joint mean covariance models with applications to longitudinal data unconstrained parameterisation summary we provide unconstrained parameterisation for and model a covariance using covariates the cholesky decomposition of the inverse of a covariance matrix is used to associate a unique unit lower triangular and a unique diagonal matrix with each covariance matrix the entries of the lower triangular and the log of the diagonal matrix are unconstrained and have meaning as regression coefficients and prediction variances when regressing a measurement on its predecessors an extended generalised linear model is introduced for joint modelling of the vectors of predictors for the mean and covariance subsuming the joint modelling strategy for mean and variance heterogeneity gabriel s antedependence models dempster s covariance selection models and the class of graphical models the likelihood function and maximum likelihood estimators of the covariance and the mean parameters are studied when the observations are normally distributed applications to modelling nonstationary dependence structures and multivariate data are discussed and illustrated using real data a graphical method similar to that based on the correlogram in time series is developed and used to identify parametric models for nonstationary covariances

mixed effects nonlinear regression for unbalanced repeated measures repeated measures data such as clinical pharmacokinetic data growth data and dose response data are often inherently nonlinear with respect to a given response function and are frequently incomplete and or unbalanced nonlinear random effects models together with a variety of estimation procedures have been proposed for the analysis of such data this paper is concerned with a straightforward procedure for estimating and comparing the parameters of a generalized mixed effects nonlinear regression model the asymptotic properties of the proposed estimators are given and large sample tests of hypothesis provided the results are applied to in vitro data on the water transport kinetics of hemodialyzers used in the treatment of patients with chronic renal failure

bayesian analysis of covariance matrices and dynamic models for longitudinal data summary parsimonious modelling of the within subject covariance structure while heeding its positive definiteness is of great importance in the analysis of longitudinal data using the cholesky decomposition and the ensuing unconstrained and statistically meaningful reparameterisation we provide a convenient and intuitive framework for developing conditionally conjugate prior distributions for covariance matrices and show their connections with generalised inverse wishart priors our priors offer many advantages with regard to elicitation positive definiteness computations using gibbs sampling shrinking covariances toward a particular structure with considerable flexibility and modelling covariances using covariates bayesian estimation methods are developed and the results are compared using two simulation studies these simulations suggest simpler and more suitable priors for the covariance structure of longitudinal data

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

joint models for the association of longitudinal binary and continuous processes with application to a smoking cessation trial joint models for the association of a longitudinal binary and a longitudinal continuous process are proposed for situations in which their association is of direct interest the models are parameterized such that the dependence between the two processes is characterized by unconstrained regression coefficients bayesian variable selection techniques are used to parsimoniously model these coefficients a markov chain monte carlo mcmc sampling algorithm is developed for sampling from the posterior distribution using data augmentation steps to handle missing data several technical issues are addressed to implement the mcmc algorithm efficiently the models are motivated by and are used for the analysis of a smoking cessation clinical trial in which an important question of interest was the effect of the exercise treatment on the relationship between smoking cessation and weight gain

garden of eden states in a traffic model revisited a modified mean field theory is framed to adjoin two successful mean field theories cluster approximation and paradisical mean field theory the results do not give an improvement in describing the fundamental diagram we point out that the success of paradisical mean field theory is accidental and cannot be improved systematically as the cluster approximation can

automatic estimation of multivariate spectra via smoothing splines summary the classical method for estimating the spectral density of a multivariate time series is first to calculate the periodogram and then to smooth it to obtain a consistent estimator typically to ensure the estimate is positive definite all the elements of the periodogram are smoothed the same way there are however many situations for which different components of the spectral matrix have different degrees of smoothness we propose a bayesian approach that uses markov chain monte carlo techniques to fit smoothing splines to each component real and imaginary of the cholesky decomposition of the periodogram matrix the spectral estimator is then obtained by reconstructing the spectral estimator from the smoothed cholesky decomposition components our technique produces an automatically smoothed spectral matrix estimator along with samples from the posterior distributions of the parameters to facilitate inference

