selection of important variables and determination of functional form for continuous predictors in multivariable model building in developing regression models data analysts are often faced with many predictor variables that may influence an outcome variable after more than half a century of research the best way of selecting a multivariable model is still unresolved it is generally agreed that subject matter knowledge when available should guide model building however such knowledge is often limited and data dependent model building is required we limit the scope of the modelling exercise to selecting important predictors and choosing interpretable and transportable functions for continuous predictors assuming linear functions stepwise selection and all subset strategies are discussed the key tuning parameters are the nominal p value for testing a variable for inclusion and the penalty for model complexity respectively we argue that stepwise procedures perform better than a literature based assessment would suggest concerning selection of functional form for continuous predictors the principal competitors are fractional polynomial functions and various types of spline techniques we note that a rigorous selection strategy known as multivariable fractional polynomials mfp has been developed no spline based procedure for simultaneously selecting variables and functional forms has found wide acceptance results of fp and spline modelling are compared in two data sets it is shown that spline modelling while extremely flexible can generate fitted curves with uninterpretable wiggles particularly when automatic methods for choosing the smoothness are employed we give general recommendations to practitioners for carrying out variable and function selection while acknowledging that further research is needed we argue why mfp is our preferred approach for multivariable model building with continuous covariates

meta analysis of non linear exposure outcome relationships using individual participant data a comparison of two methods non linear exposure outcome relationships such as between body mass index bmi and mortality are common they are best explored as continuous functions using individual participant data from multiple studies we explore two two stage methods for meta analysis of such relationships where the confounder adjusted relationship is first estimated in a non linear regression model in each study then combined across studies the metacurve approach combines the estimated curves using multiple meta analyses of the relative effect between a given exposure level and a reference level the mvmeta approach combines the estimated model parameters in a single multivariate meta analysis both methods allow the exposure outcome relationship to differ across studies using theoretical arguments we show that the methods differ most when covariate distributions differ across studies using simulated data we show that mvmeta gains precision but metacurve is more robust to model mis specification we then compare the two methods using data from the emerging risk factors collaboration on bmi coronary heart disease events and all cause mortality 80 cohorts 18 000 events for each outcome we model bmi using fractional polynomials of degree 2 in each study with adjustment for confounders for metacurve the powers defining the fractional polynomials may be study specific or common across studies for coronary heart disease metacurve with common powers and mvmeta correctly identify a small increase in risk in the lowest levels of bmi but metacurve with study specific powers does not for all cause mortality all methods identify a steep u shape the metacurve and mvmeta methods perform well in combining complex exposure disease relationships across studies

reducing subspaces for analytic multipliers of the bergman space abstract in douglas et al 2011 4 some incisive results are obtained on the structure of the reducing subspaces for the multiplication operator m by a finite blaschke product on the bergman space on the unit disk in particular the linear dimension of the commutant a m m is shown to equal the number of connected components of the riemann surface 1 using techniques from douglas et al 2011 4 and a uniformization result that expresses as a holomorphic covering map in a neighborhood of the boundary of the disk we prove that a is commutative and moreover that the minimal reducing subspaces are pairwise orthogonal finally an analytic arithmetic description of the minimal reducing subspaces is also provided along with the taxonomy of the possible structures of the reducing subspaces in case has eight zeros these results have implications in both operator theory and the geometry of finite blaschke products

winding angle distribution for planar random walk polymer ring entangled with an obstacle and all that spitzer edwards prager frisch model revisited using a general green function formulation we re derive both i spitzer and his followers results for the winding angle distribution of the planar brownian motion and ii edwards prager frisch results on the statistical mechanics of a ring polymer entangled with a straight bar in the statistical mechanics part we consider both cases of quenched and annealed topology among new results we compute exactly the expectation value of the surface area of the locus of points such that each of them has linking number n with a given closed random walk trajectory ring polymer we also consider the generalizations of the problem for the finite diameter disc like obstacle and winding within a cavity

reducing subspaces of non analytic toeplitz operators on weighted hardy and dirichlet spaces of the bidisk abstract we characterize reducing subspaces of a class of non analytic toeplitz operators on hilbert spaces of holomorphic functions on the bidisk including the hardy space and the dirichlet space of the bidisk these results compliment a recent characterization of the reducing subspaces for this class of non analytic toeplitz operators on the bergman space of the bidisk

