time varying coefficients models for recurrent event data when different varying coefficients admit different degrees of smoothness application to heart disease modeling we consider a class of semiparametric marginal rate models for analyzing recurrent event data in these models both time varying and time free effects are present and the estimation of time varying effects may result in non smooth regression functions a typical approach for avoiding this problem and producing smooth functions is based on kernel methods the traditional kernel based approach however assumes a common degree of smoothness for all time varying regression functions which may result in suboptimal estimators if the functions have different levels of smoothness in this paper we extend the traditional approach by introducing different bandwidths for different regression functions first we establish the asymptotic properties of the suggested estimators next we demonstrate the superiority of our proposed method using two finite sample simulation studies finally we illustrate our methodology by analyzing a real world heart disease dataset copyright 2016 john wiley sons ltd

temporal process regression summary we consider regression for response and covariates which are temporal processes observed over intervals a functional generalised linear model is proposed which includes extensions of standard models in multi state survival analysis simple nonparametric esti mators of time indexed parameters are developed using working independence estimating equations and are shown to be uniformly consistent and to converge weakly to gaussian processes the procedure does not require smoothing or a markov assumption unlike approaches based on transition intensities the usual definition of optimal estimating equations for parametric models is then generalised to the functional model and the optimum is identified in a class of functional generalised estimating equations simulations demonstrate large efficiency gains relative to working independence at times where censoring is heavy the estimators are the basis for new tests of the covariate effects and for the estimation of models in which greater structure is imposed on the parameters providing novel goodness of fit tests the methodology s practical utility is illustrated in a data analysis

on the cox model with time varying regression coefficients in the analysis of censored failure time observations the standard cox proportional hazards model assumes that the regression coefficients are time invariant often these parameters vary over time and the temporal covariate effects on the failure time are of great interest in this article following previous work of cai and sun we propose a simple estimation procedure for the cox model with time varying coefficients based on a kernel weighted partial likelihood approach we construct pointwise and simultaneous confidence intervals for the regression parameters over a properly chosen time interval via a simple resampling technique we derive a prediction method for future patients survival with any specific set of covariates building on the estimates for the time varying coefficients we also consider the mixed case and present an estimation procedure for time independent parameters in the model furthermore we show how to use an integrated function of the estimate for a specific regression coeffic

a relative survival regression model using b spline functions to model non proportional hazards relative survival a method for assessing prognostic factors for disease specific mortality in unselected populations is frequently used in population based studies however most relative survival models assume that the effects of covariates on disease specific mortality conform with the proportional hazards hypothesis which may not hold in some long term studies to accommodate variation over time of a predictor s effect on disease specific mortality we developed a new relative survival regression model using b splines to model the hazard ratio as a flexible function of time without having to specify a particular functional form our method also allows for testing the hypotheses of hazards proportionality and no association on disease specific hazard accuracy of estimation and inference were evaluated in simulations the method is illustrated by an analysis of a population based study of colon cancer

nonparametric survival analysis with time dependent covariate effects a penalized partial likelihood approach techniques are developed for nonparametric analysis of data under a cox regression like model permitting time dependent covariate effects determined by a regression function beta 0 t estimators resulting from maximization of an appropriate penalized partial likelihood are shown to exist and a computational approach is outlined weak uniform consistency with a rate of convergence and pointwise asymptotic normality of the estimators are established under regularity conditions a consistent estimator of a common baseline hazard function is presented and used to construct a consistent estimator of the asymptotic variance of the estimator of the regression function extensions to multiple covariates general relative risk functions and time dependent covariates are discussed