relating the classical covariance adjustment techniques of multivariate growth curve models to modern univariate mixed effects models summary the relationship between the modern univariate mixed model for analyzing longitudinal data popularized by laird and ware 1982 biometrics38 963 974 and its predecessor the classical multivariate growth curve model summarized by grizzle and allen 1969 biometrics25 357 381 has never been clearly established here the link between the two methodologies is derived and balanced polynomial and cosinor examples cited in the literature are analyzed with both approaches relating the two models demonstrates that classical covariance adjustment for higher order terms is analogous to including them as random effects in the mixed model the polynomial example clearly illustrates the relationship between the methodologies and shows their equivalence when all matrices are properly defined the cosinor example demonstrates how results from each method may differ when the total variance covariance matrix is positive definite but that the between subjects component of that matrix is not so constrained by the growth curve approach additionally advocates of each approach tend to consider different covariance structures modern mixed model analysts consider only those terms in a model s expectation or linear combinations and preferably the most parsimonious subset as candidates for random effects classical growth curve analysts automatically consider all terms in a model s expectation as random effects and then investigate whether covariance adjusting for higher order terms improves the model we apply mixed model techniques to cosinor analyses of a large unbalanced data set to demonstrate the relevance of classical covariance structures that were previously conceived for use only with completely balanced data

estimation of average treatment effect with incompletely observed longitudinal data application to a smoking cessation study we study the problem of estimation and inference on the average treatment effect in a smoking cessation trial where an outcome and some auxiliary information were measured longitudinally and both were subject to missing values dynamic generalized linear mixed effects models linking the outcome the auxiliary information and the covariates are proposed the maximum likelihood approach is applied to the estimation and inference on the model parameters the average treatment effect is estimated by the g computation approach and the sensitivity of the treatment effect estimate to the nonignorable missing data mechanisms is investigated through the local sensitivity analysis approach the proposed approach can handle missing data that form arbitrary missing patterns over time we applied the proposed method to the analysis of the smoking cessation trial

finite mixture varying coefficient models for analyzing longitudinal heterogenous data this paper aims to develop a mixture model to study heterogeneous longitudinal data on the treatment effect of heroin use from a california civil addict program each component of the mixture is characterized by a varying coefficient mixed effect model we use the bayesian p splines approach to approximate the varying coefficient functions we develop markov chain monte carlo algorithms to estimate the smooth functions unknown parameters and latent variables in the model we use modified deviance information criterion to determine the number of components in the mixture a simulation study demonstrates that the modified deviance information criterion selects the correct number of components and the estimation of unknown quantities is accurate we apply the proposed model to the heroin treatment study furthermore we identify heterogeneous longitudinal patterns copyright 2011 john wiley sons ltd

brief paper decentralized control design of interconnected chains of integrators a case study we develop a constructive decentralized control design procedure for a class of systems that may be loosely described as chained integrators which are dynamically coupled the design method is inspired by nested saturation control ideas and formulated by applying the singular perturbation theory we demonstrate that the proposed design provides a lyapunov function for an associated closed loop system from which semi global stability may be deduced using the proposed idea we design a semi globally stabilizing control law for a four degree of freedom spherical inverted pendulum

miscellanea random coefficient autoregressive models for longitudinal data summary stable autoregressive models with exogenous covariates can be used to mimic nonstationary stabilising growth profiles in longitudinal data if different individuals have very different trajectories the autoregressive parameters can be interpreted as random variables varying from one individual to the next if all random effects are assumed gaussian the model will be quite easy to estimate although the marginal distributions of the observations will no longer be gaussian however this kind of model can only be used when the observations are equally spaced

non local stabilization of a spherical inverted pendulum we design a non linear stabilizing control law for a four degree of freedom spherical inverted pendulum the pendulum is a slim cylindrical beam attached to a horizontal plane via a universal joint the joint is free to move in the plane under the influence of a planar force the upright position is an unstable equilibrium of the uncontrolled system because of gravity the objective is to design a controller so that it stabilizes the upright position starting from any position in the upper hemisphere with arbitrary velocity we achieve this by first transforming the original system to an appropriate upper triangular form and then designing a controller which incorporates a high gain design with the method of non linear forwarding the control law is evaluated through computer simulations

smoothing spline models with correlated random errors abstract spline smoothing techniques are commonly used to estimate the mean function in a nonparametric regression model their performances depend greatly on the choice of smoothing parameters many methods of selecting smoothing parameters such as generalized maximum likelihood gml generalized cross validation gcv and unbiased risk ubr have been developed under the assumption of independent observations they tend to underestimate smoothing parameters when data are correlated in this article i assume that observations are correlated and that the correlation matrix depends on a parsimonious set of parameters i extend the gml gcv and ubr methods to estimate the smoothing parameters and the correlation parameters simultaneously i also relate a smoothing spline model to three mixed effects models these relationships show that the smoothing spline estimates evaluated at design points are best linear unbiased prediction blup estimates and that the gml estimates of the smoothing parameters an

