cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

on stability of lms methods and characteristic roots of delay differential equations we investigate the use of linear multistep lms methods for computing characteristic roots of systems of linear delay differential equations ddes with multiple fixed discrete delays these roots are important in the context of stability and bifurcation analysis we prove convergence orders for the characteristic root approximations and analyze under what condition for the steplength the discrete integration scheme retains certain delay independent stability properties of the original equations unlike existing results we concentrate on the recovery of both stability and instability we illustrate our findings with a number of numerical test results

proportional hazards estimate of the conditional survival function summary we introduce a new estimator of the conditional survival function given some subset of the covariate values under a proportional hazards regression the new estimate does not require estimating the base line cumulative hazard function an estimate of the variance is given and is easy to compute involving only those quantities that are routinely calculated in a cox model analysis the asymptotic normality of the new estimate is shown by using a central limit theorem for kaplan meier integrals we indicate the straightforward extension of the estimation procedure under models with multiplicative relative risks including non proportional hazards and to stratified and frailty models the estimator is applied to a gastric cancer study where it is of interest to predict patients survival based only on measurements obtained before surgery the time at which the most important prognostic variable stage becomes known

random effects cox models a poisson modelling approach summary we propose a poisson modelling approach to nested random effects cox proportional hazards models an important feature of this approach is that the principal results depend only on the first and second moments of the unobserved random effects the orthodox best linear unbiased predictor approach to random effects poisson modelling techniques enables us to justify appropriate consistency and optimality the explicit expressions for the random effects given by our approach facilitate incorporation of a relatively large number of random effects the use of the proposed methods is illustrated through the reanalysis of data from a large scale cohort study of particulate air pollution and mortality previously reported by pope et al 1995

a comparison of certain bootstrap confidence intervals in the cox model abstract we study bootstrap confidence intervals for three types of parameters in cox s proportional hazards model the regression parameter the survival function at fixed time points and the median survival time at fixed values of a covariate several types of bootstrap confidence intervals are studied and the type of interval is determined by two factors one factor is the method of drawing the bootstrap sample we consider three such methods 1 ordinary resampling from the empirical cumulative distribution function 2 resampling conditional on the covariates and 3 resampling conditional on the covariates and the censoring pattern another factor is the method of forming the confidence interval from a given sample the methods considered are the percentile hybrid and bootstrap t all the methods of forming confidence intervals are compared to each other and to the standard asymptotic method via a monte carlo study the data sets for this monte carlo study are simulated conditionally on the c

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

the relative frailty variance and shared frailty models the relative frailty variance among survivors provides a readily interpretable measure of how the heterogeneity of a population as represented by a frailty model evolves over time we discuss the properties of the relative frailty variance show that it characterizes frailty distributions and that suitably rescaled it may be used to compare patterns of dependence across models and data sets in shared frailty models the relative frailty variance is closely related to the cross ratio function which is estimable from bivariate survival data we investigate the possible shapes of the relative frailty variance function for the purpose of model selection and review available frailty distribution families in this context we introduce several new families with contrasting properties including simple but flexible time varying frailty models the benefits of the approach we propose are illustrated with two applications to bivariate current status data obtained from serological surveys

explained randomness in proportional hazards models a coefficient of explained randomness analogous to explained variation but for non linear models was presented by kent the construct hinges upon the notion of kullback leibler information gain kent and o quigley developed these ideas obtaining simple multiple and partial coefficients for the situation of proportional hazards regression their approach was based upon the idea of transforming a general proportional hazards model to a specific one of weibull form xu and o quigley developed a more direct approach more in harmony with the semi parametric nature of the proportional hazards model thereby simplifying inference and allowing for instance the use of time dependent covariates a potential drawback to the coefficient of xu and o quigley is its interpretation as explained randomness in the covariate given time an investigator might feel that the interpretation of the kent and o quigley coefficient as a proportion of explained randomness of time given the covariate is preferable one purpose of this note is to indicate that under an independent censoring assumption the two population coefficients coincide thus the simpler inferential setting for xu and o quigley can also be applied to the coefficient of kent and o quigley our second purpose is to point out that a sample based coefficient in common use in the sas statistical package can be interpreted as an estimate of explained randomness when there is no censoring when there is censoring the sas coefficient would not seem satisfactory in that its population counterpart depends on an independent censoring mechanism however there is a quick fix and we argue in favour of its use

survival analysis with time varying regression effects using a tree based approach nonproportional hazards often arise in survival analysis as is evident in the data from the international non hodgkin s lymphoma prognostic factors project a tree based method to handle such survival data is developed for the assessment and estimation of time dependent regression effects under a cox type model the tree method approximates the time varying regression effects as piecewise constants and is designed to estimate change points in the regression parameters a fast algorithm that relies on maximized score statistics is used in recursive segmentation of the time axis following the segmentation a pruning algorithm with optimal properties similar to those of classification and regression trees cart is used to determine a sparse segmentation bootstrap resampling is used in correcting for overoptimism due to split point optimization the piecewise constant model is often more suitable for clinical interpretation of the regression parameters than the more flexible spline models the utility of the algorithm is shown on the lymphoma data where we further develop the published international risk index into a time varying risk index for non hodgkin s lymphoma

stability of matrix second order systems new conditions and perspectives modeling of many dynamic systems results in matrix second order differential equations in the paper the stability issues of matrix second order dynamical systems are discussed in the literature only sufficient conditions of stability and or instability for a system in matrix second order form are available in this paper necessary and sufficient conditions of asymptotic stability for time invariant systems in matrix second order form under different types of dynamic loadings conservative nonconservative are derived and a physical interpretation is carried out the stability conditions in the sense of lyapunov the jw axis behavior of eigenvalues are also analyzed as the conditions are gained directly in terms of physical parameters of the system the effect of different loadings on the system stability is made transparent by dealing with the stability issues directly in matrix second order form

a new method for computing delay margins for stability of linear delay systems this note is concerned with stability properties of linear time invariant delay systems we consider delay systems of retarded type modeled both as a high order scalar differential difference equation and as a set of first order differential difference equations expressed in state space form we provide a computational method that can be used to compute a delay interval such that the system under consideration is stable for all delay values that lie in the computed interval this method requires computing only the eigenvalues and generalized eigenvalues of certain constant matrices and it can be implemented efficiently based on this method we further state a simple necessary and sufficient condition concerning stability independent of delay for each of the two types of the models

bivariate survival models induced by frailties abstract when two observed survival times depend via a proportional hazards model on the same unobserved variable called a frailty this common dependence induces an association between the observed times this article considers the class of bivariate survival distributions that can arise in this way extends it to allow negative association and shows that the observable bivariate distribution determines the unobserved frailty distribution up to a scale parameter a cross ratio function easily estimated even from censored data is the key to both the characterization results and inferential procedures and diagnostic plots that are introduced and illustrated by real examples the models considered are the natural counterpart for survival data of the latent variable models that have long been used in factor analysis for continuous data or in latent structure analysis for binary data the main distinguishing feature of survival data is the possibility of censoring which makes the standard methods difficu

reml estimation for survival models with frailty a method of estimation for generalised mixed models is applied to the estimation of regression parameters in proportional hazards models for failure times when there are repeated observations of failure on each subject the subject effect is incorporated into the model as a random frailty term best linear unbiased predictors are used as an initial step in the computation of maximum likelihood and residual maximum likelihood estimates

semiparametric estimation of random effects using the cox model based on the em algorithm consider a survival experiment where individuals within a certain subset of the population share a common unobservable random frailty such a frailty could be an unobservable genetic or early environmental effect if individuals were in sibling groups or an environmental effect if individuals were grouped by households suppose that if the frailty omega is known the cox proportional hazards model for the observable covariates is valid with the consequence of the random effect being a multiplicative factor on the hazard rate assuming tht the random frailties follow a gamma distribution estimates of the fixed and random effects are obtained by using an em algorithm based on a profile likelihood construction the method developed is applied to the framingham heart study to examine the risks of smoking and cholesterol levels adjusting for potential random effects

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

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

semiparametric bayesian analysis of survival data abstract this review article investigates the potential of bayes methods for the analysis of survival data using semiparametric models based on either the hazard or the intensity function the nonparametric part of every model is assumed to be a realization of a stochastic process the parametric part which may include a regression parameter or a parameter quantifying the heterogeneity of a population is assumed to have a prior distribution with possibly unknown hyperparameters careful applications of some recently popular computational tools including sampling based algorithms are used to find posterior estimates of several quantities of interest even when dealing with complex models and unusual data structures the methodologies developed herein are motivated and aimed at analyzing some common types of survival data from different medical studies here we focus on univariate survival data in the presence of fixed and time dependent covariates multiple event time data for repeated nonfatal events

posterior propriety and computation for the cox regression model with applications to missing covariates full likelihood bayesian approaches in which a gamma process prior is specified for the cumulative baseline hazard we examine characterizations of posterior propriety under completely observed data settings as well as for missing covariates latent variables are introduced to facilitate a straightforward gibbs sampling scheme in the bayesian computation a real dataset is presented to illustrate the proposed methodology

