dealt with distributional regression, which
introduces separate predictors for all parameters of the distribution. This work
aims at combining the setup of a Bayesian distributional model with a survival
model, linking both, the conditional mean and conditional variance to the risk ofevent and thus developing a new approach towards distributional joint modelling.
Keywords: Joint Modelling; Mixed Models; Time-to-event data; Bayesian Inference.

1

Introduction

Amongst the many statistical methods that have been developed for analysing
data from longitudinal studies, the term joint modelling refers to the statistical modelling of data in which each subject provides data on two qualitatively
different kinds of outcome variable: a time-sequence of repeated measurements;
and a (possibly right-censored) time-to-event variable. In most cases the model
used for the longitudinal data is a mean regression model, which very often lacks
in flexibility. Distributional regression gives a more complete overview over the
dependent variable. It has recently been implemented thoroughly in a Bayesianframework by Klein et al (2014). Instead of only modelling the expectation of aconditional distribution further parameters of the distribution are assigned preThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

306

JM

dictors. In case of the Gaussian distribution the variance parameter σ 2 is modeledby a (potentially additive) predictor ησ2 . Data-sets with heteroscedastic characteristics are modeled more accurately when using distributional regression. The
latter is often the case in longitudinal studies and the structure of the variancepotentially holds information on the pattern of the time events analysed in thesurvival analysis. Our interest in this model class was triggered when analysing
lung function decline in cystic fibrosis patients from a Danish cystic fibrosis registry. It has been shown that the onset of certain pulmonary infections is associated with an acceleration of the loss of lung function, when using the onset of
infection as a covariate in a longitudinal model (Qvist et al. 2015). The onset of
infection however, could be seen as a process influenced by the same covariates
as the lung function decline itself and hence should be modeled as a related yet
separate process. A further interest lies in better understanding how the variability in lung function is related to lung function decline and other outcomesin cytic fibrosis. This contribution hence deals with including this distributional
regression feature into joint modelling and developing a Monte Carlo Markov
Chain (MCMC) algorithm.

2

Distributional Regression in Joint Modelling

2.1

Joint Modelling

There are many different approaches in combining time-to-event outcomes withlongitudinal data in joint models. The type we are going to refer to here, isa model as suggested by Faucett et al (1996) which is based on the following
likelihoods:f (Yij |ηij,· , σ)∝f (si , δi |Yi )

=

mi
1 X
(yij − (ηlij + ηlsij ))2
exp
−
σ mi
2σ 2 j=1

1!

λ0 (si )δi exp (ηsi + αηlsij )
Z si

· exp
λ0 (u) exp (ηsi + αηlsij ) du ,0

where Yij is the j-th outcome for individual i. The event time/censoring timeis denoted by si and δi refers to the censoring indicator. The predictors forthe two parts of the model are composed of so called sub-predictors η·ij refer
to longitudinal, survival and shared sub-predictors (l, s and ls). In case of thelongitudinal process, the composed predictor is ηlij + ηlsij , in case of the timeto-event process the predictor is composed of ηsij + αηlsij , where α is the socalled association parameter, measuring the connection between the two partsof the model. Faucett et all (1996) suggest a MCMC algorithm to estimate thecorresponding parameters.

2.2

Distributional Regression

Distributional regression models – also known as generalized additive models forlocation, shape and scale (GAMLSS) – are based on not only measuring theWaldmann et al.

307

impact on the mean but on more parameters of the conditional distribution. In
case of the Gaussian distribution the second parameter which is of interest is thevariance. The classical likelihood for the Gaussian distribution is hence extended
by a predictor ησ2 . In order to ensure positivity of the variance, an exponential
link function is used, such that σ̂ 2 = exp(η̂σ2 ), with ησ2 being a additive predictor
including a variety of different types of effects. Klein et al (2014) suggested an
MCMC algorithm using iteratively weighted least squares proposal densities.

2.3

Distributional Regression in Joint Modelling

Combining the two above explained concepts leads to an extended set of subpredictors, now including sub-predictors specifically designed for quantifying theimpact of the covariates on the variation in the data and the connection ofthem to the survival probabilities. Thus the set of longitudinal sub-predictors is
ηl,µ (xlij,µ ),ηl,σ (xlij,σ ) and the set of shared sub-predictors: ηls,µ (xlsij,µ ),ηls,σ (xlsij,σ ).
The survival sub-predictor does not have to be extended. Furthermore the model
is completed by a variance specific association parameter ασ2 . The extended likelihoods are hence:f (Yi |η·,i,· )∝1
exp ησ2 ,i,l +, ησ2 ,i,ls· exp −f (si , δi |η·,i,· )

=

ni
X

1
2 exp ησ2 ,i,l + ησ2 ,i,ls!
(yij − (ηµ,i,l + ηµ,i,ls ))2

j=1

δi


λ0 (si ) exp δi αµ ηµ,i,ls + ασ2 ησ2 ,i,ls + ηµ,i,s + ησ2 ,i,s
Z si

· exp
λ0 (u) exp αµ ηµ,i,ls + ασ2 ησ2 ,i,ls + ηµ,i,s + ησ2 ,i,s du0

The MCMC algorithm was constructed as a mixture of the one done by Faucett
et al (1996) and Klein et al (2014).

3

Simulations

In order to evaluate the algorithm we first simulated a typical setting, similar
to the simulation study in Waldmann et al (2017) were done. The sub-predictor
were constructed as linear functions. The shared mean sub-predictor included afixed and random time-effect, as well as a random intercept. We simulated 300
individuals of which each had a maximum of 5 observations. The baseline hazard
was chosen to be constant and we varied the association parameters. We did
not include a survival sub-predictor. Results for one simulation setup are shownin Figure 1. The black lines indicate the true value, the boxplots display themeans of 100 simulation runs. The results show that our algorithm captures thesimulated effects very closely.

308

JM

Longitudinal Sub−Predictor

Impact on the mean

Impact on the variance
1.0

2

1.0

●

0.5

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1

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0

0.0

−1

−0.5

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Shared Sub−Predictor

●

1

1.0

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ασ2

●

●

●
●

0
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−1
−0.5

0.05
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−2

−1.0

time−effect

0.00

λ

FIGURE 1. Results for the regression parameters from all four sub-predictors
as well as association parameters and fixed baseline hazard. The solid black lines
show the true values, the boxplots the MCMC sample means of 100 repetitionsof the same model.

4Lung Function Decline in Cystic Fibrosis Patients

In order to establish if our model is applicable to real life data, we choose asubset of a data set taken from the Danish cystic fibrosis registry. After choosing
the patients that have at least two observations, before the infection the data
set contained a total of 6268 of 489 patients of which 53 were infected withpseudomonas aeruginosa (PA) in the course of the study. The model calculated
aims at explaining both, the progression of lung function, measured in Forced
Expiratory Volume in 1 second (FEV1) and the risk of onset of PA. The covariates
chosen for the model were preselected based on previous studies (Waldmann et al(2017)). As covariates for both longitudinal sub-predictor height and weight of thepatient as well as three binary covariates indicating, if the patient had one of threedifferent additional lung infections were chosen. For the shared sub-predictors weconsidered time and sex for the mean predictor and pancreatic insufficiency andyear of birth for the variance. The boxes for the MCMC samples of the covariates
are displayed in the first two rows of Figure 2. Boxes of variables that do notinclude the value 0 in the 95% interval of the sample (i.e. which are considered tohave a significant influence) are black, the rest is gray. The results for the meanregression is similar to previous findings: while patients with higher weight havebetter lung function, height, additional infections and diabetes have a negative
impact. The variance is not influenced by weight and the additional infection,
whereas height and diabetes have a negative influence on the variance, i.e. thelung function varies less for taller patients and for patients having diabetes. The
second row has to be interpreted taking into account the association parameterswhich are displayed in the last row. The association with the mean shared subpredictor is negative, such that we can conclude that the time impact, which is
negative for the lung function itself is positive for the risk of being infected. The

Waldmann et al.
βlµ

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Intercept

HeightWeightInfection

Diabetes●●●●

Intercept

HeightWeightβlsµDiabetes

βlsσ2
●

0

Infection

0.35

●

●
●
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0.30
−1

0.25
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●

0.10
●
●
●
●
●

0.05

●
●
●
●
●

0.00
Time

Sex

Pancreatic Insufficiency

αµ
●
●
●

Birthyear

ασ2
8

●
●

●
●

7

●
●
●
●
●
●

−0.5
6

5
−1.0
4

3
−1.5

●
●
●
●
●
●
●
●
●
●

αµ

●
●
●

2

●
●
●

ασ2

FIGURE 2. Results from a model for lung function decline and risk of lunginfection in cystic fibrosis patients. The upper four plots contain the boxplots of
the MCMC samples for the linear model parameters (black if 95% of the MCMC
sample do not include zero, gray otherwise). The two plots in the last row show
the boxplots of the MCMC samples for the association parameters.

association parameter for the variance shared sub-predictor is positive. Hence theincrease in variance for higher years of birth and pancreatic insufficiency increases
the risk of infection.

310

5

JM

Conclusion

In this work, we show how to combine distributional regression and joint modelling in order to extract more information about data generating processes inlongitudinal and time-to-event data. The results for the cystic fibrosis study canbe seen as a first hint in the direction of answering the question of the impact ofvarying lung function on susceptibility for lung infections rather than the lungfunction level itself. To answer this question with more ultimate certainty, however, a more refined model has to be constructed.

References
Faucett, C. and Thomas, D. (1996). Simultaneously Modelling Censored Survival
Data and Repeatedly Measured Covariates: a Gibbs Sampling Approach.
Statistics in Medicine, 15, 1663 – 1685.
Qvist, T., Taylor-Robinson, D., Waldmann, E., Olesen, H., Rønne Hansen, C., Mathiesen, M. I., Høiby, N., Katzenstein, T. L. , Smyth, R. L. , Diggle P., Pressler
T. (2016). Infection and lung function decline; comparing the harmful effects of cystic fibrosis pathogen. Journal of Cystic Fibrosis 15(3), 380–
385.
Klein N., Kneib T., Klasen S., Lang S. (2014). Bayesian Structured Additive Distributional Regression for Multivariate Responses. Journal of the Royal
Statistical Society: Series C, 64, 569 – 591.
Waldmann E., Taylor-Robinson D., Klein N., Kneib T., Pressler T., Schmid M.,
Mayr A.(2017): Boosting Joint Models for Longitudinal and Time-to-Event
Data. Biometrical Journal. accepted.

Incorporating cyclical effects andtime-varying covariates in models for
single-source capture-recapture data
Thomas Husken1 , Maarten Cruyff1 , Peter van der Heijden1212

Utrecht University, the Netherlands
University of Southampton, England

E-mail for correspondence: t.f.husken@uu.nl
Abstract: The objective of capture-recapture analysis is to estimate the sizeof an elusive population, for which the zero-truncated Poisson model is a basicmodel. We extend this model to the more general recurrent events model toinclude cyclical effects and time-varying covariates. An application to police dataon victims of domestic violence provides strong evidence for the presence of weekly
and seasonal cyclical effects on the rate of police reports.
Keywords: single-source capture-recapture; cyclical effects; time-varying covariates; recurrent events model; zero-truncation.

1

Introduction

The zero-truncated Poisson model is a well-established model for the analysis
of single-source capture-recapture data. Such data typically arise when each observation of a member of an elusive population is recorded in a registration file.
Counting the number of records for each individual population member yields azero-truncated count distribution, because population members with a zero countare not in the register. Under the assumption that the counts follow a Poisson distribution, an estimate of the Poisson parameter can be obtained that in turn canbe used to estimate the frequency of the zero count. Relevant covariates can be
included to model individual differences in Poisson parameters, this leads to thezero-truncated Poisson regression model (TPR) (see Cruyff & van der Heijden,
2013; van der Heijden et al., 2003).
Like any type of events data, single-source capture-recapture data can exhibit
seasonal or cyclical patterns. For example, a homeless person may be more likely
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

312Time-varying and cyclical effects in capture-recapture models

to stay in a homeless shelter during winter than in the summer and a problematic
drug user may have a higher probability of being admitted to the hospital in theweekend than during the week. However, the TPR is unable to incorporate thesetypes of cyclical effects.
In this paper we present a method that allows for the inclusion of cyclical effects insingle-source capture-recapture data. In this method, the TPR is extended to thezero-truncated recurrent events model (TREM), which includes a time dimension.
This dimension makes the model more general than the TPR, since it allows forthe inclusion of time-varying covariates and cyclical effects. The resulting model
can accommodate a wide variety of effects: time-invariant, cyclical, time-varying,and interactions thereof.

2

MethodThe TREM is an extension of the TPR that allows for the modelling of timevarying covariates and cyclical effects in single-source capture-recapture data.
The likelihood of the TREM is given by(y −Λi (τ ) )
n
i
Y
Y
e
λij
L(β) =,(1)1−e−Λi (τ )i=1 j=1
P
where Λi = τt=1 λit , and ln λit = β0 + β1 xit1 + · · · + βp xitp (Cook & Lawless,
2007, p. 273-278). Here, xitp specifies the value covariate p takes at time point t
for person i. Additionally, yi is the total number of captures over the observation
period for individual i. Note that this allows for time-varying covariates since
xitp may vary over time.
Cyclical effects are modelled by adding a cosine term to the linear predictor of
the TREM:
2π
ln λit = β0 + β1 xit1 + · · · + βp xitp + α cos
t−θ ,
(2)
k
where α is the amplitude and θ the horizontal shift. The period k is a constantand determines how often the cyclical component peaks. The cosine term inEquation (2) is non-linear and therefore difficult to estimate, but can be rewritten
to a linear function. The most common method is using a trigonometric identity
to parametrise the cyclical effect as






2π
2π
2π
α cos
t − θ = βcos cos
t + βsin sin
t ,
(3)
k
k
k
from Cryer & Chan (1994, Ch. 3, p. 34). The final expression can be easily
included in the linear predictor of the recurrent events model, where cos( 2πt)kand sin( 2πt) are entered as covariates. The interpretation of the two cyclicalkregression coefficients is not very intuitive, but they can be transformed back interms of α and θ:q
2 + β2 ,α = βcossin(4)θ = arctan2(βsin , βcos ),Husken et al.

313

which is also known as a polar transformation. The linear parametrisation of
Equation (3) has the advantage that interaction effects with time-invariant covariates can be included almost in the classical manner. This provides the option
to include main effects of time-invariant covariates, cyclical effects, and interactions between these two simultaneously.
Parameter estimates of the TREM are obtained by optimizing the loglikelihood
given by
yi
n X
n
n
X
X
X
l(β) =
ln λij −Λi −ln[1 − e−Λi ].
(5)
i=1 j=1i=1i=1Analytical closed form expressions for the score function and Hessian of the
TREM can be derived from the loglikelihood (see Hu & Lawless, 1996). These expressions are then used to set up a Newton-Raphson algorithm. Standard errors
are obtained through the observed information matrix.
Given the parameter estimates β̂ of the TREM, the Horvitz-Thompson population size estimate is obtained as
N̂ =

n
X

Ii
,
1
−
e−Λ̂i
i=1

(6)

where Ii = 1 if case i is observed in the sample and Ii = 0 otherwise. The
denominator 1 − e−Λ̂i is the probability that a population member is observedin the sample. The variance of N̂ is calculated through the Delta method, aspresented in van der Heijden et al. (2003).

3

Application to domestic violence data

The application is a data set of domestic violence victims from the Netherlandsin the period 2004 - 2006. The response of interest is the number of times a
police report was filed for domestic violence for a certain individual. Although
information on perpetrators of domestic violence is also available, we do not
focus on that group in this paper. Hence, our population of interest is definedas victims of domestic violence. There are a total of 56,575 observed victims ofdomestic violence in the period 2004 - 2006. These data are made available by
the Dutch national police.
The variables gender and age are available as subject-specific covariates. Genderis included as a time-invariant covariate. Age is modelled with time-varying linear
and quadratic contrasts, meaning that an individual can move from one age group
to another during the observation period. The age categories are: 0-17, 18-29, 3039, 40-49, 50+.
Cyclical effects with periods 366 and 7 are included, representing seasonal andweekly effects, respectively. Interaction effects between the cyclical week effectand the linear and quadratic age effects allow each age group to have a different
cyclical week effect. Furthermore, a linear effect of time is added in the final
model to allow for an increase or decrease of capture probabilities over the time
period of three years. Finally, an interaction effect of the cyclical season effectand the linear time effect is included so that the cyclical seasonal is allowed tovary over time.

314

Time-varying and cyclical effects in capture-recapture models

TABLE 1. Regression coefficients and point and interval estimate of the population size for the TREM model fit to the domestic violence data

VariableIntercept
GenderAge

Coding(male = 0, female = 1)
LinearQuadratic

cos366sin366cos7
sin7
Age (Linear)*cos7
Age (Linear)*sin7
Age (Quadratic)*cos7
Age (Quadratic)*sin7
Time
Time*cos366
Time*sin366

β̂
-8.63
0.63
0.26
-0.50

SE
0.03 ∗∗∗
0.03 ∗∗∗
0.03 ∗∗∗
0.02 ∗∗∗

-0.06
-0.02
0.05
-0.04
-0.03
0.03
0.12
0.01
0.34
0.01
0.16

0.01 ∗∗∗
0.01 ∗∗
0.01 ∗∗∗
0.01 ∗∗∗
0.01 ∗
0.01
0.01 ∗∗∗
0.01
0.01 ∗∗∗
0.01
0.01 ∗∗∗

0.0

07

6
Fe
br

ua
r

y−

−2

20

00

6
be
r

−2
00
N

ov
em

us
t

20
ay
−
M

Au
g

Fe
br

ua
r

y−

−2

20

06

06

5
00

5
be
r

−2
00
N

ov
em

us
t

20
ay
−
M

Au
g

Fe
br

ua
r

y−

−2

20

05

05

4
00

4
be
r

−2
00
N

ov
em

us
t

20
ay
−
M

Au
g

Fe
br

ua
r

y−

20

04

04

−0.5

Change in ln λit

0.5N̂
211,155
95%-CI
206,460 - 215,848
Note: ∗∗∗ = p < 0.001, ∗∗ = p < 0.01, ∗ = p < 0.05.

Time

FIGURE 1. Fitted general cyclical trend for the domestic violence application

315

0.05

Age Groups

0.00

0−17
18−29
30−39
40−49
50+

1
do

29

−0

1
zo

25

−0

1
−0
21
w
o

za

17
−

01

1
3−
0
di
1

vr

09
−

01

1
a
m

do

01

05

−0

−0

1

−0.10

−0.05

Change in ln λit

0.10

Husken et al.

Time

FIGURE 2. Fitted cyclical week effect and its interaction with age for the domestic violence application for the month January 2004

Table 1 shows the regression coefficients and point and interval estimate of the
population size for the model fitted to the data. The effects of gender and age
are significant. Women are more likely to be mentioned as a victim in a police
report for domestic violence than men, and there is a quadratic effect of age.
Both the cyclical main effects are significant, indicating the presence of seasonal
and weekly variation in capture probabilities. Additionally, the interaction effects
of age (linear and quadratic) with the cosine terms of the cyclical week effect are
significant, so that the cyclical week effect is different for each age group.
A positive effect of time is found, indicating that the capture probabilities increaseover the course of the observation period. One of the two interaction componentsof time with the cyclical season effect is also significant, so that the cyclical season
effect is different over the three years. The population size estimate of domestic
violence victims in the time period 2004 - 2006 is 211,155 (95%-CI: 206,460 215,848).
The general cyclical trend of the fitted model (omitting cyclical week effects)
is presented in Figure 1. This trend consists of three components: the cyclicalseason main effect, the linear effect of time, and the interaction between the two.
In general, we can say that the cyclical season effect is stronger in 2004 and 2006
than in 2005, and that the capture probabilities increase over the course of theobservation period. Additionally, the cyclical season effect in 2006 peaks in May,
while in the 2004 and 2005 the effect peaks in September.
In Figure 2, the cyclical week effect and the interaction of this effect with age ispresented. These effects are plotted for the month January in 2004, and repeat
throughout the length of the observation window. The age group 30-39 is thereference group in this analysis, represented by the green curve. The strength ofthe cyclical week effect is lowest for this reference group. The cyclical week effectis strongest for the 50+ age group, while the strength of the cyclical effects of
the other groups is somewhere in between. For all age groups, the cyclical week
effect peaks after the weekend. The groups 40-49 and 50+ peak on Wednesday,
and the other groups on Monday (18-29) and Tuesday (0-17 and 30-39).

316

Time-varying and cyclical effects in capture-recapture models

References
Cook, R.J. & Lawless, J.F. (2007). The statistical analysis of recurrent events.
Springer Science & Business Media.
Cruyff, M.J. & van der Heijden, P.G.M. (2013). Sensitivity Analysis and Calibration of Population Size Estimates Obtained with the Zero-TruncatedPoisson Regression Model. Statistical Modelling, 14(5): 361-373
Cryer, J.D. & Chan, K. (2008). Time series analysis. Princeton university press.
Hu, X. J. & Lawless, J. F. (1996). Estimation of rate and mean functions fromtruncated recurrent event data. Journal of the American Statistical Association, 91(433):300-310.
van der Heijden, P.G.M., Bustami, R., Cruyff, M.J., Engbersen, G. & van Houwelingen, H.C.
(2003). Point and interval estimation of the population size using the truncated Poisson regression model. Statistical Modelling, 3(4): 305-322.

Statistical modelling of cell movement
Diana Giurghita1 , Dirk Husmeier1
1

University of Glasgow, United Kingdom

E-mail for correspondence: d.giurghita.1@research.gla.ac.uk
Abstract: In this paper we demonstrate an application of the unscented Kalmanfilter in the context of cell movement, using a model defined in terms of stochastic
differential equations (SDEs).
Keywords: Chemotaxis, Cancer, Stochastic Differential Equations, Unscented
Kalman Filter.

1

Introduction

Many important biological processes, such as wound healing, tissue development
and cancer cell invasion, are based on the collective movement of cells. One of
the main mechanisms for directed cell movement is chemotaxis, where cells follow
chemical gradients (chemoattractants) present in their environment. These gradients might arise from the presence of a local source of chemoattractant or due tolocal depletion of the chemical in the environment (Tweedy et al. 2016). An example of the former scenario is the migration of breast tumour cells that respondto the epidermal growth factor released by macrophages. In an attempt to acquirea deeper understanding of the mechanisms behind cell movement many population based models have been formulated using partial differential equations, withvery few of them attempting to fit these models to actual data.
In this paper, we propose a model that describes the movement of any individual
cell being driven by an external resource gradient using SDEs of the form:
dXt = σdBtX ,dYt =αβ exp[−β(Yt − γt)]dt + σdBtY{1 + exp[−β(Yt − γt)]}2(1)

Equations (1) describe the evolution in time of the x and y coordinates of a cellin 2D space. σdBtX and σdBtX are Brownian motion terms which in this model
represent the intrinsic randomness in a cell’s movement. The coordinate in the y
direction has a drift term that is described by three parameters: α - the amplitude
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

318

Statistical modelling of cell movementof the resource gradient, β - the steepness of the gradient and γ which indicates
how fast the gradient changes over time. The strength of the random component
in the cell movement equations is indicated by the diffusion coefficient σ.

FIGURE 1. Three frames from the video recording Dictyostelium cells movement.
In this paper, we present our approach to fitting this model to cell movement datausing a non-linear Bayesian filter. We provide some insight into the particularitiesof this model using simulated data and we discuss the results of this analysis froma real data set, describing the movement of Dictyostelium cells (see Figure 1).

2

Methods

Inference in non-linear dynamical systems poses numerous challenges due tothe stochastic nature of the data, intractable likelihoods and unidentifiable parameters. Recent developments have tackled this problem using likelihood-freemethods (sequential Monte Carlo ABC) or computational methods (particle
Markov Chain Monte Carlo) (Golightly & Wilkinson, 2011), however these canbecome too computationally expensive as the number of time points or parameters increases. The unscented Kalman filter (UKF) is an online Bayesian filtering
method that can easily be scaled up to higher dimensions (Julier & Uhlmann,
1997). Intuitively, the UKF starts from the initial distribution of the state vector,
drawn from a multivariate normal distribution, which is then iterated through a
prediction and updating step for each measurement available using the transitionand observation models.
We introduce the UKF by referring to a general state-space model:
xt = f (xt−1 , t ), yt = g(xt , νt )(2)

where xt represents the vector of the hidden states, yt are the measurements, t is
the process noise at time t, νt is the observation noise at time t and the functions
f and g represent the transition and, respectively, the observation models. The
model parameters θ can be included as dynamical variables in the hidden states
vector xt , which means they will be estimated at every time point along with theobserved system states (Sitz et al., 2002). The advantage of the method comesfrom the fact that the probability distribution of the predictor step: p(xt |y1:t−1 )and the probability distribution of the updating step p(xt |yt , y1:t−1 ) can be obtained in closed form using properties of the Gaussian distribution (Julier &Uhlmann, 1997):

Giurghita and Husmeier319

p(xt |y1:t−1 ) ≈ N (xt |µ̄t , Σ̄t )(3)p(xt |yt , y1:t−1 ) ≈ N (xt |µt , Σt )(4)

where µ̄t and Σ̄t are the prediction mean and covariance at time t and µt andΣt are the update mean and covariance at time t (see Julier & Uhlmann, 1997
for full derivations). Therefore, the algorithm essentially updates the mean andcovariance of the Gaussian distribution of the state vector at each iteration.
The approximation of the Gaussian distribution is made using the unscented
transform, which consists of a set of deterministically chosen sigma-points thatare passed through the non-linear function and weighted to obtain the mean andcovariance of the Gaussian. The unscented transformation is used twice for each
iteration of the algorithm: in the prediction and respectively in the update step.

3

Simulation results
x-coordinate

y-coordinate

10

0.2

Y 6

-1500

-loglikmin(-loglik)
α=5

True value
UKF estimate+/- 1 std dev

8X

Profile Likelihood

4

0

-1600

2
0

2

4

6

8

0

10

8

5

α 6

β 3

4

1

0

2

4

6

8

10
-1700

-1800

0

2

4

6

8

10

4

0

2

4

6

8

-log likelihood

-0.2

10

3
-1900

2

γ 2

σ

1
0

0

0

2

4

6

8

10

0

2

4

6

8

10

3

4

5 α 6

7

8

-2000

Time

Time

FIGURE 2. Simulation results: UKF tracking of cell coordinates: x and y, andparameters: α, β, γ, σ for time interval [0, 10]. Parameter estimates include ±1
standard error bounds. On the right hand side, negative log profile likelihood
plot for α, obtained by fixing the other three parameters at their true values.
We apply the Euler-Maruyama discretisation to bring the system in Equation (1)into the standard state-space model described in Section 2:
Xt

=

Yt

=

Xt−1 + σ∆BtXαβ exp[−β(Yt−1 − γt)]Yt−1 +
dt + σ∆BtY{1 + exp[−β(Yt−1 − γt)]}2

(5)
(6)

320

Statistical modelling of cell movement

Here ∆BtX and ∆BtY are just sums of random normal increments between timet − 1 and t. Equations (5) and (6) thus define the transition function f from
Equation 2. In this scenario, we assume the process is observed with a small
amount of Gaussian noise νt ∼ N (0, 0.12 ), so g from Equation 2 is just theidentity function.
We then fit the UKF to a synthetic data set using the following parameters:
α = 5, β = 2, γ = 0.5, σ = 0.1. The results summarised in Figure 2 indicate good
agreement between the estimated UKF path and the true cell path. The UKF also
provides good estimates for the parameters: β̂ = 1.88, γ̂ = 0.51, σ̂ = 0.04 withrelatively small standard errors: 0.17, 0.83, 0.06, except for α̂ where the estimatesindicate a more substantial deviation from the true parameter (bias: 0.44 andstandard error is 0.35).
A potential source of bias as the one observed in Figure 2 can be investigatedby looking at the likelihood i.e.: marginal likelihood with respect to the hidden
states. In order to do that, we first derive the probability of the observed systemat time t conditional on the state of the system at time t − 1 by integrating out
the latent variable xt :Zp(yt |yt−1 ) =p(yt |xt )p(xt |yt−1 )dxt
(7)Z=
N (yt |xt , Rt )N (xt |µ̄t , Σ̄t )dxt
(8)
Using the Gaussian convolution integral results (Bishop, 2006) we simplify (8) top(yt |yt−1 ) = N (yt |µ̄t , Σ̄t + Rt ), where µ̄t and Σ̄t are the predicted mean andcovariance at time t, and Rt is the measurement noise covariance matrix at timet. The log likelihood is then:YX
L = logp(yt |yt−1 ) =log N (yt |µ̄t , Σ̄t + Rt )
(9)
t

∝

t

X
{log det(2πΣt ) + 0.5(yt − µ̄t )T Σ−1t (yt − µ̄t )},

(10)

t

Where Σ̄t +Rt = Σt . We evaluate the marginal log likelihood in (10) by considering a grid of values for each parameter in the model and fitting the UKF with eachparameter combination. The results summarising the profile likelihood for the αparameter in Figure 2 can be used to calculate the Cramer-Rao lower bound,which provides an indication of the intrinsic uncertainty specific to the problem.
In this case, the minimum standard deviation attainable by an estimator of α is0.14. Considering the standard error obtained from the UKF estimation for α is0.35, this then indicates that the estimated value of the parameter is reasonably
close to the true value.

4

Real data application

Dictyostelium cells are widely used in experiments as proxies for understanding the mechanisms of human disease because of their similarities to important
human cells (leukocytes and cancer cells) in terms of biology and response to
chemotaxis (Tweedy et al., 2016). The data consists of two cell paths corresponding to Dictyostelium cells locations extracted from a time series of high

Giurghita and Husmeier

321

y-coordinate

UKF estimate of two cell paths
Cell 1 path
UKF estimate
Cell 2 path

400

200

0
300

350

400

450

500

550

600

x-coordinate

450

0

50

100

150

200

300

0

50

100

150

150

400

x-coordinate

0
200

100

Cell 2

360

50

320

0

50

100

150

200

0

Time

50

100

150

y-coordinate

x-coordinate

Cell 1525

y-coordinate

600

600

0
200

Time

FIGURE 3. UKF tracking of two cells with different movement patterns.

resolution microscopy images (see Figure 1). We emphasise that the main interest for biological applications is the inference of the parameters. However, since
the true parameters for the real data are unknown, we use the tracking of thecell trajectories as a proxy for assessing the accuracy of inference (see Figure 3).
As can be seen from the left panel of Figure 3, we have picked two cells with very
different behaviour (one dominated by drift, the other dominated by diffusion).
In both cases, the path reconstructed with the UKF is very accurate.

5

Conclusions and future work

In this paper, we demonstrate the application of the UKF, a Bayesian filtering
technique that adequately trades off accuracy versus computational efficiency,
to a real-world problem potentially relevant to cancer research: the movement ofDictyostelium cells, which has not been tackled at individual cell level before. Our
results indicate that the UKF can be successfully used for parameter inference
and tracking cells displaying various movement patterns. Future work will extend
this work by applying the UKF to a population of cells. Additionally, we plan tofit models describing alternative movement mechanisms, such as the self-induced
gradient model described by Tweedy et al. (2016) and employ model selection
criteria to choose the best model.

322

Statistical modelling of cell movement

Acknowledgments: The research described in this article is part of the research programme of SoftMech, the Centre for multiscale soft tissue mechanics with application to heart & cancer, funded by the Engineering and Physical Sciences Research Council (EPSRC) of the UK, grant reference number
EP/N014642/1.

References
Bishop, C. (2006) Pattern Recognition and Machine Learning. Springer
Golightly, A. & Wilkinson, D. J. (2011) Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo.
Interface focus, 1(6), 807 – 820
Julier, S. J. & Uhlmann, J. K. (1997) A New Extension of the Kalman Filter toNonlinear Systems. AeroSense’97, 182 – 193
Sitz, A., Schwarz, U., Kurths, J. & Voss, H. U. (2002) Estimation of parameters
and unobserved components for nonlinear systems from noisy time series
Phys. Rev. E, 66(1), 016 – 210
Tweedy L., Knecht D.A., Mackay G.M. & Insall R.H. (2016) Self-Generated Chemoattractant Gradients: Attractant Depletion Extends the Range and Robustness of Chemotaxis. PLoS Biol, 14(3)Regularisation of Generalised Linear Mixed
Models with autoregressive random effectJocelyn Chauvet1 , Catherine Trottier2 , Xavier Bry1
1
2

University of Montpellier, France
University of Paul-Valéry - Montpellier 3, France

E-mail for correspondence: jocelyn.chauvet@umontpellier.fr
Abstract: We address regularised versions of the Expectation-Maximisation(EM) algorithm for Generalised Linear Mixed Models (GLMM) in the context ofpanel data (measured on several individuals at different time-points). A random
response y is modelled by a GLMM, using a set X of explanatory variables andtwo random effects. The first one introduces the dependence within individuals
on which data is repeatedly collected while the second one embodies the seriallycorrelated time-specific effect shared by all the individuals. Variables in X areassumed many and redundant, so that regression demands regularisation. In thiscontext, we first propose a L2 -penalised EM algorithm, and then a supervised
component-based regularised EM algorithm as an alternative.
Keywords: Regularised EM algorithm; Generalised Linear Mixed Model; Autoregressive random effect; Panel data analysis.

1

Introduction

One of the main purposes of panel data analysis is to account for the dependence
induced by repeatedly measuring an outcome on each individual over time. Besides, due to the fact that it is nowadays increasingly possible to collect large
amounts of data, the potentially high level of correlation among explanatory
variables should be taken into account. To this end, ridge-, lasso- and componentbased regularisations have recently been highlighted.
In the Linear Mixed Models (LMM) framework, Eliot et al. (2011) proposed
to extend the classical ridge regression to longitudinal biomarker data. Theysuggested a variant of the EM algorithm to maximise a ridge-penalised likelihood.
This variant includes a new step to find the best shrinkage parameter - in the
Generalised Cross-Validation (GCV) sense - at each iteration.
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

324Regularised EM algorithms for GLMM with AR(1) random effectWith a view towards variable selection, Schelldorfer et al. (2014) proposed a
L1 -penalised algorithm for fitting a high-dimensional Generalised Linear Mixed
Models (GLMM), using Laplace approximation and an efficient coordinate gradient descent.
In the GLM framework, in order both to regularise the linear predictor and to facilitate its interpretation, Bry et al. (2013) developed a PLS-type method – Supervised Component-based Generalised Linear Regression (SCGLR) – which yields
explanatory components. Chauvet et al. (2016) extended SCGLR to GLMM byusing an adaptation of Schall’s algorithm (Schall (1991)).
To the best of our knowledge, the random effects in the previous strategies are
assumed normally distributed with independent levels. However, in the panel
data framework, the question naturally arises of the autocorrelation of the timespecific random effect. Consequently, our objective is twofold: on the one hand,to extend the Mixed Ridge Regression of Eliot et al. (2011) to the GLMMs withan AR(1) random effect; and on the other hand, to present the main ideas of a
new version of SCGLR which handles the high dimensional case.

2

Model hypotheses

In this section, we recall the main hypotheses of the GLMM framework and we
introduce the random effect distributions. For the sake of simplicity, we consider
balanced panel data with N individuals, each of them observed at the same T
time-points. We denote by n = N × T the total number of observations. Let
X be the n × p fixed effects design matrix, and U the n × q random effects
design matrix. Let also Y be the n-dimensional random response vector, β thep-dimensional vector of fixed effects, and ξ the q-dimensional vector of random
effects. We observe a realisation y of Y , but ξ is not observed. We conventionallyassume that:(i) the Yi | ξ, i ∈ {1, . . . , n} are independent and their distribution belongs tothe exponential family;(ii) the conditional mean µi = E(Yi | ξ) depends on β and ξ throughTthe link function g and the linear predictor ηi = xT
i β +ui ξ, with ηi = g(µi ).
Less conventionally, we consider two random effects ξ1 and ξ2 with different rolesand distributions:(i) ξ1 is the individual-specific random effect. Assuming individuals are independent, we suppose:ξ1 ∼ NN 0, σ12 IN ,with σ12 the unknown “individual” variance component.
(ii) ξ2 is the serially correlated time-specific effect common to all the individuals, which can be viewed as some latent phenomenon not measured in the
explanatory variables. As these effects tend to persist over time, we model
them with a stationary order 1 autoregressive process (AR(1)), i.e. for eacht ∈ {1, . . . , T − 1},
ξ2,t+1 = ρξ2,t + νt ,
iidνt ∼ N 0, σ22 ,Chauvet et al.

325

where ρ is the unknown parameter of the AR(1) and σ22 the unknown“temporal” variance component. Such time-specific effects arise naturally
for instance in an economic context (e.g. all companies share a common
economic climate which tend to persist over time), or in biology (e.g. theecological environment is often too complex to be directly observed throughthe explanatory variables).
T

Finally, ξ1 and ξ2 are assumed independent. Denoting ξ = (ξ1T , ξ2T ) ,U1 = IN ⊗ 1T , U2 = 1N ⊗ IT and U = [U1 |U2 ], linear predictor η can be matricially written:
η = Xβ + U ξ.

3

Methods

Owing to the GLMM dependence structure, the Fisher scoring algorithm wasadapted by Schall (1991). We, in turn, adapt Schall’s algorithm by introducing
a regularised EM at each step in order to take into account the high level ofcorrelation in X and the unconventional random effects distributions. Two steps
appear in our method: the linearisation step and the estimation step.
Linearisation step. For each i ∈ {1, . . . , n}, a classic order 1 linearisation of
yi around µi is given by: g(yi ) ' zi = g(µi ) + (yi − µi )g 0 (µi ). Matricially, thisapproximation provides a working variable z entering the following linearisedmodel
M : z = Xβ + U ξ + e,
2with Var(e | ξ) = Diag [g 0 (µi )] Var(Yi | ξ)= Γ.
i=1,...,nEstimation step. Instead of solving Henderson’s system associated with M
seen as a LMM (as proposed by Schall (1991)), we rather propose a regularised
EM step. We suggest an adaptation of the L2 -penalised EM algorithm of Eliot etal. (2011) for low dimensional data (p < n), and a supervised component-based
regularised EM algorithm for the high dimensional case (p  n), because then,interpretable dimension reduction is needed.

3.1

The low dimensional case

Our estimation step is based on Green (1990), who popularised the use of the
EM algorithm for penalised likelihood estimation, and Golub et al. (1979), who
encouraged the use of the GCV for efficiently choosing the ridge parameter λ.
However, contrary to the homoskedastic LMM considered in Eliot et al. (2011), M
contains heteroskedastic errors. We will then opt for the modified GCV criterion
suggested by Andrews (1991), p. 372.
Denoting θ = β, σ12 , σ22 , ρ , we present the current iteration of our L2 -penalised
EM algorithm for GLMM with AR(1) random effect.
(1) Linearisation step. Set:
M[t] : z [t] = Xβ + U ξ + e, with Var(e | ξ) = Γ[t] .

326Regularised EM algorithms for GLMM with AR(1) random effect(2) Estimation step.
(2.a) Denoting L the complete log-likelihood of the linearised model, define
the associated complete penalised log-likelihood Lpen by:
λLpen (θ; z, ξ) = L(θ; z, ξ) − β T β.
2[t](2.b) Denoting zb[t] the fitted values and Sλ the “hat-matrix” satisfying[t] [t]the equality zb[t] = Sλ z , set:

2
[t]



n−1 z [t] − Sλ z [t] [t] −1 Γ
λ[t] ←− arg min GCV(λ) =
.
h

i
2
λ 

[t]


1 − n−1 tr Sλ(2.c) EM step. Set: hi
Qpen θ, θ[t] = Eξ|z Lpen (θ; z [t] , ξ) | θ[t] , λ[t] ,θ[t+1] ← arg max Qpen θ, θ[t] .
θ(3) Updating step. Set ξ [t+1] = Eξ|z ξ | θ[t+1] , and update working variablez [t+1] and variance-covariance matrix Γ[t+1] .
Steps (1)-(3) are repeated until stability of parameters β, σ12 , σ22 and ρ is reached.

3.2

The high dimensional caseIn the p  n case, we need to decompose the linear predictor on a small numberof interpretable dimensions. To that end, we propose to iteratively maximise acomponent-based regularised Q−function.
Let C = XU be the set of principal components of X with non-zero eigenvalues
and f = Cw the component we currently seek. Let also φ denote a structural
relevance (SR) criterion (see Bry and Verron (2015)):
!1
p h

il l
X
2
j
cor x , fφ(w) =, l > 1.
j=1

s ∈ [0, 1] being a parameter tuning the relative importance of the SR with respect
to L, the Q−function would then be:hi
Qreg θ, θ[t] = Eξ|z Lreg (θ; z, ξ) | θ[t] , with
Lreg (θ; z, ξ) = (1 − s)L(θ; z, ξ) + sφ(w).
Parameters s and l are tuned by cross-validation and higher rank components arecomputed like rank 1 component, after adding extra orthogonality constraints.

4

Numerical results

In order to evaluate the performance of our L2 -penalised EM algorithm, we conducted simulation studies in the canonical Poisson case. We present some graphical diagnoses in FIGURE 1, which aim at answering three questions: (1) Is theconvergence assured? (2) How good are the estimations? (3) Are they sensitive
to the value of ρ? The answers to these questions is given in the figure’s caption.

Chauvet et al.
(a)

100

327

(b)
100

10−5

(t)

− σ21 ||

10−4

(t+1)

10−6

10−8

10

|| σ21|| β(t+1) − β(t) ||

10−2

10−10

−10

10−12

10−15

0

100

200

300

400

500

0

Number of iterations

100

200

300

400

500

Number of iterations

(c)

(d)

Mean Squared Errors

0.3

β

0.2

σ21
σ22
ρ

0.1

Estimated autocorrelations

1.0

0.5

0.0

−0.5

−1.0
20

30

40

50

60

70

80

Number of time points

90

100

−0

10

.9
− 5
−00.9
.8
− 5
−00.8
.7
− 5
−00.7
.6
− 5
−00.6
.5
− 5
−00.5
.4
− 5
−00.4
.3
− 5
−00.3
.2
− 5
−00.2
.1
− 5
−00.1
.0
5
0. 0
05
0
0. .1
15
0
0. .2
25
0
0. .3
35
0
0. .4
45
0
0. .5
55
0
0. .6
65
0
0. .7
75
0
0. .8
85
0
0. .9
95

0.0

Simulated autocorrelations

FIGURE 1. Graphical diagnoses relative to the L2 -penalised EM algorithm.
(a),(b): 40 trajectories of the L2 -convergence criterion for parameters β andσ12 (A similar behaviour is observed for parameters σ22 and ρ). About a hundred
iterations is necessary to achieve convergence. (c): MSEs of parameters β, σ12 , σ22
and ρ on simulated data where N = 10 and T ∈ {10, 20, . . . , 100}. As expected,
MSEs of β, σ22 and ρ decrease towards zero. In contrast, since N is fixed, the MSEof σ12 is constant. (d): Boxplots of estimated ρ according to real value.

References
Andrews, D.W. (1991). Asymptotic optimality of generalized CL, cross-validation,and generalized cross-validation in regression with heteroskedastic errors.
Journal of Econometrics, 47, 359 – 377.
Bry, X., Trottier, C., Verron, T., and Mortier, F. (2013). Supervised component
generalized linear regression using a pls-extension of the fisher scoring algorithm. Journal of Multivariate Analysis, 119, 47 – 60.
Bry, X. and Verron, T. (2015). THEME: THEmatic model exploration
through multiple co-structure maximization. Journal of Chemometrics, 29,637 – 647.
Chauvet, J., Trottier, C., Bry, X., and Mortier, F. (2016). Extension to mixed
models of the Supervised Component-based Generalised Linear Regression.
COMPSTAT: Proceedings in Computational Statistics.
Eliot, M., Ferguson, J., Reilly, M.P., and Foulkes, A.S. (2011). Ridge Regression
for Longitudinal Biomarker Data. The International Journal of Biostatistics, 7, 1, Article 37.

328

Regularised EM algorithms for GLMM with AR(1) random effectGolub, G.H., Heath, M., and Wahba, G. (1979). Generalized cross- validation asa method for choosing a good ridge parameter. Technometrics, 21, 215 –223.
Green, P.J. (1990). On use of the EM for penalized likelihood estimation. Journalof the Royal Statistical Society, Series B, 52, 443 – 452.
Schall, R. (1991). Estimation in generalized linear models with random effects.
Biometrika, 78, 719 – 727.
Schelldorfer, J., Meier, L., and Bühlmann, P. (2014). Glmmlasso: an algorithm
for high-dimensional generalized linear mixed models using l1 - penalization.
Journal of Computational and Graphical Statistics, 23, 460 – 477.

Optimal Number of Clusters Based on Inter
Cluster Elements Mapping
K. M. Matawie1 , A. Mehar1 , A. Maeder212Western Sydney University, Australia
Western Sydney University and Flinders University, Australia

E-mail for correspondence: k.matawie@westernsydney.edu.au
Abstract: To estimated the optimal number of clusters and evaluate the associated quality of the formed clusters is one of the major issue in cluster analysis. In
this paper, we present and extend the MMM index to find the optimal number ofclusters and their quality. The new index determines the combined mapped elements information from related clusters by comparing the occurrence of common
elements across the sets of clusters from successive k number of clusters. This requires comparing the k resultant clusters in each set with respect to ’forward’ and’backward’ mapping of common elements for adjacent and non-adjacent clusters(all possible distant) at all possible k. This method will also provide indicators
for the similarity and overlapped (dissimilarity) of mapped elements. The optimal or best estimated number of clusters and their quality will be decided using
the combination of maximum average similarity and minimum average overlap
measures. The evaluation and performance of this index is illustrated and tested
using real dataset.
Keywords: k-means, clustering, cluster quality, forward and backward mapping,similarity and overlap

1Introduction

Unsupervised clustering is a data analysis technique which has no a priori information available to determine the intrinsic structure in the dataset. Due to thislack of a priori knowledge, and the consequential need to infer or discover structure in the data, most clustering algorithms tried to find similar objects within adataset according to their characteristics: this is usually accomplished by using
some well-established distance measures (e.g. Euclidean, Maximum, Manhattan).
The labelling of objects using distance measures allows the dataset to be partiThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

330

optimal number of clusters

tioned into clusters, where characteristics of objects in the same cluster are moresimilar than they are to objects in other clusters. This approach has been utilized successfully in various types of problems across many fields. As describedin Ng and Han (1994) clustering algorithms are classified into two major types
partitional and hierarchical clustering. We will be focusing here on partitional
clustering algorithms and in particular the k-means algorithm. The main challenge in using the k-means algorithm is the optimal (best) k value (number ofclusters) giving well-structured or quality clusters. To determine the best k value
often requires a large number of steps in a given dataset, when particularly iterative or exhaustive search is applied. For this reason a number of techniques havebeen developed based on use of heuristics such as to minimise the distance from
centroid of a cluster to each object in the cluster, or to maximise the distance
between clusters (intra and inter cluster distances), or to use an average scatter ofclusters and total separation between clusters. They are to some extend limited
in effectiveness by the assumptions made about what constitutes the best clustering, this is also called the unsupervised machine learning. The MMM approach
Mehar at al.(2010,2013) and Matawie et al(2015) allows this optimisation to beundertaken for k clusters using knowledge of the clustering solutions obtained
from k to k+1 ’forward’ and k+1 to k ’backward’ for only adjacent distancesin consecutive k situation. In this paper extended MMM will be presented with
better evaluation criterion based on the forward and backward changes in cluster
membership over the range of all successive k values adjacent and non-adjacentclusters. In this work it is assumed that minimum k is 2 ( kmin = 2) , andmaximum K is 16 (kmax ≤ 16) and r=1 to 14 , limiting k and therefore to themost practical finite range. The details and development of the enhanced MMM
is explained in details in section 2. The forward and backward inter cluster elements mapping when k=2 with all possible k+r , where r=1,2,......,14 , is given
in Figure-1 below with bold and light arrowheads indicating the forward andbackward directions respectively.

FIGURE 1. Forward and backward mapping of elements when k=2 for different
k+r distant, and r=1,2,...., K-k.

2

MMM Index Extended

In this section we will extend the approach of developing the MMM index described in Mehar at al.(2010) by including information from adjacent and nonadjacent (more distant) mapping of the common elements between different clusters. To develop a practical implementation of this approach we need to consider

Matawie et al.

331

different resultant of k number of clusters from the k-means algorithm, over arange of successive k values always starting with minimum number of clusters
kmin = 2 to maximum kmax = K , where K = N-1 and N= total number of
elements (objects or observations) in the dataset D. Following are notations anddetails used to define and describe the approach. Assume we have dataset D containing N number of observations (elements). Each observation may have numberof variables that may be used to determine information about the relationshipbetween them. Using a k-means clustering algorithm on D with a certain set ofparameters to control the behaviour of the k-means algorithm such as choosing
variables, number of variables, number of clusters (centres), number of maximum iterations, number of initial random starting sets of seeds and algorithm
to obtain the required partitioning of D into k clusters, k ≤ K . Suppose at
any k number of clusters, we define the first set of clusters to be C(k)i where k∈ 2,3,......,K and i =1,2,......,k. We define another set of clusters C(k+r)j , wherer ∈ 1,2,.....,K-k and j = 1,2,....,k+r. As k get larger and closer to K the K-k
set of clusters will have fewer elements, per cluster, than the preceding set ofclusters until reaching one element in each K cluster ( when K = N-1). Let us
define m(k,k+r)ij as the number of common elements in the forward inter cluster mapping from the source cluster C(k)i to the target cluster C(k+r)j i.e. thenumber of elements of C(k)i ∩ C(k+r)j . These forward mapped number of elements are used to construct forward mapped inter cluster matrices [M(k,k+r)ij ]for i=1,2,....,k and j = 1,2,....,k+r, i.e. the mapped elements from a particularcluster C(k)i at k to all different sets of target clusters C(k+r)j , j=1,2,...., k+r.
Similarly but in reverse, the backward inter cluster mapping m(k+r,k)ji is defined
to be the numbers of common elements from the target to source cluster, andthe backward mapped inter cluster matrices is defined to be [M(k+r,k)ji ]. Thesebackward mapped inter cluster matrices are also defined as transpose of the M
forward inter cluster mapped matrices. These M matrices are rectangular in size(k,(k+r)) and ((k+r),k) respectively. We now define the forward inter cluster
proportion of elements as [P(k,k+r)ij ] matrices from a source cluster C(k)i to all
the target sets of clusters C(k+r)j . Each row (vector) in P matrix is summed up
to 1 and the proportion matrix can be computed asm(k,k+r)ij
P(k,k+r)ij = Pk+r
j=1 m(k,k+r)ij(1)

Equation (1) simply shows the inter cluster proportion of the elements mappedfrom the source cluster C(k)i to the target cluster C(k+r)j . Similarly, we can obtain
the backward inter clusters proportion [P(k+r,k)ji ] matrices. The inner product of
the forward and backward proportion matrices is computed and constructed as
it will determine the mutual similarity between clusters. This inner product will
result in a matrix of (k,k) combined mapped proportion matrices called O(k,k)
for different k+r distances(see equation (2) below). These combined mapped proportion matrices O provide the combined similarity at each entry of the diagonal,
while the off diagonal entries are the dissimilarity or overlap proportions of theclusters.
[O(k,k) ] = [P(k,k+r)ij ][P(k+r,k)ji ](2)

It is important to note, that the inner product of the forward and backward are
different due to the cardinality difference as ((k, k+r),(k+r, k)) is not equal to332

optimal number of clusters((k+r, k),(k, k+r)), also the backward to forward would be less informative thanforward to backward at a given k value.
Finally, we define the combined mapped elements matrices as [Q(k,k) ] to be calculated from the inner product of matrix [k] by matrix [O(k, k)] for each k and k+r
distance, where [k] is a (k,k) diagonal matrix with diagonal entries determinedby the size or number element of each cluster from k-means algorithm at k. The
diagonal entries of [Q(k,k) ] matrix q(ii) at each k with different k+r distances are
the number of elements belong or remained in the same cluster (within cluster)while the off diagonal q(ij) entries are the number of elements belong or moved
from other clusters ( representing the overlap at k).

2.1

Computing Cluster Similarity and Overlap

In this section similarity, overlap, average similarity and average overlap are computed from Q, the combined mapped elements matrix. We define the similarity
as the trace value of the Q matrix at each k that is mapped to different k+r
distances. Equations (3) and (4) below shows the similarity and overlap of the Q
matrices respectively;

T raceQ(k,k+r) =

k
X

qii

(3)

i=1

Overlap(k,k+r) = N − T raceQ(k,k+r)(4)

In addition, the traces of each k to different k+r mapped distance would be used
to define and compute the average similarity and overlap of k to k+r, equation(4) and (5) below calculate these averages.
PK−kAverageT racek = µk =r=1

T race(k,k+r)
K −k

AverageOverlapk = N − µk

(5)
(6)

where, k = 2,3,.....,K and r = 1,2,3,....K-k.
The best (optimal) estimated number of clusters K* can be found to be themaximum average of the traces from equation (4) above;K ∗ = M ax(AverageT racek )(7)Equation 7 is considered as a criteria for the best estimated number of cluster,
however, there are more details needed to cover the situations when the best
estimated number of clusters are fully or partially separated.

3

Real Data Example

To illustrate the proposed method we will use a well-known Ruspini real datasetwith four clusters that is known in advance. This particular dataset is also widelyMatawie et al.

333

used in the literature to illuminate and evaluate clustering number and structure
by many research papers, and was first used and analysed by Ruspini (1970)
to investigate fuzzy clustering. This data is a two dimensional numerical datasetthat includes 75 elements, Figure 3(a and b) shows the scatter plot of this dataset
and the k-means clusters when k=4 respectively. Forward and backward mapping
proportion and combined matrices at different k and for all k+r distances (up toK=16) are calculated and their Trace, AverageTrace and number of overlapped
elements at each k are also calculated and presented in Figure 3 (c and d). The
average trace at each k is given by the solid black line in Figure 3 (c) showingthat the average trace was maximum at k= 2, 3 and 4 and was equal to the total
number of the element (N=75). This indicates there is a continuous potential tosplit into new and fully separated clusters up until k=4, and as k increased andmoved away from 4 the trace averages start decreasing and fluctuating but always
below the value at k=4. The overlap calculations given in figure 3(d) showed zero
overlap up until k=4 and it increased after k=4 by which we strongly nominate
k=4 as the best estimated number of clusters.

(a)

(b)

(c)

(d)

FIGURE 2. Plot (a) Rusipini data scatter plot, (b) k-means clusters, (c andd)the calculations of trace and overlap at different k to k+1.

4

Conclusion

A good choice of K is an important feature for building meaningful homogenous
and well separated clusters when applying the k-means clustering algorithm fora dataset. In this study we extended and enhanced the MMM index approachbased on the results from a k-means clustering algorithm, and demonstrated
its effectiveness to explore the clustering structure for finding a best (correct)
value of K. This approach aimed to maximize the mapping similarity based on
the forward and backward inter cluster elements mapping that shows to be more
descriptive, informative and analytical by which the best and stable set of clusters
are reached. Development of this approach is still in progress and it will cover
its advantage and effectiveness when applied on variety of simulated and real
datasets including the evaluation and comparison with other existing validationindexes.

334

optimal number of clusters

References
Matawie, K., Mehar, M.A. and Maeder, A. (2015). An approach to determine
clusters overlap for k-means clustering. International Workshop on Statistical Modelling, Linz- Austria, 163 – 166.
Mehar, A., Maeder, A., Matawie, K. and Ginige, A. (2010). Blended Clusteringfor Health Data Mining. Springer Berlin Heidelberg, 130 – 137.
Mehar, A.M., Matawie, K. and Maeder, A. (2013). Determining an optimal valueof K in K-means clustering. IEEE International Conference on Bioinformatics and Biomedicine, Shanghai- china, 51 – 55.
Ng, R.T. and Han, J. (1994). Efficient and Effective Clustering Methods for Spatial Data Mining. Proceedings of 20th International Conference on Very
Large Data Bases (VLDB), 144 – 155.
Ruspini, E.H. (1970). Numerical methods for fuzzy clustering. Information Sciences,2(3), 319 – 350.

Nowcasting infectious disease outbreaks using
constrained P-spline smoothing
Jan van de Kassteele1 , Paul Eilers2 , Jacco Wallinga13
1

National Institute for Public Health and the Environment, the Netherlands
Erasmus Medical Center, the Netherlands
3
Leiden University Medical Center, the Netherlands
2E-mail for correspondence: jan.van.de.kassteele@rivm.nl
Abstract: During an infectious disease outbreak it is crucial to have timely
information on epidemic trends. However, the reporting of new cases is subject
to delay. The real-time assessment of the expected number of cases based onpartial information is called nowcasting. We present a nowcasting model basedon two dimensional P-spline smoothing with additional constraints. Our aim
is to predict the number of occurred-but-not-yet-reported cases. We force theunderlying time-varying delay distribution to be unimodal, and to be zero at apredefined maximum delay. We illustrate our method on a large measles outbreakin the Netherlands. We show that even with very limited information our model
is able to predict the number of occurred-but-not-yet-reported cases very well.
Keywords: Smoothing; Constrained P-splines; Asymmetric penalty; Infectious
disease outbreaks

1

Introduction

During an infectious disease outbreak the National Institute for Public Healthand the Environment - RIVM has the responsibility to real-time monitor thenumber of cases, in order to inform relevant health authorities. However, there
is a delay between the first day of symptoms onset and the time that the case isregistered. A consequence is that the epidemic curve of reported cases drops tozero at the present day.
The assessment of the current situation based on imperfect or partial information
is called nowcasting (Donker et al., 2011; Höhle and an der Heiden, 2014). Whenthe delay distribution is known, it is possible to estimate the number of newcases in real-time, e.g. by dividing the number of reported cases by the fraction ofThis paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

336

Nowcasting infectious disease outbreaks

reported cases. In practice however, it is very difficult to obtain stable estimates,
especially when the number and fraction of reported cases is low, or even zero.
Typically, no cases are reported during the weekend.
Statistical modeling techniques can make an improvement here. The number ofreported cases can be tabulated by calendar time and delay. If we set a maximum
delay (where all cases have been reported), we get the so-called reporting trapezium. The aim is to predict the number of occurred-but-not-yet-reported cases
outside the trapezium, while accounting for a time dependent delay distribution.
We model the expected number of reported cases in the reporting trapezium bya smooth function of calendar time and delay using two dimensional P-splines.
However, to safely extrapolate the surface outside the reporting trapezium, especially at the beginning of the outbreak, additional constraints are required. First,
the surface is expected to be unimodal in the delay direction. Second, it should
go to zero at the maximum delay. Our approach is fast; one run takes about 30
seconds.

2

Methods

The number of reported cases by calendar time and delay is a stochastic variable
Y and is assumed to follow a Negative Binomial distribution with reporting
intensity µ = E(Y) and overdispersion parameter θ. The log-intensity is modeledby a linear predictor η = log(µ) = Xβ. To separate the calender time and delay
trend surface from weekday effects, we write model matrix X as a partitioned
matrix [Xs |Xw ] and coefficients β as a partitioned vector [βs |βw ]. Here Xs =Xd ⊗ Xt , where Xt and Xd are nt × kt and nd × kd B-spline basis matrices for
marginal variables calender time t and delay d respectively. Xw is a nt nd × kw
matrix with weekday expanded to a set of dummy variables. Hence, weekday
effects are expressed as deviations from the trend surface.
Smoothness of the trend surface Xs βs is achieved by quadratic penalization of
the coefficients. The penalized log-likelihood function becomes`∗ = `(βs , βw , θ|y) − 12 λt βs0 D0t Dt βs − 21 λd βs0 D0d Dd βs .
We assume that outside the reporting trapezium the delay distribution is constantand the tails of it are linear (on a log-scale), so we have Dt = Ikd ⊗D(1) and Dd =D(2) ⊗ Ikt , which are first and second difference operator matrices in respectively
time and delay direction. λt and λd are unknown smoothing parameters.
An unimodal function estimate in the delay direction can be achieved by a priori
only allowing negative values for the second order differences between the coefficients in the delay direction (Eilers, 2005). The modified penalized log-likelihood
function then becomes`∗∗ = `∗ − 21 κu βs0 D0u Vu Du βswhere Du = D(2) ⊗ Ikt . Typically κu = 106 and Vu = diag(vu ) is a matrix withasymmetric weights
1 if Du βs > 0vu =.
0 if Du βs < 0Van de Kassteele et al.

337

In order to ensure a reporting intensity near zero at the maximum delay, we set
an additional constraint Xs βs < g at the boundary, where g is a vector withnegative numbers, say -5. The modified penalized log-likelihood function thenbecomes`∗∗∗ = `∗∗ − 12 κb (Xs βs − g)0 Vb (Xs βs − g),where Vb = diag(bvb ) is a matrix with asymmetric weights
1 if Xs βs > gvb =,0 if Xs βs < gand b is a vector with elements equal to 1 at the locations where the boundary
constraint is applied, and 0 otherwise. Typically κb = 106 .
Finally, two additional constraints are applied. To obtain stable estimates of0
βw with κw = 1 is added to logthe weekday effects, a ridge penalty − 21 κw βw
likelihood function. To make estimation of the trend surface numerically stable,a small ridge penalty − 12 κs βs0 βs with κs = 10−3 is added.
Given λt , λd and θ, the coefficients β are found by penalized iterative weighted
least squares, by repeatedly solving the system
 


βs
κb X0 Vb g
(X0 WX + P)−1
= X0 Wz +,βw
0kw
where z = η + W−1 (y − µ) is the working variable and weight matrix W =diag(rw). r is a vector with elements equal to 1 if the element lies within the2reporting trapezium, and 0 otherwise and w = µ2 /(µ + µθ ) is a weight vector,
corresponding to the Negative Binomial distribution.
The corresponding penalty matrix P is given by
P = blockdiag(λt D0t Dt + λd D0d Dd +
κu D0u Vu Du + κb X0s Vb Xs + κs Ikt kd , κw Ikw ).
The overdispersion parameter θ is found maximizing the log-likelihood given thecurrent estimates of β within the IWLS algorithm. Smoothing parameters λt andλd are found by minimizing the BIC.
The nowcast is then achieved as follows. First generate 1000 Monte Carlo samplesof β and θ. Given µ = exp(Xβ) and θ, draw 1000 samples from the Negative
Binomial distribution for each combination of calendar time and delay outside
the reporting trapezium. By summarizing the already reported number of cases
and the predicted number of cases over the delays, we obtain the predictive distribution by date. The time dependent delay distribution is found by conditioning
the trend surface exp(Xs βs ), in a nt × nd matrix, on its column sums.

3

Application: nowcasting a measles outbreak

During May 2013 - March 2014, the Netherlands was affected by a large measles
outbreak. Figure 1 shows the situation halfway the outbreak on October 1, 2013.
Although zero cases have been reported on October 1 (dark bars), we are still
able to predict the number of occurred-but-not-yet-reported cases (light bars).
As more information is available in the past, the prediction interval gets smaller.

338

Nowcasting infectious disease outbreaks

Delay in days

Number of cases

40

Reported
NowcastNot yet reported

40
30
20

30
0.01

20

0.03

0.02
0.04

0.05

10

10

0.07

0.06

0

0

●

1

11

21

1

May

11

21

Jun

1

11

21

Jul

1

11

21

Aug

1

11

21

Sep

1

11
Oct

0.02
1

11

21

May

0.04
1

0.05

0.03

0.01

11

21

Jun

1

11

21

Jul

●

1

11

21

Aug

1

11

21

Sep

1

11
Oct

FIGURE 1. Left panel: Nowcast of the measles outbreak on October 1, 2013. The
black line is the nowcast (incl. 95% prediction interval) to a maximum delay of
six weeks back. Right panel: Time dependent delay distribution.

The right panel shows the estimate of the time dependent delay distribution. The
triangle on the right has been obtained by extrapolation. The highest reporting
probabilities occur with a delay between five and nine days.

4

Conclusions

We have presented a nowcasting model based on constrained P-splines. The modeltakes care of a time dependent delay distribution, right truncation and day-ofthe-week effects. Without the additional constraints, stable extrapolation outside
the reporting trapezium is almost impossible. Even with very limited information,
the model is able to predict the number of occurred-but-not-yet-reported cases
very well on a daily resolution.

References
Donker T., van Boven, M., . . . , and Wallinga, J. (2011). Nowcasting pandemic influenza A-H1N1 2009 hospitalizations in the Netherlands. Eur. J. Epidemiol., 26 (3), 195 – 201.
Eilers, P.H.C. (2005). Unimodal smoothing. J. Chemometrics, 19 (5-7), 317 –328.
Höhle, M. and an der Heiden, M. (2014). Bayesian nowcasting during the STEC
O104:H4 outbreak in Germany. Biometrics, 70 (4), 993 – 1002.

A general statistical framework to study thediversification of species
Francisco Richter1 , Rampal Etienne1 , Ernst Wit1
1

University of Groningen, The Netherlands

E-mail for correspondence: f.richter@rug.nl
Abstract: The mechanisms that control the diversification of species are poorly
understood. Sophisticated diversification models have been developed, but they
have been developed on a case-by-case basis and no general method to study the
combined effect of ecological factors exists.
Such a general method has remained elusive for several reasons. Firstly, evolutionary processes have extremely complex dynamics. Secondly, decay and fossilization
degrade crucial evidence useful for phylogenetic analyses. Thirdly, diversification
processes have many potential explanatory variables, which increases the dimensionality of the models enormously.
To overcome these issues, we propose a general diversification model expressing
the evolutionary species diversification dynamics as a combination of two generalized linear models. The fact that we typically only have data on currently
existing species can be described as a missing data problem and we developed an
MCEM-type algorithm for it.
We show that our method performs well for cases where an exact solution isavailable, and discuss potential future usage of our approach.
Keywords: Diversification models; GLM; EM.

1

Introduction

Biodiversity, the term used to describe the wide variety of species on Earth, isdeclining at enormous rates. To conserve biodiversity, we must understand themechanisms how it comes about and how it is maintained, in assemblages of
species, so-called ecological communities.
Figure 1 shows different sources of information regarding evolutionary processes.
It is our aim to incorporate the sources of high-dimensional data under a unified
statistical framework in order to overcome the main challenges that evolutionary
biologists currently face. Particularly, the lack of information of extinct species
and the huge complexity of current stochastic differential diversification models
This paper was published as a part of the proceedings of the 32nd International Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,
Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remains
with the author(s). Permission to reproduce or extract any parts of this abstract
should be requested from the author(s).

340

A general statistical framework to study the diversification of species

FIGURE 1. Inserting feedback with spatial ecological interactions into the phylogenetic data we are able to infer diversifications dynamics.

are bottlenecks for a proper inference on a general scenario. The framework wepropose is shown to be an efficient and general method with the potential toprovide practical solutions for a large number of open questions in evolutionary
biology and ecology.

2

Methodology

A phylogenetic time-tree is defined as a graph (V, E) with the following properties:
1. It is undirected and acyclic, i.e, a tree,
2. All nodes have degrees 1,2 or 3, depending on the biological meaning of
the node. The tree is binary,
3. It has a time-dimension.

Nodes
speciation
extinction
extant species

t1

t2

t3

t4

t5

FIGURE 2. Phylogenetic time-tree.

tpRichter et al.

341

If we consider the space of trees T , extending the ideas of Gavryushkin (Gavryushkin
et. al. 2016), we define the space of phylogenetic trees as
S = {(rt(T ), τ )|T ∈ T , τ ∈ RN }where rt(T ) is the ranked topology of the tree T and τ is the vector containing
the waiting times between nodes (speciations or extinctions).
Assuming that the evolutionary mechanics follows a birth-death process (Nee et.
al. 1994), the Markov nature of the dynamical system means that the likelihood isexactly the product of the conditional densities (Reynolds 1973), in other words,
a multiplication of exponentially distributed waiting times and a multinomial
event selection variables

L(θ|Y ) =NYσi (θ)e−σi (θ)τi

i=1

N
Y
ρi (θ)
e−σi (θ)τi ρi (θ),=σi (θ)
i=1(1)

where σi (θ) and ρi (θ) are linear functions of the speciation and extinction rates
λ(θ) and µ(θ), which in turn are monotone functions of many potential explanatory (ecological) variables.
We have shown that for complete phylogenetic trees this approach is equivalent
to current diversification models. However, assuming knowledge on the complete
phylogenies is not realistic because information of extinct species is almost neveravailable. Figure 2 shows the missing information on phylogenetic trees, wheremost of the times we only see the tree on the right with extant species only. To
overcome that issue we implement an EM algorithm considering extinct species
as missing data.

FIGURE 3. Phylogenetic trees where we can visualize the loss of information onreconstructed trees. At the left we have a tree with all extinct species whereas
the right plot shows the same tree with only observable species.
Thus, if we define D as the observed phylogenetic tree and D+ the extinct-extant
species tree, we iteratively performs
E-step: Compute Q(θ|θ∗ ) = Eθ∗ [logP (D+ |θ)|D],M-step: Choose θ to be the value of θ ∈ Ω which maximizes Q(θ|θ∗ )However, the expectationEθ∗ [logP (D+ |θ)|D] =Z

log fD+ (x; θ)fD+ |D (x|D, θ∗ )dx
X (y)

342

A general statistical framework to study the diversification of species

has not a close form due to the complexity of the time-tree space. Thus, anapproximation via Monte-Carlo method is neededNEθ∗ [logP (D+ |θ)|D] ≈1 X
logP (Di+ |θ)
N i=1

90
60
30
0
0.6

0.8

1.0

150
100
50
0
0.05

MLE from incomplete tree

0.10

0.15

300
200
100
0
40

0.20

1.4

0.16

0.8

0.6

1.0

60

70

80

70

70

60

60

0.16

0.12

0.8
0.08

0.12

^
KC

^
λC

1.0

^C
µ

MLE of complete tree

1.2

MLE of complete tree

1.2

50

MLE from incomplete tree

MLE from incomplete tree

0.08

MLE of complete tree

0.4

count

count

count

for sampling sets of complete-trees conditioned to the observed tree. However,
the marginal distribution needed for sampling is in fact a complex combinationof non-homogeneous Poisson processes and an approximated sampling method
is needed. On figure 4 we can see the estimations for the Diversity-dependence(DD) model described on next section. Because the simulated sampled trees areapproximations in the sampling space and not completely unbiased, we incorporate an importance sampling correction when needed.

50

40

50

40

0.6

0.4

0.6

0.8

^
λI

1.0

0.04

0.4

0 255075100

count

0.05

0.10

0.15

0.20

^I
µ

0.04

40
0 306090120

count

50

60

^
KI

70

80

30
0 100200300

count

FIGURE 4. Parameter estimations for the DD model. Every point represents apylogenetic tree, the y-axes shows the MLE of the complete tree and the x-axes
shows the MLE of a set of reconstructed trees using the reconstruction algorithm. From the histograms we can see that the estimations are precise, however while
the complete tree estimations are unbiased, the incomplete estimations are a bit
biased. To correct that we implement include importance weights on the final
estimations.

Application: Diversity-dependence model
The existence of upper limits to diversity has been discussed extensively in thelast decades. Etienne et. al. (2012) propose a diversity-dependence model as astandard model for macro-evolutionary dynamics, with diversification rates defined by
λi = λ0 − (λ0 − µ0 )ni,Kµi = µ0

where n corresponds to the number of species, and µ0 , λ0 , K are parameters ofinterest. µ0 and λ0 are initial extinction and speciation rates and K is the socalled carrying capacity. ,Pni
λ0 −µ02,
If we define σi = N
j=1 λ0 −(λ0 −µ0 ) K +µ0 = ni (λ0 +µ0 )−ni β0 for β0 =Kand ρi = Ei (λ0 − ni β0 ) + (1 − Ei )µ0 . ni is defined as the number of species attime ti , E is a binary vector containing the topology of the tree. For equation 1,
the log-likelihood function of the phylogenetic tree isRichter et al.

l(λ0 , µ0 , K|Y ) = −

m
X
i

+

m
X
i

ni ti [λ0 (1 −

343

ni
ni
) + µ0 (1 + )]
K
K



ni
ni
log λ0 Ixi (1 − ) + µ0 (Ixi
+ I1−xi )K
KWe incorporate the Diversity-dependence model within our framework with satisfactory results, reproducing the evolutionary dynamics and estimations efficiently
using a general approach, easily extensible to many more variables and biologicalscenarios. In figure 5 we can see the expected number of species versus time fortwo clades: Dendroica and Foraminifera. Here we can see the influence on theso-called carrying capacity on their evolutionary dynamics.

FIGURE 5. Expectation of lineages through time for the obtained parameters
under the proposed framework. Here we can observe two clades with different
responses to diversification. While Foraminifera shows an upper limit on diversification, Dendroica shows opposite behavior.

3

Conclusions and further work

In this manuscript we propose an alternative and general method to the usual
diversification models. This framework is a methodological innovation with manypotential applications, including been able to study the effects of climate, species
interaction, protracted speciation on evolutionary dynamics, among many others.
Further work will be the implementation of a differential geometric path finding
method (Augugliaro et al., 2013) in order to deal efficiently with high-dimensional
covariates.

Acknowledgments: This work is part of the research programme Mathematics of planet Earth with project number 657.014.005, which is financed by the
Netherlands Organisation for Scientific Research (NWO).

References
Augugliaro, L., Mineo, A. M., and Wit, E. C. (2013). Differential geometric least
angle regression: a differential geometric approach to sparse generalized lin-

344

A general statistical framework to study the diversification of species
ear models. Journal of the Royal Statistical Society: Series B (Statistical
Methodology), 75(3):471498.

Etienne, R. S. et. al. (2011).Diversity-dependence brings molecular phylogenies closer
to agreement with the fossil record. Proceedings of the Royal Society of London B: Biological Sciences, page rspb20111439.
Etienne, R. S. and Rosindell, J. (2012). Prolonging the past counteracts the pullof the present: protracted speciation can explain observed slowdowns in diversification., Systematic Biology, 61(2):204213.
Gavryushkin, Alex, and Alexei J. Drummond. (2016). The space of ultrametricphylogenetic trees., Journal of theoretical biology 403 (2016): 197-208.
Nee, S., May, R. M., and Harvey, P. H. (1994). The reconstructed evolutionary
process., Philosophical Transactions of the Royal Society of London B:
Biological Sciences, 344(1309):305311.
Reynolds, John F. (1973) On estimating the parameters of a birth-death process.
Australian & New Zealand Journal of Statistics 15.1 (1973): 35-43.

IndexAgresti, Alan, 252
Alexander, Craig, 270
Amati, Viviana, 161
Ampountolas, Konstantinos, 76

Evers, Ludger, 76, 270
Faschingbauer, Florian, 97
Fasola, Salvatore, 200
Fellinghauer, Carolina, 229
Ferguson, Elaine A., 70
Fernandes, Cristiano, 119
Franco-Villoria, Maria, 212
Frühwirth-Schnatter, Sylvia, 246
Fu, Wei, 299

Baayen, R.Harald, 21
Bartolucci, Francesco, 167
Bass, Mark, 229
Behrouzi, Pariya, 47
Berger, Moritz, 231
Bernardi, Mauro, 35
Blas, Betsabe, 280
Bogaerts, Kris, 10
Bowman, Adrian, 41
Brockhaus, Sarah, 109
Bruyneel, Luk, 206
Bry, Xavier, 323
Burke, Kevin, 290

Gebetsberger, Manuel, 137
Giurghita, Diana, 317
Gledhill, Jacob, 217
Goeman, Jelle, 7Gomes, José Clelto Barros, 184
Gray, Elizabeth, 217
Gregorczyk, Marco, 173
Greven, Sonja, 109
Groll, Andreas, 103
Groß, Jürgen, 131

Carroll, Raymond J., 280
Cepeda-Cuervo, Edilberto, 64
Chatterji, Somnath, 229
Chauvet, Jocelyn, 323
Cieza, Alarcos, 229
Cilluffo, Giovanna, 264
Colebank, Mitchel, 190
Cook, Scott J., 280
Cruyff, Maarten, 311

Ha, Il Do, 294
Haider, Mansoor A., 190
Hambuckers, Julien, 113
Heller, Gillian, 236
Heumann, Christian, 179
Hill, Nicholas A., 190
Hohberg, Maike, 125
Husken, Thomas, 311
Husmeier, Dirk, 52, 70, 190, 317

da Silva Junior, Antonio Hermes
Marques, 217

Jacobs, Rianne, 80
Edwards, Matthew, 149
Eilers, Paul, 335
Eilers, Paul H. C., 258
Einbeck, Jochen, 217, 275
Etienne, Rampal, 339

Kateri, Maria, 91
Kim, Jong-Min, 294
Kirch, Claudia, 149
Klasen, Stephan, 125

345

346

INDEX

Klein, Nadja, 305
Kneib, Thomas, 103, 113, 125
Knein, Nadja, 97
Komárek, Arnošt, 206
Komárek, Arnošt, 10Pedeli, Xanthi, 196
Petrella, Lea, 35
Pohle, Jennifer, 58

La Grutta, Stefania, 200, 264
Landau, Katja, 125
Langrock, Roland, 3, 58, 113
Lazarus, Alan, 52
Lesaffre, Emmanuel, 10, 80, 206,
242

Richter, Francisco, 339
Rue, Håvard, 212
Rusá, Šárka, 206
Russo, Cibele Maria, 184Möller, Annette, 131
Ma, Jun, 236
Macaulay, Vincent, 41
MacKenzie, Gilbert, 290
Maeder, A., 329
Manuguerra, Maurizio, 236
Mariñas, Irene, 41
Marino, Maria Francesca, 167
Matawie, K. M., 329
Matthiopoulos, Jason, 70
Mayr, Andreas, 97
Mayr, Georg J., 137, 143
Mehar, A., 329
Meier, Alexander, 149
Meira-Machado, Luı́s, 284
Melo, Eduardo, 119
Melo, Mariana, 119
Meyer, Renate, 149
Muggeo, Vito M.R., 200, 264
Núñez-Antón, Vicente, 64
Neocleous, Tereza, 270
Olufsen, Mette S., 190
Opitz, Madeleine, 109
Pandolfi, Silvia, 167
Papamarkou, Theodore, 52
Paun, L. Mihaela, 190
Pazira, Hassan, 85

Qureshi, M. Umar, 190

Sabariego, Carla, 229
Sahu, Sujit, 229
Sampaio, Beatriz, 284
Sangalli, Laura M., 4
Schönenberger, Felix, 161
Schaffer, Sophia, 109
Schauberger, Gunther, 231
Schmid, Matthias, 97
Sering, Tino, 21
Shafiee Kamalabad, Mahdi, 173
Shaul, Cyrus, 21
Signorelli Mirko, 155
Simon, Thorsten, 143
Simone, Rosaria, 223
Simonoff, Jeffrey S., 299
Sinha, Samiran, 280
Snijders, Tom A. B., 161
Sofroniou, Nick, 217
Sohn, Alexander, 113
Solari, Aldo, 7Sottile, Gianluca, 264
Stöcker, Almond, 109
Staerk, Christian, 91
Stauffer, Reto, 137
Stolfi, Paola, 35
Stuart-Smith, Jane, 270
Tarantola, Claudia, 252
Taylor-Robinson, David, 305
Teunis, Peter, 80
Thomas Janek, 97
Trottier, Catherine, 323

INDEX
Tutz, Gerhard, 223, 231
Umlauf, Nikolaus, 103, 143
van de Kassteele, Jan, 80, 335
van der Heijden, Peter, 311
Vanbelle, Sophie, 242
Varin, Cristiano, 196
Venkatasubramaniam, Ashwini, 76
Ventrucci, Massimo, 212
von Bronk, Benedikt, 109

Wagner, Helga, 246
Waldmann, Elisabeth, 305
Wallinga, Jacco, 335
Wilson, Paul, 275
Wit, Ernst C., 47, 85, 155, 339

Zahid, Faisal Maqbool, 179
Zeileis, Achim, 137, 143
Zucchini, Walter, 125347

International Workshop on Statistical Mo
32rd edition3 – 7 July 2017
Groningen, The Netherlands
is sponsored by

We are very grateful to the following organisations for sponsoring the32nd IWSM 2017:

International
Workshop• The Statistical
Modelling
Society on Statistical Modelling

32rd edition
International Workshop on Statistical M
3 – 7 July 2017
32rd edition
Groningen, The Netherlands
NWO Star Cluster

3 – 7 July 2017
• Rijksuniversiteit Groningen
Groningen, Theon
Netherlands
International
Workshop
Statistiis(RUG)
sponsored by
NWO Star Cluster

32rd
editionis sponsored by
3 – 7 July 2017
• Johann Bernoulli Institute, RUG International Workshop on Statistical Mod
32rd The
editionNetherl
Groningen,
3 – 7 July 2017
Groningen, The Netherlands

is sponsored by• Netherlands Organisation for Scientific Research (NWO), (NWO sponsors our Open Access Session)NWO Star Cluster

is sponsored by• STAR clusterronde, NWO
NWO Star Cluster• Koninklijke Nederlandse Akademie van Wetenschappen
NWO Star Cluster• Toyota Motor CorporationNWO Star Cluster

to be continued

NWO Star Cluster

32rd edition3 – 7 July 2017
Groningen, The Netherlands
350

INDEX

is sponsored by• C.R. Rao Stichting, Groningen
NWO Star Cluster

ational Workshop on Statistical Modelling• Center for Language and Cognition, RUG
32rd edition3 – 7 July 2017
Groningen, The Netherlands
is sponsoredby and Cognitive Neurosciences, RUG• Behavioural

NWO Star Clu

NWO
Star Cluster• R StudioInternational Workshop on Statistical Modelling
32rd edition3 – 7 July 2017
Groningen, The Netherlands
is sponsored by• Statistical Analysis Software (SAS)• The welcome reception
is sponsored by:
NWO Star Cluster– The Province of Groningen– The Muncipality of Groningen
NWO Star Cluster

NWO Star Cluster– The Rijksuniversiteit Groningen (RUG)