nonlinear modeling of serial immunologic data a case study abstract this article concerns the analysis of serial immunologic data from a clinical trial of immunosuppressive chemotherapy in the treatment of multiple sclerosis the goal of the analysis is to relate levels of drug dose to levels of an immunologic outcome variable i propose a new nonlinear model for the analysis of such data the model assumes that the mean function is the solution of an ordinary differential equation in time parameters of which are related to the dose via a regression function the defining differential equation is that which gives rise to the generalized logistic function a flexible form that includes a number of popular growth models we fit the model which accounts for random effects and time series autocorrelation by maximum likelihood results suggest strong treatment effects in two active drug groups and a small but significant effect in a placebo group these findings agree well with previously reported analyses of clinical outcomes from the trial an empirical comparis

stabilization of feedforward systems approximated by a non linear chain of integrators we prove that for any odd integer p and any strictly positive integer n feedforward systems which are approximated at the origin by a chain of integrators of degree p and length n can be globally asymptotically stabilized by bounded smooth time invariant state feedbacks our proof is based on the construction of a lyapunov function and the feedback laws we obtain are given by explicit formulas

reply to comment on garden of eden states in a traffic model revisited a consistent theory is presented the kolmogorov conditions are satisfied with the state dependent normalizations however such a consistent theory does not provide a better description to the flow density relation eliminating the garden of eden states as done in the paradisical mean field theory will not succeed when one goes beyond the simple mean field theory

bayesian semiparametric regression for longitudinal binary processes with missing data longitudinal studies with binary repeated measures are widespread in biomedical research marginal regression approaches for balanced binary data are well developed whereas for binary process data where measurement times are irregular and may differ by individuals likelihood based methods for marginal regression analysis are less well developed in this article we develop a bayesian regression model for analyzing longitudinal binary process data with emphasis on dealing with missingness we focus on the settings where data are missing at random mar which require a correctly specified joint distribution for the repeated measures in order to draw valid likelihood based inference about the marginal mean to provide maximum flexibility the proposed model specifies both the marginal mean and serial dependence structures using nonparametric smooth functions serial dependence is allowed to depend on the time lag between adjacent outcomes as well as other relevant covariates inference is fully bayesian using simulations we show that adequate modeling of the serial dependence structure is necessary for valid inference of the marginal mean when the binary process data are mar longitudinal viral load data from the hiv epidemiology research study are analyzed for illustration

shrinkage estimators for covariance matrices estimation of covariance matrices in small samples has been studied by many authors standard estimators like the unstructured maximum likelihood estimator ml or restricted maximum likelihood reml estimator can be very unstable with the smallest estimated eigenvalues being too small and the largest too big a standard approach to more stably estimating the matrix in small samples is to compute the ml or reml estimator under some simple structure that involves estimation of fewer parameters such as compound symmetry or independence however these estimators will not be consistent unless the hypothesized structure is correct if interest focuses on estimation of regression coefficients with correlated or longitudinal data a sandwich estimator of the covariance matrix may be used to provide standard errors for the estimated coefficients that are robust in the sense that they remain consistent under misspecification of the covariance structure with large matrices however the inefficiency of the sandwich estimator becomes worrisome we consider here two general shrinkage approaches to estimating the covariance matrix and regression coefficients the first involves shrinking the eigenvalues of the unstructured ml or reml estimator the second involves shrinking an unstructured estimator toward a structured estimator for both cases the data determine the amount of shrinkage these estimators are consistent and give consistent and asymptotically efficient estimates for regression coefficients simulations show the improved operating characteristics of the shrinkage estimators of the covariance matrix and the regression coefficients in finite samples the final estimator chosen includes a combination of both shrinkage approaches i e shrinking the eigenvalues and then shrinking toward structure we illustrate our approach on a sleep eeg study that requires estimation of a 24 x 24 covariance matrix and for which inferences on mean parameters critically depend on the covariance estimator chosen we recommend making inference using a particular shrinkage estimator that provides a reasonable compromise between structured and unstructured estimators

comment on garden of eden states in a traffic model revisited recently huang and lin suggested a combination of two successful mean field theories the 2 cluster approximation and paradisical mean field for the nagel schreckenberg cellular automaton model of traffic flow we argue that this new approximation is inconsistent since it violates the kolmogorov conditions

maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix the positive definiteness constraint is the most awkward stumbling block in modelling the covariance matrix pourahmadi s 1999 unconstrained parameterisation models covariance using covariates in a similar manner to mean modelling in generalised linear models the new covariance parameters have statistical interpretation as the regression coefficients and logarithms of prediction error variances corresponding to regressing a response on its predecessors in this paper the maximum likelihood estimators of the parameters of a generalised linear model for the covariance matrix their consistency and their asymptotic normality are studied when the observations are normally distributed these results along with the likelihood ratio test and penalised likelihood criteria such as bic for model and variable selection are illustrated using a real dataset

garden of eden states in traffic models we investigate the allowed configurations in the stationary state of the cellular automaton model for single lane traffic it is found that certain states in the configuration space cannot be reached if one uses parallel dynamics these so called garden of eden goe states do not exist for random sequential dynamics and are responsible for the strong short ranged correlations found in parallel dynamics by eliminating the goe states we obtain a simple and effective approximative description of the model for the exact solution is recovered for this elimination leads to much higher values of the flux compared to the mean field result which are in good agreement with monte carlo simulations

discrete stochastic models for traffic flow we investigate a probabilistic cellular automaton model which has been introduced recently this model describes single lane traffic flow on a ring and generalizes the asymmetric exclusion process models we study the equilibrium properties and calculate the so called fundamental diagrams flow versus density for parallel dynamics this is done numerically by computer simulations of the model and by means of an improved mean field approximation which takes into account short range correlations for cars with a maximum velocity of 1 the simplest nontrivial approximation gives the exact result for higher velocities the analytical results obtained by iterated application of the approximation scheme are in excellent agreement with the numerical simulations

robust global stabilization of a class of uncertain feedforward nonlinear systems abstract the problem of global asymptotically stabilizing a certain class of uncertain feedforward nonlinear systems is considered the control law is obtained by nesting saturation functions whose amplitude can be rendered arbitrarily small with respect to previous works on the subject the design procedure is able to deal with uncertain possibly time varying parameters ranging within the prescribed compact sets which can affect also the linear approximation of the system the small gain theorem for nonlinear systems which are input to state stable with restrictions is shown to be a key tool for designing a state feedback saturated control law

modeling the random effects covariance matrix for longitudinal data with covariates measurement error longitudinal data occur frequently in practice such as medical studies and life sciences generalized linear mixed models glmms are commonly used to analyze such data it is typically assumed that the random effects covariance matrix is constant among subjects in these models in many situations however the correlation structure may differ among subjects and ignoring this heterogeneity can lead to biases in model parameters estimate recently lee et al developed a heterogeneous random effects covariance matrix for glmms for error free covariates covariates measured with error also happen frequently in the longitudinal data set up eg blood pressure and cholesterol level ignoring this issue in the data may produce bias in model parameters estimate and lead to wrong conclusions in this paper we propose an approach to properly model the random effects covariance matrix based on covariates in the class of glmms where we also have covariates measured with error the resulting parameters from the decomposition of random effects covariance matrix have a sensible interpretation and can be easily modeled without the concern of positive definiteness of the resulting estimator the performance of the proposed approach is evaluated through simulation studies which show that the proposed method performs very well in terms of bias mean squared error and coverage rate an application of the proposed method is also provided using a longitudinal data from manitoba follow up study

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

bayesian nonparametric longitudinal data analysis abstractpractical bayesian nonparametric methods have been developed across a wide variety of contexts here we develop a novel statistical model that generalizes standard mixed models for longitudinal data that include flexible mean functions as well as combined compound symmetry cs and autoregressive ar covariance structures ar structure is often specified through the use of a gaussian process gp with covariance functions that allow longitudinal data to be more correlated if they are observed closer in time than if they are observed farther apart we allow for ar structure by considering a broader class of models that incorporates a dirichlet process mixture dpm over the covariance parameters of the gp we are able to take advantage of modern bayesian statistical methods in making full predictive inferences and about characteristics of longitudinal profiles and their differences across covariate combinations we also take advantage of the generality of our model which provides for estimation