varying coefficient models we explore a class of regression and generalized regression models in which the coefficients are allowed to vary as smooth functions of other variables general algorithms are presented for estimating the models flexibly and some examples are given this class of models ties together generalized additive models and dynamic generalized linear models into one common framework when applied to the proportional hazards model for survival data this approach provides a new way of modelling departures from the proportional hazards assumption

multivariate survival analysis using piecewise gamma frailty in this note we propose a frailty model called piecewise gamma frailty for correlated survival data with random effects having a nested structure in frailty models a dependence function defined as a hazard ratio of one member given the failure time of another member in a unit is determined by the distributional assumptions on frailty in the piecewise gamma frailty model the nested structure of random effects or frailty allows the dependence function to vary over the time periods this model includes existing models such as the piecewise exponential model breslow 1974 biometrics 30 89 100 and the gamma frailty model clayton 1978 biometrika 65 141 151 oakes 1982 journal of the royal statistical society series b 44 414 428 as special cases a study of familial aggregation of epilepsy is used to illustrate the proposed method

bayesian models for multivariate current status data with informative censoring des donnees multivariees d etat consistent en un ensemble d indicatrices associees a la realisation de plusieurs evenements au moment d un examen nous nous interessons aux inferences sur les distributions jointes des temps d evenements les methodes conventionnelles d analyse des donnees de durees multiples ne peuvent etre utilisees parce que tous les temps d evenements sont censures et la censure peut etre informative nous prenons en compte la correlation des temps d evenements chez un sujet au moyen d une variable latente propre au sujet conditionnellement a laquelle les divers temps d evenements sont supposes independants nous supposons egalement que chaque evenement contribue independamment au risque de censure des fonctions en escalier non parametriques sont utilisees pour caracteriser les distributions de base des differents temps d evenements et des temps d examen les covariables et les effets specifiques du sujet sont incorpores grace au modele lineaire generalise un algorithme de monte carlo par chaines de markov est decrit pour l estimation des distributions a posteriori des inconnues les methodes sont illustrees par une application sur des donnees de sites multiples de tumeurs dans une etude de carcinogenicite animale

a bayesian justification of cox s partial likelihood summary in this paper we establish both naive and formal bayesian justifications of cox s 1975 partial likelihood and its various modifications we extend the original work of kalbfieisch 1978 who showed that the partial likelihood is a limiting marginal posterior under noninformative priors for baseline hazards we extend the result to scenarios with timedependent covariates and time varying regression parameters we establish results for continuous time as well as grouped survival data in addition we present a bayesian justification of a modified partial likelihood for handling ties we also present tools for simplification of the gibbs sampling algorithm for implementing partial likelihood based bayesian inference in various practical applications

two level proportional hazards models nous proposons une extension du modele de taux proportionnels a un modele a deux niveaux avec un terme constant aleatoire et des coefficients aleatoires les parametres de ce modele a plusieurs niveaux sont estimes par une combinaison des algorithmes em et de nuwton raphson meme pour des echantillons de 50 groupes la methode fournit pour les coefficients des effets fixes des estimateurs a peu pres sans biais et normalement distribues deux methodes distinctes information observee et information de la vraisemblance des profils seront utilisees pour estimer les erreurs standards ce travail a ete entrepris dans le but de comprendre les determinants des habitudes contraceptive des femmes nepaleses dans l enquete sur les familles de la vallee chitwan axinn barber et ghimire 1997 nous avons utilise ce modele de taux proportionnels a deux niveaux pour etudier la facon dont le niveau d education et les possibilites d acces a l education des enfants varient avec l initiation a un usage permanent de la contraception

quantification of the prentice criteria for surrogate endpoints in a recent discussion on the prentice criteria for surrogate endpoints 1989 and the role of meta analyses in this context alonso et al expand on a number of points among these is a suggestion to base a measure of the degree at which an endpoint meets the prentice criteria on a likelihood reduction factor although new in this context the suggestion has been studied elsewhere and the measure of alonso et al as presented requires some modification if it is to be useful in general situations as it stands it depends upon the censoring mechanism even when independent of the failure mechanism this is however easily fixed the measure suggested by alonso et al can be seen to fit in with a well established theory so that once corrected for the presence of censoring it can be employed with confidence of particular interest is the wide availability of a standard sas program producing a coefficient which can be immediately transformed to provide an estimate of the population quantity behind the alonso et al proposal

a frequency sweeping framework for stability analysis of time delay systems for time delay systems the asymptotic behavior analysis of the critical imaginary roots w r t the infinitely many critical delays is an open problem in order to find a general solution we will exploit the link between the asymptotic behavior of critical imaginary roots and the asymptotic behavior of frequency sweeping curves from a new analytic curve perspective as a consequence we will establish a frequency sweeping framework with three main results 1 a finer regularity singularity classification for time delay systems will be obtained 2 the general invariance property will be proved and hence the asymptotic behavior of the critical imaginary roots w r t the infinitely many critical delays can be adequately studied 3 the complete stability problem can be fully solved moreover the frequency sweeping framework is extended to cover a broader class of time delay systems finally the geometric insights of frequency sweeping curves are investigated consequently some deeper properties on the asymptotic behavior of time delay systems and the link to frequency sweeping curves are found

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

semiparametric bayesian analysis of multiple event time data abstract multiple event time data e g carcinogenic growths in different times and locations multiple attacks of cardiac arrest arise in various medical studies a bayesian analysis of such data based on proportional intensity model of multiple event time data is presented in this paper the bayesian structure is somewhat analogous to that used by kalbfleisch in a proportional hazard model an unobserved random frailty component is used in the proportional intensity model to take care of heterogeneity among the intensity processes in different subjects the monte carlo method of sampling from multivariate distributions the so called gibbs sampler is used to sample from the joint posterior distribution of the unknown parameters the methodology developed here is exemplified with the well known data set on rat tumors of gail santner and brown

selecting factors predictive of heterogeneity in multivariate event time data in multivariate survival analysis investigators are often interested in testing for heterogeneity among clusters both overall and within specific classes we represent different hypotheses about the heterogeneity structure using a sequence of gamma frailty models ranging from a null model with no random effects to a full model having random effects for each class following a bayesian approach we define prior distributions for the frailty variances consisting of mixtures of point masses at zero and inverse gamma densities since frailties with zero variance effectively drop out of the model this prior allocates probability to each model in the sequence including the overall null hypothesis of homogeneity using a counting process formulation the conditional posterior distributions of the frailties and proportional hazards regression coefficients have simple forms posterior computation proceeds via a data augmentation gibbs sampling algorithm a single run of which can be used to obtain model averaged estimates of the population parameters and posterior model probabilities for testing hypotheses about the heterogeneity structure the methods are illustrated using data from a lung cancer trial

nonparametric survival estimation using prognostic longitudinal covariates one of the primary problems facing statisticians who work with survival data is the loss of information that occurs with right censored data this research considers trying to recover some of this endpoint information through the use of a prognostic covariate which is measured on each individual we begin by defining a survival estimate which uses time dependent covariates to more precisely get at the underlying survival curves in the presence of censoring this estimate has a smaller asymptotic variance than the usual kaplan meier in the presence of censoring and reduces to the kaplan meier 1958 journal of the american statistical association 53 457 481 in situations where the covariate is not prognostic or no censoring occurs in addition this estimate remains consistent when the incorporated covariate contains information about the censoring process as well as survival information because the kaplan meier estimate is known to be biased in this situation due to informative censoring we recommend use of our estimate

score tests for homogeneity of regression effect in the proportional hazards model a simple model containing the proportional hazards regression model as a special case is presented the purpose of the model is to provide a framework in which specific alternatives to the proportional hazards assumption may be tested rank invariant score tests for linear quadratic or exponential trends can for instance all be undertaken within this framework in the case of the two sample problem the required calculations are shown to take a particularly simple form special consideration is given to the two sample case in which there is an inversion of the regression effect i e where the hazard functions cross at some given point both of the motivating examples are concerned with this problem computational aspects are relatively straightforward and some discussion on this is provided

the robust inference for the cox proportional hazards model abstract we derive the asymptotic distribution of the maximum partial likelihood estimator for the vector of regression coefficients under a possibly misspecified cox proportional hazards model as in the parametric setting this estimator converges to a well defined constant vector in addition the random vector n 1 2 is asymptotically normal with mean 0 and with a covariance matrix that can be consistently estimated the newly proposed robust covariance matrix estimator is similar to the so called sandwich variance estimators that have been extensively studied for parametric cases for many misspecified cox models the asymptotic limit or part of it can be interpreted meaningfully in those circumstances valid statistical inferences about the corresponding covariate effects can be drawn based on the aforementioned asymptotic theory of and the related results for the score statistics extensive studies demonstrate that the proposed robust tests and interval estimation procedures

analysis of episodic data with application to recurrent pulmonary exacerbations in cystic fibrosis patients we consider a special type of recurrent event data termed recurrent episode data arising in episodic illness studies when an event occurs it lasts for a random length of time a naive recurrent event analysis disregards the length of the episodes which may contain important information about the severity of the disease the associated medical costs and quality of life bivariate gap time models have been suggested in which length of episodes and time between episodes are modeled jointly these models are useful but may obscure the overall effects of treatment and other prognostic factors the analysis can be further complicated if covariate effects change over time as may occur when the effects vary across episodes this article reviews the existing methods applied to recurrent episode data and approaches the problem using the recently developed temporal process regression novel endpoints are constructed that summarize both episode frequency and the length of episodes and time between episodes

difference feedback can stabilize uncertain steady states the paper is concerned with the stabilization of uncertain steady states by the state difference feedback the feedback method has a peculiar feature that it uses only the difference between the present state x t and the past state x t t considering exact information on the steady state is unavailable hitherto a condition is known under which such stabilization can not be realized the article conversely shows that the state difference feedback can stabilize only if the exclusion condition is not true furthermore a dynamic output difference feedback is shown to be able to stabilize under quite a mild condition that the steady state is not associated with zero eigenvalues the ability of the method is illustrated by using a cart pendulum system which moves along a one dimensional varying slope

matrix polynomials similar operators and the imaginary axis eigenvalues of a matrix delay equation we present a new approach to determining the imaginary axis eigenvalues of a matrix delay equation with a full rank delay coefficient matrix the approach requires computation of the generalized eigenvalues of a pair of matrices which are a quarter of the size used in currently known matrix based or operator approaches these matrices have square of n for both row and column length improved from twice those lengths for an equation having matrix n times n coefficients with rank n delay matrix considering problem dimension and computational demands this brings matrix based approaches more in synchrony with the well known scalar substitution approach given the symmetries evident in our approach problems previously done with computation alone can now be considered with insight and sometimes completed with simple calculations some of the computational and mathematical features of the approach are displayed in a section of examples

a note on the uncertainty of a survival probability estimated from cox s regression model a partir d un modele de cox pour des donnees de survie censurees il est possible d estimer la probabilite de survie a l annee t pour un individu de covariables z 0

a comparison of frailty models for multivariate survival data this paper reviews some of the main approaches to the analysis of multivariate censored survival data such data typically have correlated failure times the correlation can be a consequence of the observational design for example with clustered sampling and matching or it can be a focus of interest as in genetic studies longitudinal studies of recurrent events and other studies involving multiple measurements we assume that the correlation between the failure or survival times can be accounted for by fixed or random frailty effects we then compare the performance of conditional and mixture likelihood approaches to estimating models with these frailty effects in censored bivariate survival data we find that the mixture methods are surprisingly robust to misspecification of the frailty distribution the paper also contains an illustrative example on the times to onset of chest pain brought on by three endurance exercise tests during a drug treatment trial of heart patients

the central limit theorem under random censorship let f n be the kaplan meier estimator of a distribution function f computed from randomly censored data we show that under optimal integrability assumptions on a function the kaplan meier integral df n when properly standardized is asymptotically normal

mixed effect poisson log linear models for clinical and epidemiological sleep hypnogram data bayesian poisson log linear multilevel models scalable to epidemiological studies are proposed to investigate population variability in sleep state transition rates hierarchical random effects are used to account for pairings of subjects and repeated measures within those subjects as comparing diseased to non diseased subjects while minimizing bias is of importance essentially non parametric piecewise constant hazards are estimated and smoothed allowing for time varying covariates and segment of the night comparisons the bayesian poisson regression is justified through a re derivation of a classical algebraic likelihood equivalence of poisson regression with a log time offset and survival regression assuming exponentially distributed survival times such re derivation allows synthesis of two methods currently used to analyze sleep transition phenomena stratified multi state proportional hazards models and log linear models with gee for transition counts an example data set from the sleep heart health study is analyzed supplementary material includes the analyzed data set as well as the code for a reproducible analysis

a general framework for random effects survival analysis in the cox proportional hazards setting summary the use of random effects modeling in statistics has increased greatly in recent years the introduction of such modeling into event time analysis has proceeded more slowly however previously random effects models for survival data have either required assumptions regarding the form of the baseline hazard function or restrictions on the classes of models that can be fit in this paper we develop a method of random effect analysis of survival data the hierarchical cox model that is an extension of cox s original formulation in that the baseline hazard function remains unspecified this method also allows an arbitrary distribution for the random effects we accomplish this using markov chain monte carlo methods in a bayesian setting the method is illustrated with three models for a dataset with times to multiple occurrences of mammory tumors for 48 rats treated with a carcinogen and then randomized to either treatment or control this analysis is more satisfying than standard approaches such as studying the first event for each subject which does not fully use the data or assuming independence which in this case would overestimate the precision

predictive capability of proportional hazards regression a measure of the predictive capability of a proportional hazards regression is derived the measure is based on the residuals appropriate to proportional hazards regression a population version is presented and can be seen not to depend on the censoring mechanism under the provision that any such censoring be independent or conditionally independent of the failure mechanism given the covariate for the special case of a weibull regression model for which the covariate distribution follows binary uniform normal or exponential laws we derive analytic results these alone give credence to the measure which can be seen to reflect strength of regression effect as quantified by the parameter estimate although on a scale between 0 and 1 independently of the intercept or shape parameter of the particular weibull law and only weakly dependent on the covariate distribution extensions to partial and multiple measures of predictive ability are straightforward an example is provided

a measure of explained variation for event history data there is no shortage of proposed measures of prognostic value of survival models in the statistical literature they come under different names including explained variation correlation explained randomness and information gain but their goal is common to define something analogous to the coefficient of determination r 2 in linear regression none however have been uniformly accepted none have been extended to general event history data including recurrent events and many cannot incorporate time varying effects or covariates we present here a measure specifically tailored for use with general dynamic event history regression models the measure is applicable and interpretable in discrete or continuous time with tied data or otherwise with time varying time fixed or dynamic covariates with time varying or time constant effects with single or multiple event times with parametric or semiparametric models and under general independent censoring observation for single event survival data with neither censoring nor time dependency it reduces to the concordance index we give expressions for its population value and the variance of the estimator and explore its use in simulations and applications a web link to r software is provided

confidence bands for survival curves under the proportional hazards model summary in this paper we show how to construct simultaneous confidence bands for the subjectspecific survival curve under the cox proportional hazards model the idea is to approximate the distribution of the normalized cumulative hazard estimator by a zero mean gaussian process whose distribution can be easily generated through simulation numerical studies indicate that the proposed bands are appropriate for practical use a liver disease example is presented

imaginary characteristic roots of neutral systems with commensurate delays abstract this paper concentrates on stability analysis of neutral linear time invariant lti delay systems with multiple commensurate time delays a new numerical procedure is offered for the determination of purely imaginary characteristic roots of neutral delay systems which plays a crucial role in assessing the system stability based on simple linear algebra it is shown that the imaginary characteristic roots of such a system can be found by calculating the generalized eigenvalues of an associated matrix pair if the system is of retarded type we just need to determine the eigenvalues of a single matrix the results extend previously known work on neutral delay systems subject to a single delay for neutral systems subject to multiple commensurate delays in the light of the main result of this paper we present a new method to determine the largest first time delay interval for which a neutral delay system preserves its stability the paper is closed by showing numerical examples that illustrate the applicability and effectiveness of the proposed method

a multilevel excess hazard model to estimate net survival on hierarchical data allowing for non linear and non proportional effects of covariates the excess hazard regression model is an approach developed for the analysis of cancer registry data to estimate net survival that is the survival of cancer patients that would be observed if cancer was the only cause of death cancer registry data typically possess a hierarchical structure individuals from the same geographical unit share common characteristics such as proximity to a large hospital that may influence access to and quality of health care so that their survival times might be correlated as a consequence correct statistical inference regarding the estimation of net survival and the effect of covariates should take this hierarchical structure into account it becomes particularly important as many studies in cancer epidemiology aim at studying the effect on the excess mortality hazard of variables such as deprivation indexes often available only at the ecological level rather than at the individual level we developed here an approach to fit a flexible excess hazard model including a random effect to describe the unobserved heterogeneity existing between different clusters of individuals and with the possibility to estimate non linear and time dependent effects of covariates we demonstrated the overall good performance of the proposed approach in a simulation study that assessed the impact on parameter estimates of the number of clusters their size and their level of unbalance we then used this multilevel model to describe the effect of a deprivation index defined at the geographical level on the excess mortality hazard of patients diagnosed with cancer of the oral cavity copyright 2016 john wiley sons ltd

revisiting simple linear regression with autocorrelated errors this paper studies properties of ordinary and generalised least squares estimators in a simple linear regression with stationary autocorrelated errors explicit expressions for the variances of the regression parameter estimators are derived for some common time series autocorrelation structures including a first order autoregression and general moving averages applications of the results include confidence intervals and an example where the variance of the trend slope estimator does not increase with increasing autocorrelation

a class of bayesian shared gamma frailty models with multivariate failure time data nous proposons ici dans le cadre des donnees de survie une nouvelle classe de modeles a fragilite gamma partagee en appliquant la transformation de box cox a la fonction de hasard ainsi qu au produit du risque de base et de la fragilite cette nouvelle classe de modeles offre un large eventail de formes et relations entre la fonction de hasard et le risque de base et inclut en tant que cas particuliers le modele de cox usuel a fragilite gamma ainsi qu un nouveau modele additif a fragilite gamma de par la contrainte que le risque soit non negatif ce modele a fragilite gamma partagee presente un defi calculatoire dans le paradigme bayesien les distributions conjointes a priori sont definies par une distribution conditionnelle marginale ou la distribution conditionnelle est univariee cela permet d integrer les contraintes non lineaires sur les parametres la partie marginale de la distribution a priori quant a elle est libre de toute contrainte les distributions a priori permettent de programmer facilement l integralite des distributions conditionnelles requises pour l echantillonnage de gibbs tout en integrant les contraintes un jeu reel de donnees permet d illustrer cette classe de modeles a fragilite gamma partagee

three measures of explained variation for correlated survival data under the proportional hazards mixed effects model measures of explained variation are useful in scientific research as they quantify the amount of variation in an outcome variable of interest that is explained by one or more other variables we develop such measures for correlated survival data under the proportional hazards mixed effects model because different approaches have been studied in the literature outside the classical linear regression model we investigate three measures r 2 rres2 and 2 that quantify three different population coefficients we show that although the three population measures are not the same they reflect similar amounts of variation explained by the predictors among the three measures we show that r 2 which is the simplest to compute is also consistent for the first population measure under the usual asymptotic scenario when the number of clusters tends to infinity the other two measures on the other hand all require that in addition the cluster sizes be large we study the properties of the measures both analytically and through simulation studies we illustrate their different usage on a multi center clinical trial and a recurrent events data set copyright 2016 john wiley sons ltd

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

stability robustness of retarded lti systems with single delay and exhaustive determination of their imaginary spectra in this paper we consider the stability robustness of the general class of vector lti linear time invariant equations with a single delay dot bf x t bf a bf x t bf b bf x t tau bf x in bf r n the robustness is against the uncertain but constant delay tau in bf r we first present a set of novel propositions and state that the solution must start from the complete knowledge of imaginary spectra of the system and the corresponding delays the propositions claim that such spectra form a set of manageably small number of members and this number is upper bounded by n 2 regardless of the composition of bf a and bf b matrices they also claim that the infinite dimensional system at hand has an outstanding discipline regarding these imaginary spectra this discipline invites the recently developed concept called the cluster treatment of characteristic roots ctcr the ctcr procedure requires a complete and precise determination of the imaginary spectra of the system there are many procedures in the literature to achieve this they are in fact some variations of the five main methods of different levels of precision and complexity there is however no study known to the authors for presenting a comparison among these methods this paper addresses this need we first offer an overview of each of the five methods and then compare their numerical performances over an example case study

conditional akaike information under generalized linear and proportional hazards mixed models we study model selection for clustered data when the focus is on cluster specific inference such data are often modelled using random effects and conditional akaike information was proposed in vaida blanchard 2005 and used to derive an information criterion under linear mixed models here we extend the approach to generalized linear and proportional hazards mixed models outside the normal linear mixed models exact calculations are not available and we resort to asymptotic approximations in the presence of nuisance parameters a profile conditional akaike information is proposed bootstrap methods are considered for their potential advantage in finite samples simulations show that the performance of the bootstrap and the analytic criteria are comparable with bootstrap demonstrating some advantages for larger cluster sizes the proposed criteria are applied to two cancer datasets to select models when the cluster specific inference is of interest

the problem of a covariate time qualitative interaction in a survival study we introduce a test for the equality of two survival distributions against the specific alternative of crossing hazards although this kind of alternative is somewhat rare designing a test specifically aimed at detecting such departures from the null hypothesis in this direction leads to powerful procedures upon which we can call in those few cases where such departures are suspected furthermore the proposed test and an approximate version of the test are seen to suffer only moderate losses in power when compared with their optimal counterparts should the alternative be one of proportional hazards our interest in the problem is motivated by clinical studies on the role of acute graft versus host disease as a risk factor in leukemic children and we discuss the analysis of this study in detail the model we use in this work is a special case of the one introduced by anderson and senthilselvan 1982 applied statistics 31 44 51 we propose overcoming an inferential problem stemming from their model by using the methods of davies 1977 biometrika 64 247 254 1987 biometrika 74 33 43 backed up by resampling techniques we also look at an approach relying directly on resampling techniques the distributional aspects of this approach under the null hypothesis are interesting but practically its behaviour is such that its use cannot be generally recommended outlines of the necessary asymptotic theory are presented and for this we use the tools of martingale theory

time dependent effects of fixed covariates in cox regression a nonparametric modification is proposed for cox s proportional hazards model cox 1972 journal of the royal statistical society series b 34 187 220 where the covariates are fixed but their effects are allowed to vary in time parameters are introduced for the covariate effects at the uncensored survival times estimates are obtained by maximizing the penalized partial log likelihood that arises if a penalty function of the first order differences of consecutive parameters is subtracted from the partial log likelihood the choice of the smoothing parameter which determines the weight of the penalty is based on akaike s information criterion the methods are illustrated in a set of ovarian cancer data and in a set of kidney transplantation data

imaginary axis eigenvalues of matrix delay equations with a certain alternating coefficient structure abstract in this paper we examine the imaginary axis eigenvalues of matrix delay differential equations with coefficients alternating between hermitian and skew hermitian such as occurring in many mechanical systems we show that given this special structure and a certain sign condition the dimension of the eigenvalue problem is greatly reduced we also give some theorems on eigenvalue crossing directions in this context and show how reduction of dimension and crossing theorems work to provide insight on mechanical systems with time delay

a semiparametric recurrent events model with time varying coefficients we consider a recurrent events model with time varying coefficients motivated by two clinical applications a random effects gaussian frailty model is used to describe the intensity of recurrent events the model can accommodate both time varying and time constant coefficients the penalized spline method is used to estimate the time varying coefficients laplace approximation is used to evaluate the penalized likelihood without a closed form the smoothing parameters are estimated in a similar way to variance components we conduct simulations to evaluate the performance of the estimates for both time varying and time independent coefficients we apply this method to analyze two data sets a stroke study and a child wheeze study

hierarchical proportional hazards regression models for highly stratified data summary in clinical trials conducted over several data collection centers the most common statistically defensible analytic method a stratified cox model analysis suffers from two important defects first identification of units that are outlying with respect to the baseline hazard is awkward since this hazard is implicit rather than explicit in the cox partial likelihood second and more seriously identification of modest treatment effects is often difficult since the model fails to acknowledge any similarity across the strata we consider a number of hierarchical modeling approaches that preserve the integrity of the stratified design while offering a middle ground between traditional stratified and unstratified analyses we investigate both fully parametric weibull and semiparametric models the latter based not on the cox model but on an extension of an idea by gelfand and mallick 1995 biometrics51 843 852 which models the integrated baseline hazard as a mixture of monotone functions we illustrate the methods using data from a recent multicenter aids clinical trial comparing their ease of use interpretation and degree of robustness with respect to estimates of both the unit specific baseline hazards and the treatment effect

testing goodness of fit for proportional hazards model with censored observations abstract a numerical omnibus test of fit for the two sample proportional hazards model is proposed for randomly censored observations the test is derived from cox s partial likelihood and does not need a dummy time dependent covariate or any partition of the time axis consistency of the test is established examples are provided for illustration

bayesian semiparametric dynamic frailty models for multiple event time data many biomedical studies collect data on times of occurrence for a health event that can occur repeatedly such as infection hospitalization recurrence of disease or tumor onset to analyze such data it is necessary to account for within subject dependency in the multiple event times motivated by data from studies of palpable tumors this article proposes a dynamic frailty model and bayesian semiparametric approach to inference the widely used shared frailty proportional hazards model is generalized to allow subject specific frailties to change dynamically with age while also accommodating nonproportional hazards parametric assumptions on the frailty distribution are avoided by using dirichlet process priors for a shared frailty and for multiplicative innovations on this frailty by centering the semiparametric model on a conditionally conjugate dynamic gamma model we facilitate posterior computation and lack of fit assessments of the parametric model our proposed method is demonstrated using data from a cancer chemoprevention study

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

proportional hazards model with random effects we propose a general proportional hazards model with random effects for handling clustered survival data this generalizes the usual frailty model by allowing a multivariate random effect with arbitrary design matrix in the log relative risk in a way similar to the modelling of random effects in linear generalized linear and non linear mixed models the distribution of the random effects is generally assumed to be multivariate normal but other preferably symmetrical distributions are also possible maximum likelihood estimates of the regression parameters the variance components and the baseline hazard function are obtained via the em algorithm the e step of the algorithm involves computation of the conditional expectations of functions of the random effects for which we use markov chain monte carlo mcmc methods approximate variances of the estimates are computed by louis formula and posterior expectations and variances of the individual random effects can be obtained as a by product of the estimation the inference procedure is exemplified on two data sets

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

bias of some commonly used time series estimates summary we study the bias of yule walker and least squares estimates for univariate and multivariate autoregressive processes we obtain explicit formulae for the large sample bias of yule walker estimates in the scalar first and second order cases and for least squares estimates in the general case both simulations and theory indicate that yulewalker estimates are inferior to least squares estimates for strongly autocorrelated processes yule walker estimates can be severely biased even for comparatively largesample sizes