efficient semiparametric estimation of short term and long term hazard ratios with right censored data the proportional hazards assumption in the commonly used cox model for censored failure time data is often violated in scientific studies yang and prentice 2005 proposed a novel semiparametric two sample model that includes the proportional hazards model and the proportional odds model as sub models and accommodates crossing survival curves the model leaves the baseline hazard unspecified and the two model parameters can be interpreted as the short term and long term hazard ratios inference procedures were developed based on a pseudo score approach although extension to accommodate covariates was mentioned no formal procedures have been provided or proved furthermore the pseudo score approach may not be asymptotically efficient we study the extension of the short term and long term hazard ratio model of yang and prentice 2005 to accommodate potentially time dependent covariates we develop efficient likelihood based estimation and inference procedures the nonparametric maximum likelihood estimators are shown to be consistent asymptotically normal and asymptotically efficient extensive simulation studies demonstrate that the proposed methods perform well in practical settings the proposed method successfully captured the phenomenon of crossing hazards in a cancer clinical trial and identified a genetic marker with significant long term effect missed by using the proportional hazards model on age at onset of alcoholism in a genetic study

estimation of the survival function for gray s piecewise constant time varying coefficients model gray s extension of cox s proportional hazards ph model for right censored survival data allows for a departure from the ph assumption via introduction of time varying regression coefficients tvc using penalized splines gray s work focused on estimation inference and residual analyses but no estimator for the survival function has been proposed we derive a survival function estimator for one important member of the class of tvc models a piecewise constant time varying coefficients pc tvc model we also derive an estimate for the confidence limits of the survival function accuracy in estimating underlying survival times and survival quantiles is assessed for both cox s and gray s pc tvc model using a simulation study featuring scenarios violating the ph assumption finally an example of the estimated survival functions and the corresponding confidence limits derived from cox s ph and gray s pc tvc model respectively is presented for a liver transplant data set in the second part of the thesis we examine the effect of model misspecification for two classes of regression models for right censored survival data additive and multiplicative models for the conditional hazard rate a particular attention is given to data exhibiting time varying regression coefficients the class of multiplicative models is represented by cox ph model and gray s tvc model respectively and for additive models we use aalen s linear model both gray s tvc model and aalen s linear model incorporate time varying coefficients a simulation study is performed to cross analyze survival data which follows either a multiplicative or an additive model for the conditional hazard rate the effect of misspecifying the true model for the conditional hazard rate is assessed by looking at the power of the individual models to detect an existing effect bias and mean square error observed for each conditional model based estimator of survival we also show that aalen s model formulae is a first order taylor series approximation of that of gray s model which explains the comparably higher flexibility on part of the aalen s model as compared to the cox ph when the gray s tvc model for the data is misspecified

dynamic stochastic approximation for multi stage stochastic optimization in this paper we consider multi stage stochastic optimization problems with convex objectives and conic constraints at each stage we present a new stochastic first order method namely the dynamic stochastic approximation dsa algorithm for solving these types of stochastic optimization problems we show that dsa can achieve an optimal mathcal o 1 epsilon 4 rate of convergence in terms of the total number of required scenarios when applied to a three stage stochastic optimization problem we further show that this rate of convergence can be improved to mathcal o 1 epsilon 2 when the objective function is strongly convex we also discuss variants of dsa for solving more general multi stage stochastic optimization problems with the number of stages t 3 the developed dsa algorithms only need to go through the scenario tree once in order to compute an epsilon solution of the multi stage stochastic optimization problem as a result the memory required by dsa only grows linearly with respect to the number of stages to the best of our knowledge this is the first time that stochastic approximation type methods are generalized for multi stage stochastic optimization with t ge 3

survival analysis with temporal covariate effects summary we propose a natural generalization of the cox regression model in which the regression coefficients have direct interpretations as temporal covariate effects on the survival function under the conditionally independent censoring mechanism we develop a smoothing free estimation procedure with a set of martingale based equations our estimator is shown to be uniformly consistent and to converge weakly to a gaussian process a simple resampling method is proposed for approximating the limiting distribution of the estimated coefficients second stage inferences with time varying coefficients are developed accordingly simulations and a real example illustrate the practical utility of the proposed method finally we extend this proposal of temporal covariate effects to the general class of linear transformation models and also establish a connection with the additive hazards model

iterated residuals and time varying covariate effects in cox regression summary the cox proportional hazards model which is widely used for the analysis of treatment and prognostic effects with censored survival data makes the assumption that the hazard ratio is constant over time nonparametric estimators have been developed for an extended model in which the hazard ratio is allowed to change over time estimators based on residuals are appealing as they are easy to use and relate in a simple way to the more restricted cox model estimator after fitting a cox model and calculating the residuals one can obtain a crude estimate of the time varying coefficients by adding a smooth of the residuals to the initial constant estimate treating the crude estimate as the fit one can re estimate the residuals iteration leads to consistent estimation of the nonparametric time varying coefficients this approach leads to clear guidelines for residual analysis in applications the results are illustrated by an analysis of the medical research council s myeloma trials and by simulation

functional association models for multivariate survival processes we consider multivariate temporal processes that are continuously observed within overlapping time windows the intended application is in censored multistate and multivariate survival settings where point processes are continuously observed these data differ from other functional data like longitudinal data which are discretely observed at irregular times existing functional approaches to survival processes use intensity models which require smoothing and depend critically on the choice of smoothing parameters similarly to discretely observed data in this article we study functional mean and association regression models for the point processes with unspecified time varying coefficients the continuous observation scheme is exploited the coefficients may be estimated nonparametrically by extending generalized estimating equations to continuously observed data the estimators automatically converge at the parametric rate without smoothing unlike with discretely observed data uniform consistenc

testing and estimation of time varying cause specific hazard ratios with covariate adjustment summary in the evaluation of efficacy of a vaccine to protect against disease caused by a genetically diverse infectious pathogen it is often important to assess whether vaccine protection depends on variations of the exposing pathogen this problem can be viewed within the framework of a if competing risks model where the endpoint event is pathogen specific infection and the cause of failure is the strain type determined after the infection is diagnosed the cox model with time dependent coefficients is used to relate the cause specific outcomes to explanatory variables to allow for time varying treatment effects the strain specific vaccine efficacy can be defined in terms of one minus the cause specific hazard ratios we develop inferential methods for testing whether the vaccine affords some protection against at least one pathogen strain and for testing equal vaccine protection against the strains adjusting for covariate effects we also consider estimation of covariate adjusted time varying strain specific vaccine efficacy the methods are applied to a dataset from an oral cholera vaccine trial and the performances of the proposed tests are studied through simulations these techniques apply more generally for testing and estimation of time varying cause specific hazard ratios

efficient estimation for the cox model with varying coefficients summary a proportional hazards model with varying coefficients allows one to examine the extent to which covariates interact nonlinearly with an exposure variable a global partial likelihood method in contrast with the local partial likelihood method of fan et al 2006 is proposed for estimation of varying coefficient functions the proposed estimators are proved to be consistent and asymptotically normal semiparametric efficiency of the estimators is demonstrated in terms of their linear functionals evidence in support of the superiority of the method is presented in numerical studies and real examples

proportional hazards models with frailties and random effects we discuss some of the fundamental concepts underlying the development of frailty and random effects models in survival one of these fundamental concepts was the idea of a frailty model where each subject has his or her own disposition to failure their so called frailty additional to any effects we wish to quantify via regression although the concept of individual frailty can be of value when thinking about how data arise or when interpreting parameter estimates in the context of a fitted model we argue that the concept is of limited practical value individual random effects frailties whenever detected can be made to disappear by elementary model transformation in consequence unless we are to take some model form as unassailable beyond challenge and carved in stone and if we are to understand the term frailty as referring to individual random effects then frailty models have no value random effects models on the other hand in which groups of individuals share some common effect can be used to advantage even in this case however if we are prepared to sacrifice some efficiency we can avoid complex modelling by using the considerable power already provided by the stratified proportional hazards model stratified models and random effects models can both be seen to be particular cases of partially proportional hazards models a view that gives further insight the added structure of a random effects model viewed as a stratified proportional hazards model with some added distributional constraints will for group sizes of five or more provide no more than modest efficiency gains even when the additional assumptions are exactly true on the other hand for moderate to large numbers of very small groups of sizes two or three the study of twins being a well known example the efficiency gains of the random effects model can be far from negligible for such applications the case for using random effects models rather than the stratified model is strong this is especially so in view of the good robustness properties of random effects models nonetheless the simpler analysis based upon the stratified model remains valid albeit making a less efficient use of resources

partially linear additive hazards regression with varying coefficients please see the supplementary materials for a correction to this article to explore the nonlinear interactions between some covariates and an exposure variable we propose the partially linear additive hazards model for survival data in a semiparametric setting we construct a local pseudoscore function to estimate the varying and constant coefficients and establish the asymptotic normality of the proposed estimators moreover we develop the weak convergence property for the local estimator of the baseline cumulative hazard function we conduct simulation studies to empirically examine the finite sample performance of the proposed methods and use real data from a breast cancer study for illustration

bayesian estimators for conditional hazard functions summary this article introduces a new bayesian approach to the analysis of right censored survival data the hazard rate of interest is modeled as a product of conditionally independent stochastic processes corresponding to 1 a baseline hazard function and 2 a regression function representing the temporal influence of the covariates these processes jump at times that form a time homogeneous poisson process and have a pairwise dependency structure for adjacent values the two processes are assumed to be conditionally independent given their jump times features of the posterior distribution such as the mean covariate effects and survival probabilities conditional on the covariate are evaluated using the metropolis hastings green algorithm we illustrate our methodology by an application to nasopharynx cancer survival data

erosion of regression effect in a survival study lack of persistence or erosion of the regression effect is an alternative to proportional hazards of particular interest in many medical applications such a departure from proportional hazards is often the most likely direction in which the model may be inadequate questions such as is the effect of treatment only transitory or to what extent does an initially measured prognostic variable maintain its impact frequently arise in the context of a simple changepoint model we propose a test of the null hypothesis of proportional hazards against the specific alternative of erosion of the regression effect the particular changepoint model used can be viewed as a first approximation to a more complex reality an approximation that enables us to avoid specifically modeling the functional form that any erosion might take practical guidelines for carrying out the test are provided the approach is illustrated in the context of a study on risk factors for breast cancer survival

the mizon richard encompassing test for the cox and aalen additive hazards models summary the cox hazards model cox 1972 journal of the royal statistical society series b 34 187 220 for survival data is routinely used in many applied fields sometimes however with too little emphasis on the fit of the model a useful alternative to the cox model is the aalen additive hazards model aalen 1980 in lecture notes in statistics 2 1 25 that can easily accommodate time changing covariate effects it is of interest to decide which of the two models that are most appropriate to apply in a given application this is a nontrivial problem as these two classes of models are nonnested except only for special cases in this article we explore the mizon richard encompassing test for this particular problem it turns out that it corresponds to fitting of the aalen model to the martingale residuals obtained from the cox regression analysis we also consider a variant of this method which relates to the proportional excess model martinussen and scheike 2002 biometrika 89 283 298 large sample properties of the suggested methods under the two rival models are derived the finite sample properties of the proposed procedures are assessed through a simulation study the methods are further applied to the well known primary biliary cirrhosis data set

regression analysis with multiplicative and time varying additive regression coefficients with examples from breast and colon cancer regression analysis may be used to simplify the representation of mortality rates when there are many significant prognostic covariates or to adjust for confounding effects the principal request of the regression model in this range of use is to have unbiased parameter estimates a model with constant multiplicative and time varying additive regression coefficients is discussed the model allows some covariate effects to be multiplicative while allowing others to have a time varying additive effect thus it is a mix of classical cox regression and aalen s additive risk model a major characteristic of cancer mortality rates in contrast to general mortality rates is that hazard rates after a potentially initial increase decrease although not always tending to zero cancer diseases like breast and colon cancer have significantly increased cause specific mortality rates even 20 years after diagnosis another major feature in cancer survival analysis is that many covariate effects are time varying some covariate effects like age at diagnosis may only be significant for a limited time after diagnosis furthermore some treatment procedures may initially decrease the mortality while the long term effect may be opposite a third issue is that average covariate effects are very often not multiplicative estimation is carried out iteratively the cumulative additive regression functions are estimated non parametrically using a least squares method and the multiplicative parameters are estimated from the partial likelihood the method is applied on 3201 female breast cancer and 1372 male colon cancer patients

marginal regression analysis of recurrent events with coarsened censoring times motivated by an ongoing pediatric mental health care pmhc study this article presents weakly structured methods for analyzing doubly censored recurrent event data where only coarsened information on censoring is available the study extracted administrative records of emergency department visits from provincial health administrative databases the available information of each individual subject is limited to a subject specific time window determined up to concealed data to evaluate time dependent effect of exposures we adapt the local linear estimation with right censored survival times under the cox regression model with time varying coefficients cf cai and sun scandinavian journal of statistics 2003 30 93 111 we establish the pointwise consistency and asymptotic normality of the regression parameter estimator and examine its performance by simulation the pmhc study illustrates the proposed approach throughout the article

group analysis in functional neuroimaging selecting subjects using similarity measures standard group analyses of fmri data rely on spatial and temporal averaging of individuals this averaging operation is only sensible when the mean is a good representation of the group this is not the case if subjects are not homogeneous and it is therefore a major concern in fmri studies to assess this group homogeneity we present a method that provides relevant distances or similarity measures between temporal series of brain functional images belonging to different subjects the method allows a multivariate comparison between data sets of several subjects in the time or in the space domain these analyses assess the global intersubject variability before averaging subjects and drawing conclusions across subjects at the population level we adapt the rv coefficient to measure meaningful spatial or temporal similarities and use multidimensional scaling to give a visual representation of each subject s position with respect to other subjects in the group we also provide a measure for detecting subjects that may be outliers results show that the method is a powerful tool to detect subjects with specific temporal or spatial patterns and that despite the apparent loss of information restricting the analysis to a homogeneous subgroup of subjects does not reduce the statistical sensitivity of standard group fmri analyses

model selection for cox models with time varying coefficients summary cox models with time varying coefficients offer great flexibility in capturing the temporal dynamics of covariate effects on right censored failure times because not all covariate coefficients are time varying model selection for such models presents an additional challenge which is to distinguish covariates with time varying coefficient from those with time independent coefficient we propose an adaptive group lasso method that not only selects important variables but also selects between time independent and time varying specifications of their presence in the model each covariate effect is partitioned into a time independent part and a time varying part the latter of which is characterized by a group of coefficients of basis splines without intercept model selection and estimation are carried out through a fast iterative group shooting algorithm our approach is shown to have good properties in a simulation study that mimics realistic situations with up to 20 variables a real example illustrates the utility of the method

proportional hazards tests and diagnostics based on weighted residuals summary nonproportional hazards can often be expressed by extending the cox model to include time varying coefficients e g for a single covariate the hazard function for subject i is modelled as exp fl t zi t a common example is a treatment effect that decreases with time we show that the function 3 t can be directly visualized by smoothing an appropriate residual plot also many tests of proportional hazards including those of cox 1972 gill schumacher 1987 harrell 1986 lin 1991 moreau o quigley mesbah 1985 nagelkerke oosting hart 1984 o quigley pessione 1989 schoenfeld 1980 and wei 1984 are related to time weighted score tests of the proportional hazards hypothesis and can be visualized as a weighted least squares line fitted to the residual plot

estimating cumulative treatment effects in the presence of nonproportional hazards often in medical studies of time to an event the treatment effect is not constant over time in the context of cox regression modeling the most frequent solution is to apply a model that assumes the treatment effect is either piecewise constant or varies smoothly over time i e the cox nonproportional hazards model this approach has at least two major limitations first it is generally difficult to assess whether the parametric form chosen for the treatment effect is correct second in the presence of nonproportional hazards investigators are usually more interested in the cumulative than the instantaneous treatment effect e g determining if and when the survival functions cross therefore we propose an estimator for the aggregate treatment effect in the presence of nonproportional hazards our estimator is based on the treatment specific baseline cumulative hazards estimated under a stratified cox model no functional form for the nonproportionality need be assumed asymptotic properties of the proposed estimators are derived and the finite sample properties are assessed in simulation studies pointwise and simultaneous confidence bands of the estimator can be computed the proposed method is applied to data from a national organ failure registry

local partial likelihood estimation for lifetime data this paper considers a proportional hazards model which allows one to examine the extent to which covariates interact nonlinearly with an exposure variable for analysis of lifetime data a local partial likelihood technique is proposed to estimate nonlinear interactions asymptotic normality of the proposed estimator is established the baseline hazard function the bias and the variance of the local likelihood estimator are consistently estimated in addition a one step local partial likelihood estimator is presented to facilitate the computation of the proposed procedure and is demonstrated to be as efficient as the fully iterated local partial likelihood estimator furthermore a penalized local likelihood estimator is proposed to select important risk variables in the model numerical examples are used to illustrate the effectiveness of the proposed procedures

a partly parametric additive risk model summary aalen s additive risk model allows the influence of each covariate to vary separately over time although allowing greater flexibility of temporal structure than a cox model aalen s model is more limited in the number of covariates it can handle we introduce a partly parametric version of aalen s model in which the influence of only a few covariates varies nonparametrically over time and that of the remaining covariates is constant efficient procedures for fitting this new model are developed and studied the approach is applied to data from the medical research council s myelomatosis trials it is the first step of a taylor series expansion of a general hazard function about the zero of the covariate vector however in estimating the unknown functions in such a general model there is a variance bias trade off that may be critical in small and medium samples also after fitting the model one does not have parameters or formulae that are easily reported we propose a model that takes the additive structure of aalen s model and imposes parametric constraints to obtain a semiparametric submodel which may be more appropriate in some applications the model will be illustrated with data from clinical trials on myelomatosis covariates include treatment sex and four age strata which will be treated parametrically together with serum levels of haemoglobin and f32 microglobulin whose effects will be investigated nonparametrically the additive form can be interpreted loosely in terms of unobserved competing risks since the hazard function for the minimum of independent random vari ables is the sum of the hazard functions for the individual variables microglobulin levels are related to kidney function and tumour mass whereas haemoglobin is unaffected by kidney function hence one might anticipate that the hazard function associated with each covariate represents a different cause of death

on fitting cox s regression model with time dependent coefficients summary this paper examines goodness of fit testing in the cox regression model with timevarying regression coefficients arjas 1988 approach is used to define test statistics of the kolmogorov smirnov and cramer von mises types their asymptotic limits are shown to be well known functionals of standard brownian motion leading to the construction of formal goodness of fit tests some numerical studies are also included to illustrate the performance of the tests for moderate sample sizes

stochastic calculus as a tool in survival analysis a review this paper gives a brief survey of the uses of stochastic calculus in survival analysis the role played by martingale central limit theory in deriving asymptotic distributions of estimators and test statistics is described the nelson aalen estimator the kaplan meier estimator cox s proportional hazards model aalen s additive risk model and a goodness of fit test for cox s model are discussed sketches of the proofs of the main results are included

semiparametric approaches for joint modeling of longitudinal and survival data with time varying coefficients we study joint modeling of survival and longitudinal data there are two regression models of interest the primary model is for survival outcomes which are assumed to follow a time varying coefficient proportional hazards model the second model is for longitudinal data which are assumed to follow a random effects model based on the trajectory of a subject s longitudinal data some covariates in the survival model are functions of the unobserved random effects estimated random effects are generally different from the unobserved random effects and hence this leads to covariate measurement error to deal with covariate measurement error we propose a local corrected score estimator and a local conditional score estimator both approaches are semiparametric methods in the sense that there is no distributional assumption needed for the underlying true covariates the estimators are shown to be consistent and asymptotically normal however simulation studies indicate that the conditional score estimator outperforms the corrected score estimator for finite samples especially in the case of relatively large measurement error the approaches are demonstrated by an application to data from an hiv clinical trial