semiparametric inference in a model for association in bivanate survival data summary clayton s 1978 model for association in bivariate survival data is both of intrinsic importance and an interesting example of a semiparametric estimation problem that is a problem where inference about a parameter is required in the presence of nuisance functions we derive the asymptotic variance of clayton s estimator obtaining a simple explicit formula for uncensored data and indicating the modification required when the survival times are subject to arbitrary random censorship some comparisons are made with results derived by oakes 1982 for other estimators within this model in the absence of censoring the exact null variance of the score statistic corresponding to clayton s estimator is derived and compared with that of the locally most powerful rank test given by cuzick 1982

shared frailty models for recurrent events and a terminal event there has been an increasing interest in the analysis of recurrent event data cook and lawless 2002 statistical methods in medical research 11 141 166 in many situations a terminating event such as death can happen during the follow up period to preclude further occurrence of the recurrent events furthermore the death time may be dependent on the recurrent event history in this article we consider frailty proportional hazards models for the recurrent and terminal event processes the dependence is modeled by conditioning on a shared frailty that is included in both hazard functions covariate effects can be taken into account in the model as well maximum likelihood estimation and inference are carried out through a monte carlo em algorithm with metropolis hastings sampler in the e step an analysis of hospitalization and death data for waitlisted dialysis patients is presented to illustrate the proposed methods methods to check the validity of the proposed model are also demonstrated this model avoids the difficulties encountered in alternative approaches which attempt to specify a dependent joint distribution with marginal proportional hazards and yields an estimate of the degree of dependence

checking the marginal cox model for correlated failure time data summary correlated failure time data arise frequently in scientific investigations because there exists natural or artificial clustering of study subjects such that failure times within the same cluster are correlated it is convenient and useful to perform regression analysis by formulating the marginal distributions of the correlated failure times with the cox proportional hazards model in this paper we develop a class of graphical and numerical techniques for checking the adequacy of the marginal cox model the proposed methods are derived from cumulative sums of martingale based residuals over the failure time and or covariates the distributions of these stochastic processes under the assumed model can be approximated through simulating certain zero mean gaussian processes each observed residual pattern can then be compared objectively with a number of realisations from the approximating process an illustration with the diabetic retinopathy study is provided

a semiparametric estimation procedure of dependence parameters in multivariate families of distributions summary this paper investigates the properties of a semiparametric method for estimating the dependence parameters in a family of multivariate distributions the proposed estimator obtained as a solution of a pseudo likelihood equation is shown to be consistent asymptotically normal and fully efficient at independence a natural estimator of its asymptotic variance is proved to be consistent comparisons are made with alternative semiparametric estimators in the special case of clayton s model for association in bivariate data

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

pairwise dependence diagnostics for clustered failure time data frailty and copula models specify a parametric dependence structure for multivariate failure time data estimation of some joint quantities can be highly sensitive to the assumed parametric form and hence model fit is an important issue this paper lays out a general diagnostic framework for evaluating and selecting frailty and copula models the approach is based on the cumulative sum of residuals that are calculated in bivariate time the residuals reflect the difference between the observed and expected bivariate association structures the proposed model checking process is interpretable with a limiting distribution which can be approximated using the bootstrap simulations and a data example illustrate the practical application of the method

on the local behavior of spaces of natural images in this study we concentrate on qualitative topological analysis of the local behavior of the space of natural images to this end we use a space of 3 by 3 high contrast patches we develop a theoretical model for the high density 2 dimensional submanifold of showing that it has the topology of the klein bottle using our topological software package plex we experimentally verify our theoretical conclusions we use polynomial representation to give coordinatization to various subspaces of we find the best fitting embedding of the klein bottle into the ambient space of our results are currently being used in developing a compression algorithm based on a klein bottle dictionary

copula inference under censoring this paper discusses copula model selection procedures and goodness of fit tests under censoring the proposed methodology is based on a comparison of nonparametric and model based estimators of the probability integral transformation k new weighted estimators for k are introduced the resulting tests are compared to an existing approach by simulation and illustrated with an example involving bleeding changes in a woman s reproductive history

inferences on the association parameter in copula models for bivariate survival data we investigate two stage parametric and two stage semi parametric estimation procedures for the association parameter in copula models for bivariate survival data where censoring in either or both components is allowed we derive asymptotic properties of the estimators and compare their performance by simulations both parametric and semi parametric estimators of the association parameter are efficient at independence and the parameter estimates in the margins have high efficiency and are robust to misspecification of dependency structures in addition we propose a consistent variance estimator for the semi parametric estimator of the association parameter we apply the proposed methods to an aids data set for illustration

a class of multivanate failure time distributions summary a class of continuous multivariate lifetime distributions is proposed the dependence between individuals in a group is modelled by a group specific quantity which can be interpreted as an unobserved covariate common to the individuals in the group and assumed to follow a positive stable distribution it is possible to include covariates in the model and discuss whether the dependence is still present after specific covariates are taken into account if the conditional hazards are proportional then the hazards in the marginal distributions are also proportional but with different constants of proportionality also the hazard for the minimum in a group is proportional to the marginal hazards if the conditional distributions given the group quantity are weibull then the marginal distributions are also weibull this class can be used to test the hypothesis of independence of litter mates in the proportional hazards model

on assessing the strength of dependency between failure time variates summary the correlation p between cumulative hazard variates is considered as a measure of dependence between pairs of failure time variates such correlation is readily estimated in the absence of censorship but modelling assumptions are required for its estimation in the presence of independent right censorship a semiparametric class of bivariate failure time models in which a dependence parameter 0 is in one to one correspondence with the cumulative hazard variate correlation p is considered for this purpose a simple estimating function consisting of a weighted sum over the grid formed by the observed failure times of differences between nonparametric and model restricted estimates of covariance rates is proposed for estimation and corresponding asymptotic distribution theory is given this estimation procedure is shown to lack robustness under conditions of moderate to heavy censorship motivating consideration of dependency measures that can be defined on the support of the observed failure times two such measures the correlation between marginal martingales and a cumulative cross ratio statistic are discussed and corresponding nonparametric estimators are proposed

checking the cox model with cumulative sums of martingale based residuals summary this paper presents a new class of graphical and numerical methods for checking the adequacy of the cox regression model the procedures are derived from cumulative sums of martingale based residuals over follow up time and or covariate values the distributions of these stochastic processes under the assumed model can be approximated by zero mean gaussian processes each observed process can then be compared both visually and analytically with a number of simulated realizations from the approximate null distribution these comparisons enable the data analyst to assess objectively how unusual the observed residual patterns are special attention is given to checking the functional form of a covariate the form of the link function and the validity of the proportional hazards assumption an omnibus test consistent against any model misspecification is also studied the proposed techniques are illustrated with two real data sets

clique topology reveals intrinsic geometric structure in neural correlations detecting meaningful structure in neural activity and connectivity data is challenging in the presence of hidden nonlinearities where traditional eigenvalue based methods may be misleading we introduce a novel approach to matrix analysis called clique topology that extracts features of the data invariant under nonlinear monotone transformations these features can be used to detect both random and geometric structure and depend only on the relative ordering of matrix entries we then analyzed the activity of pyramidal neurons in rat hippocampus recorded while the animal was exploring a 2d environment and confirmed that our method is able to detect geometric organization using only the intrinsic pattern of neural correlations remarkably we found similar results during nonspatial behaviors such as wheel running and rapid eye movement rem sleep this suggests that the geometric structure of correlations is shaped by the underlying hippocampal circuits and is not merely a consequence of position coding we propose that clique topology is a powerful new tool for matrix analysis in biological settings where the relationship of observed quantities to more meaningful variables is often nonlinear and unknown

non parametric estimation and model checking procedures for marginal gap time distributions for recurrent events for recurrent events there is evidence that misspecification of the frailty distribution can cause severe bias in estimated regression coefficients am j epidemiol 1998 149 404 411 statist med 2006 25 1672 1684 in this paper we adapt a procedure originally suggested in biometrika 1999 86 381 393 for parallel data for checking the gamma frailty to recurrent events to apply the model checking procedure a consistent non parametric estimator for the marginal gap time distributions is needed this is in general not possible due to induced dependent censoring in the recurrent events setting however in biometrika 1999 86 59 70 a non parametric estimator for the joint gap time distributions based on the principle of inverse probability of censoring weights is suggested here we attempt to apply this estimator in the model checking procedure and the performance of the method is investigated with simulations and applied to danish registry data the method is further investigated using the usual kaplan meier estimator and a marginalized estimator for the marginal gap time distributions we conclude that the procedure only works when the recurrent event is common and when the intra individual association between gap times is weak

an estimating equation for parametric shared frailty models with marginal additive hazards summary multivariate failure time data arise when data consist of clusters in which the failure times may be dependent a popular approach to such data is the marginal proportional hazards model with estimation under the working independence assumption in some contexts however it may be more reasonable to use the marginal additive hazards model we derive asymptotic properties of the lin and ying estimators for the marginal additive hazards model for multivariate failure time data furthermore we suggest estimating equations for the regression parameters and association parameters in parametric shared frailty models with marginal additive hazards by using the lin and ying estimators we give the large sample properties of the estimators arising from these estimating equations and investigate their small sample properties by monte carlo simulation a real example is provided for illustration

a simple estimator for a shared frailty regression model summary we propose a simple estimation procedure for a proportional hazards frailty regression model for clustered survival data in which the dependence is generated by a positive stable distribution inferences for the frailty parameter can be obtained by using output from cox regression analyses the computational burden is substantially less than that of the other approaches to estimation the large sample behaviour of the estimator is studied and simulations show that the approximations are appropriate for use with realistic sample sizes the methods are motivated by studies of familial associations in the natural history of diseases their practical utility is illustrated with sib pair data from beaver dam wisconsin

asymptotic theory for the frailty model the frailty model is a generalization of cox s proportional hazards model which includes a random effect nielsen gill andersen and sorensen 1992 proposed an em algorithm to estimate the cumulative baseline hazard and the variance of the random effect here the asymptotic distribution of the estimators is given along with a consistent estimator of the asymptotic variance

on robustness of marginal regression coefficient estimates and hazard functions in multivariate survival analysis of family data when the frailty distribution is mis specified the shared frailty model is an extension of the cox model to correlated failure times and essentially a random effects model for failure time outcomes in this model the latent frailty shared by individual members in a cluster acts multiplicatively as a factor on the hazard function and is typically modelled parametrically one commonly used distribution is gamma where both shape and scale parameters are set to be the same to allow for unique identification of baseline hazard function it is popular because it is a conjugate prior and the posterior distribution possesses the same form as gamma in addition the parameter can be interpreted as a time independent cross ratio function a natural extension of odds ratio to failure time outcomes in this paper we study the effect of frailty distribution mis specification on the marginal regression estimates and hazard functions under assumed gamma distribution with an application to family studies the simulation results show that the biases are generally 10 and lower even when the true frailty distribution deviates substantially from the assumed gamma distribution this suggests that the gamma frailty model can be a practical choice in real data analyses if the regression parameters and marginal hazard function are of primary interest and individual cluster members are exchangeable with respect to their dependencies

pseudo invexity and duality in nonlinear programming abstract the purpose of this paper is to study various duality results in nonlinear programming for pseudo invex functions such results were known in the literature for invex functions

information topological characterization of periodically correlated processes by dilation operators giving process information through spectral considerations has been tackled for decades we propose in this paper a new way of dealing with such an objective by giving hidden information topology of the spectral measure of non stationary and periodically correlated processes we used first the kolmogorov decomposition which is a natural extension of the naimark operator theory to obtain a sequence of rotation matrices called the dilation matrices these matrices carry all the spectral information of the process and belong to so n or su n with respect to respectively the real or complex nature of the periodically correlated processes studied in order to give a topological interpretation of the positioning of these matrices on the space of rotation matrices we have applied a persistent homology technique and next exposed fundamental attributes we showed that different types of periodically correlated processes are endowed with a point cloud structure that can be easily discriminated by topological and information features

consistency in a proportional hazards model incorporating a random effect supported in part by the north atlantic treaty organization under a grant awarded in 1990 conducted at the mathematical institute of the university of utrecht the netherlands

a frailty model for informative censoring to account for the correlation between failure and censoring we propose a new frailty model for clustered data in this model the risk to be censored is affected by the risk of failure this model allows flexibility in the direction and degree of dependence between failure and censoring it includes the traditional frailty model as a special case it allows censoring by some causes to be analyzed as informative while treating censoring by other causes as noninformative it can also analyze data for competing risks to fit the model the em algorithm is used with markov chain monte carlo simulations in the e steps simulation studies and analysis of data for kidney disease patients are provided consequences of incorrectly assuming noninformative censoring are investigated

a covariate specific time dependent receiver operating characteristic curve for correlated survival data several studies for the clinical validity of circulating tumor cells ctcs in metastatic breast cancer were conducted showing that it is a prognostic biomarker of overall survival in this work we consider an individual patient data meta analysis for nonmetastatic breast cancer to assess the discrimination of ctcs regarding the risk of death data are collected in several centers and present correlated failure times for subjects of the same center however although the covariate specific time dependent receiver operating characteristic roc curve has been widely used for assessing the performance of a biomarker there is no methodology yet that can handle this specific setting with clustered censored failure times we propose an estimator for the covariate specific time dependent roc curves and area under the roc curve when clustered failure times are detected we discuss the assumptions under which the estimators are consistent and their interpretations we assume a shared frailty model for modeling the effect of the covariates and the biomarker on the outcome in order to account for the cluster effect a simulation study was conducted and it shows negligible bias for the proposed estimator and a nonparametric one based on inverse probability censoring weighting while a semiparametric estimator ignoring the clustering is markedly biased finally in our application to breast cancer data the estimation of the covariate specific area under the curves illustrates that the ctcs discriminate better patients with inflammatory tumor than patients with noninflammatory tumor with respect to their risk of death

a model for association in bivariate survival data summary a reparameterization of a model introduced by d g clayton for association in bivariate life tables is discussed inference for the parameter governing the association is con sidered when the marginal distributions are specified up to lehmann alternatives the information matrix is derived explicitly and it is shown that the parameterization is moderately successful in introducing orthogonality between the association parameter and the two scale parameters the likelihood proposed by clayton for the case that the marginal distributions are completely unknown is criticized an alternative non parametric estimator based on kendall s coefficient of concordance is proposed and its asymptotic variance evaluated if g s pr s s and h t pr t t are continuous univariate survivor functions with g o h o 1 and 0 1 then the function f s t pr s s t t given by f s t g h 1 1 is a bivariate survivor function with g and h as marginals this paper considers inference about the parameter 0 governing the association between s and t given a sample of size n from their bivariate distribution we suppose first that the marginals g s and h t are known up to lehmann alternatives and secondly that they are completely unknown the model 1 is a reparameterization of a model introduced by clayton 1978 for association in bivariate life tables section 2 describes some properties of the family including the limits as 0 1 and 0 oo a simple interpretation of 0 in terms of hazard functions and a random effects derivation of the model this section leans heavily on clayton s work section 3 discusses parametric inference for 0 including tests of the hypothesis 0 1 of independence between s and t we assume that the marginal survivor functions g s and h t are specified up to lehmann alternatives asymptotic variance formulae are derived in section 4 which also explores the extent to which the parameterization achieves orthogonality clayton 1978 proposed an ingenious method for estimating 0 in the absence of knowledge of the marginal distributions unfortunately there are difficulties with his proposed con ditional likelihood function in particular the estimated variance from the second derivative appears to be incorrect clayton s derivation is described and criticized in our section 5 in principle rank invariance properties of 1 when g s and h t are completely unknown yield a marginal likelihood for 0 although this likelihood appears to be intractable in general in section 6 it is evaluated when n 3 and shown to differ from clayton s likelihood a simpler non parametric estimator of 0 based on the coefficient of concordance is proposed in section

cumulative incidence association models for bivariate competing risks data association models like frailty and copula models are frequently used to analyze clustered survival data and evaluate within cluster associations the assumption of noninformative censoring is commonly applied to these models though it may not be true in many situations in this paper we consider bivariate competing risk data and focus on association models specified for the bivariate cumulative incidence function cif a nonparametrically identifiable quantity copula models are proposed which relate the bivariate cif to its corresponding univariate cifs similarly to independently right censored data and accommodate frailty models for the bivariate cif two estimating equations are developed to estimate the association parameter permitting the univariate cifs to be estimated either parametrically or nonparametrically goodness of fit tests are presented for formally evaluating the parametric models both estimators perform well with moderate sample sizes in simulation studies the practical use of the methodology is illustrated in an analysis of dementia associations

biometrika centenary survival analysis although papers relating to survival analysis have appeared in biometrika since the very first issue the current emphasis on the topic in this journal is fairly new especially as it relates to applications in biostatistics this survey traces the development of survival analysis throughout the twentieth century as reflected in the pages of biometrika focusing primarily on work published since 1980

bootstrap based methods for estimating standard errors in cox s regression analyses of clustered event times we propose two bootstrap based methods to correct the standard errors ses from cox s model for within cluster correlation of right censored event times the cluster bootstrap method resamples with replacement only the clusters whereas the two step bootstrap method resamples i the clusters and ii individuals within each selected cluster with replacement in simulations we evaluate both methods and compare them with the existing robust variance estimator and the shared gamma frailty model which are available in statistical software packages we simulate clustered event time data with latent cluster level random effects which are ignored in the conventional cox s model for cluster level covariates both proposed bootstrap methods yield accurate ses and type i error rates and acceptable coverage rates regardless of the true random effects distribution and avoid serious variance under estimation by conventional cox based standard errors however the two step bootstrap method over estimates the variance for individual level covariates we also apply the proposed bootstrap methods to obtain confidence bands around flexible estimates of time dependent effects in a real life analysis of cluster event times

case control survival analysis with a general semiparametric shared frailty model a pseudo full likelihood approach in this work we deal with correlated failure time age at onset data arising from population based case control studies where case and control probands are selected by population based sampling and an array of risk factor measures is collected for both cases and controls and their relatives parameters of interest are effects of risk factors on the failure time hazard function and within family dependencies among failure times after adjusting for the risk factors due to the retrospective sampling scheme large sample theory for existing methods has not been established we develop a novel technique for estimating the parameters of interest under a general semiparametric shared frailty model we also present a simple easily computed and non iterative nonparametric estimator for the cumulative baseline hazard function we provide rigorous large sample theory for the proposed method we also present simulation results and a real data example for illustrating the utility of the proposed method

martingale based residuals for survival models summary graphical methods based on the analysis of residuals are considered for the setting of the highly used cox 1972 regression model and for the andersen gill 1982 generalization of that model we start with a class of martingale based residuals as proposed by barlow prentice 1988 these residuals and or their transforms are useful for investigating the functional form of a covariate the proportional hazards assumption the leverage of each subject upon the estimates of 13 and the lack of model fit to a given subject 1 1 model consider a set of n independent subjects such that the counting process ni ni t t o for the ith subject in the set indicates the number of observed events experienced over time t the sample paths of the ni are step functions with jumps of size 1 and with ni 0 0 we assume that the intensity function for ni t is given by yi t da t zi t yi t ep z t dao t 1 where yi t is a 0 1 process indicating whether the ith subject is a risk at time t 13 is a vector of regression coefficients zi t is a p dimensional vector of covariate processes and ao is the baseline cumulative hazard function several familar survival models fit into this framework the andersen gill 1982 generalization of the cox 1972 model arises when ao t is completely unspecified the further restriction that yi t 1 until the first event or censoring and is 0 thereafter yields the cox model the parametric form ao t t yields a poisson model or an exponential if restricted to a single event per subject and ao t tp a weibull model our attention will focus primarily on the andersen gill and cox models however the

analysis of failure time data with multilevel clustering with application to the child vitamin a intervention trial in nepal we consider the problem of estimating covariate effects in the marginal cox proportional hazard model and multilevel associations for child mortality data collected from a vitamin a supplementation trial in nepal where the data are clustered within households and villages for this purpose a class of multivariate survival models that can be represented by a functional of marginal survival functions and accounts for hierarchical structure of clustering is exploited based on this class of models an estimation strategy involving a within cluster resampling procedure is proposed and a model assessment approach is presented the asymptotic theory for the proposed estimators and lack of fit test is established the simulation study shows that the estimates are approximately unbiased and the proposed test statistic is conservative under extremely heavy censoring but approaches the size otherwise the analysis of the nepal study data shows that the association of mortality is much greater within households than within villages

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

marginal regression models for multivariate failure time data abstract in this article we propose a general cox type regression model to formulate the marginal distributions of multivariate failure time data this model has a nested structure in that it allows different baseline hazard functions among distinct failure types and imposes a common baseline hazard function on the failure times of the same type we prove that the maximum quasi partial likelihood estimator for the vector of regression parameters under the independence working assumption is consistent and asymptotically normal with a covariance matrix for which a consistent estimator is provided furthermore we establish the uniform consistency and joint weak convergence of the aalen breslow type estimators for the cumulative baseline hazard functions and develop a resampling technique to approximate the joint distribution of these processes which enables one to make simultaneous inference about the survival functions over the time axis and across failure types finally we assess the small sample pro

checking the adequacy of the gamma frailty model for multivariate failure times summary multivariate failure time data arise when the sample consists of clusters and each cluster contains several dependent failure times the semiparametric gamma frailty model vaupel manton stallard 1979 clayton 1978 oakes 1982 for multivariate failure times characterises the intracluster dependence by the gamma frailty distribution while allowing the marginal distributions to be unspecified this paper develops both graphical and numerical techniques for checking the adequacy of this model the proposed techniques are based on the posterior expectation of the frailty given the observable data two examples from genetics are provided

cox regression analysis of multivariate failure time data the marginal approach multivariate failure time data are commonly encountered in scientific investigations because each study subject may experience multiple events or because there exists clustering of subjects such that failure times within the same cluster are correlated in this paper i present a general methodology for analysing such data which is analogous to that of liang and zeger for longitudinal data analysis this approach formulates the marginal distributions of multivariate failure times with the familiar cox proportional hazards models while leaving the nature of dependence among related failure times completely unspecified the baseline hazard functions for the marginal models may be identical or different simple estimating equations for the regression parameters are developed which yield consistent and asymptotically normal estimators and robust variance covarinace estimators are constructed to account for the intra class correlation simulation results demonstrate that the large sample approximations are adequate for practical use and that ignoring the intra class correlation could yield rather misleading variance estimators the proposed methodology has been fully implemented in a simple computer program which also incorporates several alternative approaches detailed illustrations with data from four clinical or epidemiologic studies are provided