operator theory in the hardy space over the bidisk iii this paper is a continuation of a project of developing a systematic operator theory inh2 d2 a large part of it is devoted to a study ofevaluation operator which is a very useful tool in the theory a number of elementary properties of the evaluation operator are exhibited and these properties are used to derive results in other topics such as interpretation of characteristic opertor function inh2 d2 spectral equivalence compactness compressions of shift operators etc even though some results reflect the two variable nature ofh2 d2 the goal of this paper is to manifest a close tie between the operator theory inh2 d2 and classical single operator theory the unilateral shift of a finite multiplicity and the bergman shift will be used as examples to illustrate some of the results

the winding angle distribution of an ordinary random walk the winding angle about a selected point of a random walk quantifies what is perhaps the simplest manifestation of entanglement a phenomenon of great importance in the study of polymers the distribution of winding angles is studied here for ordinary or non self avoiding walks new results highlight the crucial influence of the exclusion of a finite region about the selected point on the winding angle distribution the results of this analysis are compared with simulations of random walks on a lattice the agreement between the two is very good there are however small and as yet unexplained discrepancies

factorization theorems and the structure of operators on hilbert space soit 7b h un espace de hilbert complexe et soit 7b l 7b h l algebre des operateurs lineaires bornes agissant sur 7b h on demontre des theoremes de factorisation on etudie la structure des operateurs sur 7b h

szego s theorem on a bidisc g szego showed that inf l f 2wd6 2n exp xogwdo ln jo jo where ranges over analytic polynomials with mean value zeros we study extensions of the szego s theorem on the disc to the bidisc we show that the quantity is a mixed form of an arithmetic mean and a geometric one of w in some special cases

asymptotic laws of planar brownian motion on presente des resultats recents sur la distribution asymptotique des nombres de passages et d enroulements

confidence interval estimation for the changepoint of treatment stratification in the presence of a qualitative covariate treatment interaction the goal in stratified medicine is to administer the textquotedblbest textquotedbl treatment to a patient not all patients might benefit from the same treatment the choice of best treatment can depend on certain patient characteristics in this article it is assumed that a time to event outcome is considered as a patient relevant outcome and a qualitative interaction between a continuous covariate and treatment exists ie that patients with different values of one specific covariate should be treated differently we suggest and investigate different methods for confidence interval estimation for the covariate value where the treatment recommendation should be changed based on data collected in a randomized clinical trial an adaptation of fieller s theorem the delta method and different bootstrap approaches normal percentile based wild bootstrap are investigated and compared in a simulation study extensions to multivariable problems are presented and evaluated we observed appropriate confidence interval coverage following fieller s theorem irrespective of sample size but at the cost of very wide or even infinite confidence intervals the delta method and the wild bootstrap approach provided the smallest intervals but inadequate coverage for small to moderate event numbers also depending on the location of the true changepoint for the percentile based bootstrap wide intervals were observed and it was slightly conservative regarding coverage whereas the normal bootstrap did not provide acceptable results for many scenarios the described methods were also applied to data from a randomized clinical trial comparing two treatments for patients with symptomatic severe carotid artery stenosis considering patient s age as predictive marker

detecting treatment covariate interactions using permutation methods the primary objective of a randomized clinical trial usually is to investigate whether one treatment is better than its alternatives on average however treatment effects may vary across different patient subpopulations in contrast to demonstrating one treatment is superior to another on the average sense one is often more concerned with the question that for a particular patient or a group of patients with similar characteristics which treatment strategy is most appropriate to achieve a desired outcome various interaction tests have been proposed to detect treatment effect heterogeneity however they typically examine covariates one at a time do not offer an integrated approach that incorporates all available information and can greatly increase the chance of a false positive finding when the number of covariates is large we propose a new permutation test for the null hypothesis of no interaction effects for any covariate the proposed test allows us to consider the interaction effects of many covariates simultaneously without having to group subjects into subsets based on pre specified criteria and applies generally to randomized clinical trials of multiple treatments the test provides an attractive alternative to the standard likelihood ratio test especially when the number of covariates is large we illustrate the proposed methods using a dataset from the treatment of adolescents with depression study

a new approach to modelling interactions between treatment and continuous covariates in clinical trials by using fractional polynomials we consider modelling interaction between a categoric covariate t and a continuous covariate z in a regression model here t represents the two treatment arms in a parallel group clinical trial and z is a prognostic factor which may influence response to treatment known as a predictive factor generalization to more than two treatments is straightforward the usual approach to analysis is to categorize z into groups according to cutpoint s and to analyse the interaction in a model with main effects and multiplicative terms the cutpoint approach raises several well known and difficult issues for the analyst we propose an alternative approach based on fractional polynomial fp modelling of z in all patients and at each level of t other prognostic variables can also be incorporated by first constructing a multivariable adjustment model which may contain binary covariates and fp transformations of continuous covariates other than z the main step involves fp modelling of z and testing equality of regression coefficients between treatment groups in an interaction model adjusted for other covariates extensive experience suggests that a two term fractional polynomial fp2 function may describe the effect of a prognostic factor on a survival outcome quite well in a controlled trial this fp2 function describes the prognostic effect averaged over the treatment groups we refit this function in each treatment group to see if there are substantial differences between groups allowing different parameter values for the chosen fp2 function is flexible enough to detect such differences within the same algorithm we can also deal with the conceptually different cases of a predefined hypothesis of interaction or searching for interactions we demonstrate the ability of the approach to detect and display treatment covariate interactions in two examples from controlled trials in cancer

asymptotic laws for the winding angles of planar brownian motion using constrained path integrals we study the winding angle distribution of a two dimensional brownian motion around a given point by a careful analysis of the spectral properties of some schrodinger like hamiltonians we obtain a generalization of the messulam yor law various limiting cases are considered

the curvature of a hilbert module over ℂ z1 zd a notion of curvature is introduced in multivariable operator theory the curvature invariant of a hilbert module over z1 zd is a nonnegative real number which has significant extremal properties which tends to be an integer and which is hard to compute directly it is shown that for graded hilbert modules the curvature agrees with the euler characteristic of a certain finitely generated algebraic module over the appropriate polynomial ring this result is a higher dimensional operator theoretic counterpart of the gauss bonnet formula which expresses the average gaussian curvature of a compact oriented riemann surface as the alternating sum of the betti numbers of the surface and it solves the problem of calculating the curvature of graded hilbert modules the proof of that result is based on an asymptotic formula which expresses the curvature of a hilbert module in terms that allow its comparison to a corresponding asymptotic expression for the euler characteristic

interaction of treatment with a continuous variable simulation study of significance level for several methods of analysis interactions between treatments and covariates in rcts are a key topic standard methods for modelling treatment covariate interactions with continuous covariates are categorisation or linear functions both approaches are easily criticised but for different reasons multivariable fractional polynomial interactions an approach based on fractional polynomials with the linear interaction model as the simplest special case was proposed four variants of multivariable fractional polynomial interaction flex1 flex4 allowing varying flexibility in functional form were suggested however their properties are unknown and comparisons with other procedures are unavailable additionally we consider various methods based on categorisation and on cubic regression splines we present the results of a simulation study to determine the significance level probability of a type 1 error of various tests for interaction between a binary covariate treatment effect and a continuous covariate in univariate analysis we consider a simplified setting in which the response variable is conditionally normally distributed given the continuous covariate we consider two main cases with the covariate distribution well behaved approximately symmetric or badly behaved positively skewed we construct nine scenarios with different functional forms for the main effect in the well behaved case significance levels are in general acceptably close to nominal and are slightly better for the larger sample size n 250 and 500 were investigated in the badly behaved case departures from nominal are more pronounced for several approaches for a final assessment of these results and recommendations for practice a study of power is needed copyright 2013 john wiley sons ltd

the winding angle of planar self avoiding walks the problem of understanding the asymptotic statistical behaviour of the winding angle on of a self avoiding walk of n steps on a planar lattice is posed exact series expansion data for the square lattice up to n 21 are reported these data and monte carlo estimates up to n g 170 steps are fitted well by a logarithmic growth law for 06 the ratio oh appears to approach a limit of 2 9 to 3 2 which is close to the gaussian value 3 heuristic scaling arguments are consistent with simple logarithmic growth and also illuminate the logarithmic behaviour known rigorously for free brownian motion

additive and multiplicative covariate regression models for relative survival incorporating fractional polynomials for time dependent effects relative survival is used to estimate patient survival excluding causes of death not related to the disease of interest rather than using cause of death information from death certificates which is often poorly recorded relative survival compares the observed survival to that expected in a matched group from the general population models for relative survival can be expressed on the hazard mortality rate scale as the sum of two components where the total mortality rate is the sum of the underlying baseline mortality rate and the excess mortality rate due to the disease of interest previous models for relative survival have assumed that covariate effects act multiplicatively and have thus provided relative effects of differences between groups using excess mortality rate ratios in this paper we consider i the use of an additive covariate model which provides estimates of the absolute difference in the excess mortality rate and ii the use of fractional polynomials in relative survival models for the baseline excess mortality rate and time dependent effects the approaches are illustrated using data on 115 331 female breast cancer patients diagnosed between 1 january 1986 and 31 december 1990 the use of additive covariate relative survival models can be useful in situations when the excess mortality rate is zero or slightly less than zero and can provide useful information from a public health perspective the use of fractional polynomials has advantages over the usual piecewise estimation by providing smooth estimates of the baseline excess mortality rate and time dependent effects for both the multiplicative and additive covariate models all models presented in this paper can be estimated within a generalized linear models framework and thus can be implemented using standard software copyright 2005 john wiley sons ltd

the operator mz1nz2n on subspaces of bergman spaces over the biannulus abstract we consider the multiplication operator m z 1 n z 2 n on a certain closed subspace l a 2 a r 2 of weighted bergman spaces l a 2 a r 2 over the biannulus a r 2 we prove that the operator m z 1 n z 2 n is similar to 1 n 2 m z 1 z 2 on l a 2 a r 2 moreover the reducing subspaces of m z 1 n z 2 n are investigated

multiplication operators on the bergman space via analytic continuation abstract in this paper using the group like property of local inverses of a finite blaschke product we will show that the largest c algebra in the commutant of the multiplication operator m by on the bergman space is finite dimensional and its dimension equals the number of connected components of the riemann surface of 1 over the unit disk if the order of the blaschke product is less than or equal to eight then every c algebra contained in the commutant of m is abelian and hence the number of minimal reducing subspaces of m equals the number of connected components of the riemann surface of 1 over the unit disk

extended quasi likelihood with fractional polynomials in the frame of the accelerated failure time model the accelerated failure time model is frequently used in survival analysis because of its direct physical interpretation semiparametric inference methods have been extensively investigated for this model however the accelerated failure time model and the existing inference methods assume homogeneity of the survival data after taking log transformation this assumption is not always appropriate because heterogeneous data are often encountered in practice in dealing with this heterogeneity yu yu and liu proposed a parametric quasi likelihood method by assuming a known variance function which may not be realistic for real data in this paper we extend the parametric quasi likelihood method to semiparametric via relaxing its assumption and approximating the unknown variance function by using fractional polynomials approach simulations show that this novel extension performs superior to other methods in statistical properties of unbiasedness efficiency and correct coverage probability in finite samples an application to real data set in primary biliary cirrhosis demonstrates the applicability of this new methodology

winding angle distribution of 2d random walks with traps we study analytically the asymptotic behaviour of the average probability for the trajectory of a 2d brownian particle wandering in the presence of randomly distributed traps to wind n times around a given point after a time t it is shown that with where the first exponent represents a well known long time tail of the probability that a particle will not be trapped

winding angle distributions for random walks and flux lines we study analytically and numerically the winding of a flux line around a columnar defect reflecting and absorbing boundary conditions apply to marginal or repulsive defects respectively in both cases the winding angle distribution decays exponentially for large angles with a decay constant depending only on the boundary condition but not on microscopic features nonuniversal distributions are encountered for chiral defects that preferentially twist the flux line in one direction the resulting asymmetric distributions have decay constants that depend on the degree of chirality in particular strong chirality encourages entanglements and leads to broad distributions we also examine the windings of flux lines in the presence of point impurities random bonds our results suggest that pinning to impurities reduces entanglements leading to a narrow gaussian distribution textcopyright 1996 the american physical society

algebraic areas enclosed by 2d brownian curves in random media we study closed brownian trajectories when the brownian particle is subjected to a random potential for a poissonian repulsive potential the enclosed algebraic area a is a gaussian variable that scales like t3 4 when t the length of the curve goes to infinity this is intermediate between the situations where the particle is allowed to wander i everywhere on the plane a t and ii only on a bounded domain a t1 2 for the lloyd model we show that the probability distribution p a is the same as in the absence of disorder this surprising result is related to some peculiarities of the cauchy law