dynamic conditionally linear mixed models for longitudinal data we develop a new class of models dynamic conditionally linear mixed models for longitudinal data by decomposing the within subject covariance matrix using a special cholesky decomposition here dynamic means using past responses as covariates and conditional linearity means that parameters entering the model linearly may be random but nonlinear parameters are nonrandom this setup offers several advantages and is surprisingly similar to models obtained from the first order linearization method applied to nonlinear mixed models first it allows for flexible and computationally tractable models that include a wide array of covariance structures these structures may depend on covariates and hence may differ across subjects this class of models includes e g all standard linear mixed models antedependence models and vonesh carter models second it guarantees the fitted marginal covariance matrix of the data is positive definite we develop methods for bayesian inference and motivate the usefulness of these models using a series of longitudinal depression studies for which the features of these new models are well suited

modelling the random effects covariance matrix in longitudinal data a common class of models for longitudinal data are random effects mixed models in these models the random effects covariance matrix is typically assumed constant across subject however in many situations this matrix may differ by measured covariates in this paper we propose an approach to model the random effects covariance matrix by using a special cholesky decomposition of the matrix in particular we will allow the parameters that result from this decomposition to depend on subject specific covariates and also explore ways to parsimoniously model these parameters an advantage of this parameterization is that there is no concern about the positive definiteness of the resulting estimator of the covariance matrix in addition the parameters resulting from this decomposition have a sensible interpretation we propose fully bayesian modelling for which a simple gibbs sampler can be implemented to sample from the posterior distribution of the parameters we illustrate these models on data from depression studies and examine the impact of heterogeneity in the covariance matrix on estimation of both fixed and random effects

smoothing spline models for the analysis of nested and crossed samples of curves abstract we introduce a class of models for an additive decomposition of groups of curves stratified by crossed and nested factors generalizing smoothing splines to such samples by associating them with a corresponding mixed effects model the models are also useful for imputation of missing data and exploratory analysis of variance we prove that the best linear unbiased predictors blups from the extended mixed effects model correspond to solutions of a generalized penalized regression where smoothing parameters are directly related to variance components and we show that these solutions are natural cubic splines the model parameters are estimated using a highly efficient implementation of the em algorithm for restricted maximum likelihood reml estimation based on a preliminary eigenvector decomposition variability of computed estimates can be assessed with asymptotic techniques or with a novel hierarchical bootstrap resampling scheme for nested mixed effects models our methods are applied to me

cellular automation models and traffic flow a cellular automaton model for the description of traffic flow is investigated it generalizes asymmetric exclusion models which have attracted a lot of interest in the past the authors calculate the so called fundamental diagram flow versus density for parallel dynamics using an improved mean field approximation which takes into account short range correlations for maximum velocity they find that the simplest of these non trivial approximations gives the exact result for higher velocities their results are in excellent agreement with numerical data

a nonparametric prior for simultaneous covariance estimation in the modeling of longitudinal data from several groups appropriate handling of the dependence structure is of central importance standard methods include specifying a single covariance matrix for all groups or independently estimating the covariance matrix for each group without regard to the others but when these model assumptions are incorrect these techniques can lead to biased mean effects or loss of efficiency respectively thus it is desirable to develop methods to simultaneously estimate the covariance matrix for each group that will borrow strength across groups in a way that is ultimately informed by the data in addition for several groups with covariance matrices of even medium dimension it is difficult to manually select a single best parametric model among the huge number of possibilities given by incorporating structural zeros and or commonality of individual parameters across groups in this paper we develop a family of nonparametric priors using the matrix stick breaking process of dunson et al 2008 that seeks to accomplish this task by parameterizing the covariance matrices in terms of the parameters of their modified cholesky decomposition pourahmadi 1999 we establish some theoretic properties of these priors examine their effectiveness via a simulation study and illustrate the priors using data from a longitudinal clinical trial

an estimating equation approach to dimension reduction for longitudinal data sufficient dimension reduction has been extensively explored in the context of independent and identically distributed data in this article we generalize sufficient dimension reduction to longitudinal data and propose an estimating equation approach to estimating the central mean subspace the proposed method accounts for the covariance structure within each subject and improves estimation efficiency when the covariance structure is correctly specified even if the covariance structure is misspecified our estimator remains consistent in addition our method relaxes distributional assumptions on the covariates and is doubly robust to determine the structural dimension of the central mean subspace we propose a bayesian type information criterion we show that the estimated structural dimension is consistent and that the estimated basis directions are root formula see text consistent asymptotically normal and locally efficient simulations and an analysis of the framingham heart study data confirm the effectiveness of our approach

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea