survival analysis using a scale change random effects model abstract frailty models are effective in broadening the class of survival models and inducing dependence in multivariate survival distributions in proportional hazards the random effect multiplies the hazard function the scale change model incorporates unobserved heterogeneity through a random effect that enters the baseline hazard function to change the time scale we interpret this random effect as frailty or other unobserved risks that create heterogeneity in the population this model produces a wide range of shapes for univariate survival and hazard functions we extend this model to multivariate survival data by assuming that members of a group share a common random effect this structure induces association among the survival times in a group and provides alternative association structures to the proportional hazards frailty model we present parametric and semiparametric estimation techniques and illustrate these methods with an example

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

the foundations of factor analysis summary a new approach to factor analysis and related latent variable methods is proposed which is based on data reduction using the idea of bayesian sufficiency considerations of symmetry invariance and independence are used to determine an appropriate family of models the results are expressed in terms of linear functions of the manifest variables after the manner of principal components analysis the approach justifies some of the practices based on the normal theory factor model and lays a foundation for the treatment of nonnormal including categorical variables 1 background factor analysis is a widely used statistical technique but its theoretical foundations are somewhat obscure and subject to dispute it is one of a family of multivariate methods which also includes latent structure and latent trait analysis the common feature of the models underlying these methods is that the observed random variables are assumed to depend on latent that is unobserved random variables there is sometimes debate about whether these latent variables are real in any sense but they can be viewed simply as constructs designed to simplify and summarize the complex web of interrelated variables with which nature confronts us by expressing these relationships in terms of a small number of latent variables or factors the models effect a reduction in dimensionality which aids comprehension an early theoretical account of the subject is by anderson rubin 1956 and more recent and comprehensive treatments are provided by lawley maxwell 1971 and harman 1968 the present paper is an attempt to provide a coherent framework within which existing methodology can be evaluated and a base from which new methods can be developed our approach is to start from a very general statement of the problem in terms of the distributions of the random variables involved and then to invoke ideas of symmetry invariance and conditional independence to determine the class of models that it is reasonable to consider this not only exhibits the unity of the various models already in existence but resolves many of the ambiguities and obscurities with which the subject has been bedevilled an early approach on these lines is given by anderson 1959 and was recognized by birnbaum in his contribution to lord novick 1968 the same line was followed by bartholomew 1980 in spite of this the practical implications do not seem to have been made fully explicit or generally recognized

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

survival models for heterogeneous populations derived from stable distributions summary a new three parameter family of distributions on the positive numbers is proposed it includes the stable distributions on the positive numbers the gamma the degenerate and the inverse gaussian distributions the family is characterized by the laplace transform from which moments convolutions infinite divisibility unimodality and other properties are derived the density is complicated but a simple saddlepoint approximation is provided weibull and gompertz distributions are naturally mixed over some of the distributions the family is natural exponential in one of the parameters the distributions are relevant for application as frailty distributions in life table methods for heterogeneous populations desirable properties of such distributions are discussed as an example survival after myocardial infarction is considered

nonparametric estimation of a multivariate distribution in the presence of censoring summary this paper presents examples of situations in which one wishes to estimate a multivariate distribution from data that may be right censored a distinction is made between what we term homogeneous and heterogeneous censoring it is shown how a multivariate empirical survivor function must be constructed in order to be considered a nonparametric maximum likelihood estimate of the underlying survivor function a closed form solution similar to the product limit estimate of kaplan and meier is possible with homogeneous censoring but an iterative method such as the em algorithm is required with heterogeneous censoring an example is given in which an anomaly is produced if censored multivariate data are analyzed as a series of univariate variables this anomaly is shown to disappear if

local dependence functions for some families of bivariate distributions and total positivity the purpose of this paper is to investigate a very useful application of a certain local dependence function c f x y which was considered recently by holland and wang 20 an interesting property of c f x y is that the underlying joint density f x y is tp 2 that is totally positive of order 2 if and only if c f x y 0 this gives an elegant way to investigate the tp 2 property of any bivariate distribution for the saramanov family the ali mikhail haq family of bivariate distributions and the family of bivariate elliptical distributions we derive the local dependence function and obtain conditions for f x y to be tp 2 these families are quite rich and include many other large classes of bivariate distributions as their special cases similar conditions are obtained for bivariate distributions with exponential conditionals and bivariate distributions with pareto conditionals

nonparametric estimation of the concordance correlation coefficient under univariate censoring assessing agreement is often of interest in clinical studies to evaluate the similarity of measurements produced by different raters or methods on the same subjects lin s 1989 biometrics 45 255 268 concordance correlation coefficient ccc has become a popular measure of agreement for correlated continuous outcomes however commonly used estimation methods for the ccc do not accommodate censored observations and are therefore not applicable for survival outcomes in this article we estimate the ccc nonparametrically through the bivariate survival function the proposed estimator of the ccc is proven to be strongly consistent and asymptotically normal with a consistent bootstrap variance estimator furthermore we propose a time dependent agreement coefficient as an extension of lin s 1989 ccc for measuring the agreement between survival times among subjects who survive beyond a specified time point a nonparametric estimator is developed for the time dependent agreement coefficient as well it has the same asymptotic properties as the estimator of the ccc simulation studies are conducted to evaluate the performance of the proposed estimators a real data example from a prostate cancer study is used to illustrate the method

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

adaptation of bivariate frailty models for prediction with application to biological markers as prognostic indicators serial biological markers are often used in medicine as prognostic indicators for example elevated tumour antigen tests presage tumour recurrence rapid decrease in the cd4 cell counts of hiv individuals indicate high risk for developing aids methods for evaluating the usefulness of markers should address how the prediction of future risk is revised in a way that assists the making of decisions at the time such an indicator is observed proportional hazards modelling with time varying covariates a common approach seems unnatural for this purpose because its orientation is explanatory it addresses the question is the individual suffering a failure more likely to have had the marker and allows use of all information prior to the failure rather than the more relevant predictive question does the appearance of the marker alter the individual s subsequent risk enough to warrant intervention today frailty models do address this question but available methods for their analysis make censoring assumptions which are inappropriate in the predictive setting we adapt frailty models to relevant censoring processes and propose a predictively oriented test statistic and a related estimation procedure based on summing concurrent martingales each representing a landmark analysis

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

estimating survival and association in a semicompeting risks model in many follow up studies patients are subject to concurrent events in this article we consider semicompeting risks data as defined by fine jiang and chappell 2001 biometrika 88 907 919 where one event is censored by the other but not vice versa the proposed model involves marginal survival functions for the two events and a parametric family of copulas for their dependency this article suggests a general method for estimating the dependence parameter when the dependency is modeled with an archimedean copula it uses the copula graphic estimator of zheng and klein 1995 biometrika 82 127 138 for estimating the survival function of the nonterminal event subject to dependent censoring asymptotic properties of these estimators are derived simulations show that the new methods work well with finite samples the copula graphic estimator is shown to be more accurate than the estimator proposed by fine et al 2001 its performances are similar to those of the self consistent estimator of jiang fine kosorok and chappell 2005 scandinavian journal of statistics 33 1 20 the analysis of a data set emphasizing the estimation of characteristics of the observable region is presented as an 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

nonparametric estimation of bivariate failure time associations in the presence of a competing risk most research on the study of associations among paired failure times has either assumed time invariance or been based on complex measures or estimators little has accommodated competing risks this paper targets the conditional cause specific hazard ratio henceforth called the cause specific cross ratio a recent modification of the conditional hazard ratio designed to accommodate competing risks data estimation is accomplished by an intuitive non parametric method that localizes kendall s tau time variance is accommodated through a partitioning of space into bins between which the strength of association may differ inferential procedures are developed small sample performance is evaluated and the methods are applied to the investigation of familial association in dementia onset

estimating survival under a dependent truncation summary the product limit estimator calculated from data subject to random left truncation relies on the testable assumption of quasi independence between the failure time and the truncation time in this paper we propose a model for a truncated sample of pairs xi y satisfying 1 xi a possible dependency between the truncation time and the variable of interest is modelled with a parametric family of copulas the model also features a distribution function fx and a survival distribution sy associated with the marginal behaviours of x and y in the observable region y x semiparametric estimators for these two functions are proposed they do not make any parametric assumption about either fx or sy we derive an estimator for the copula parameter cx based on the conditional kendall s tau we generalise the copula graphic estimators of zheng klein 1995 to truncated variables the asymptotic distributions of all these estimators are then investigated the methods are illustrated with a real dataset on hiv infection by transfusion and by simulations

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 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

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

families of multivariate distributions abstract for many years there has been an interest in families of bivariate distributions with marginals as parameters genest and mackay 1986a b showed that several such families that appear in the literature can be derived by a unified method a similar conclusion is obtained in this article through the use of mixture models these models might be regarded as multivariate proportional hazards models with random constants of proportionality the mixture models are useful for two purposes first they make some properties of the derived distributions more transparent the positive dependency property of association is sometimes exposed and a method for simulation of data from the distributions is suggested but the mixture models also allow derivation of several new families of bivariate distributions with marginals as parameters and they indicate obvious multivariate extensions some of the new families of bivariate distributions given in this article extend known distributions by adding a parameter

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

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

the d module structure of r f modules let r be a regular ring essentially of finite type over a perfect field k an r module m is called a unit r f module if it comes equipped with an isomorphism f e m m where f denotes the frobenius map on spec r and f e is the associated pullback functor it is well known that m then carries a natural d r module structure in this paper we investigate the relation between the unit r f structure and the induced d r structure on m in particular it is shown that if k is algebraically closed and m is a simple finitely generated unit r f module then it is also simple as a d r module an example showing the necessity of k being algebraically closed is also given

on consistency of kendall s tau under censoring necessary and sufficient conditions for consistency of a simple estimator of kendall s tau under bivariate censoring are presented the results are extended to data subject to bivariate left truncation as well as right censoring

semiparametric model for bivariate survival data subject to biased sampling to better understand the relationship between patient characteristics and their residual survival after an intermediate event such as the local cancer recurrence it is of interest to identify patients with the intermediate event and then analyze their residual survival data one challenge in analyzing such data is that the observed residual survival times tend to be longer than those in the target population since patients who die before experiencing the intermediate event are excluded from the identified cohort we propose to jointly model the ordered bivariate survival data using a copula model and appropriately adjusting for the sampling bias we develop an estimating procedure to simultaneously estimate the parameters for the marginal survival functions and the association parameter in the copula model and use a two stage expectation maximization algorithm using empirical process theory we prove that the estimators have strong consistency and asymptotic normality we conduct simulations studies to evaluate the finite sample performance of the proposed method we apply the proposed method to two cohort studies to evaluate the association between patient characteristics and residual survival

analysis of dependent survival data applied to lifetimes of amalgam fillings when studying lifetimes of amalgam fillings one is faced with the fact that each patient may contribute multiple survival data this creates a dependence problem and some methods for solving it are presented here analysis for individual patients with simple approaches for combining data is discussed first then a parametric frailty model is applied to analyse variation within and between patients the results are presented in the form of an empirical bayes estimate for each patient

time dependent asymptotic behavior of the solution for wave equations with linear memory abstract in this article we consider the long time behavior of solutions for the wave equation with linear memory within the theory of process on time dependent spaces we investigate the existence of the time dependent attractor by using the operator decomposition technique and compactness of translation theorem and more detailed estimates furthermore the asymptotic structure of time dependent attractor which converges to the attractor of parabolic equation with memory is proved besides we obtain a further regular result about u t

life table methods for heterogeneous populations distributions describing the heterogeneity taking account of heterogeneity between the individuals in population based mortality studies is important a systematic way of describing heterogeneity is by an unobserved quantity called frailty entering the hazard multiplicatively until now most studies have used a gamma distributed frailty which is mathematically convenient for example the distribution among survivors is also gamma this paper shows that several other distributions have equally simple properties the main example being the inverse gaussian distribution consequences of the different distributions are examined the inverse gaussian makes the population homogeneous with time whereas for the gamma the relative heterogeneity is constant

a bivariate cure mixture approach for modeling familial association in diseases summary for modeling correlation in familial diseases with variable ages at onset we propose a bivariate model that incorporates two types of pairwise association one between the lifetime risk or the overall susceptibility of two individuals and one between the ages at onset between two susceptible individuals for estimation we consider a two stage estimation procedure similar to that of shih 1998 biometrics 54 1115 1128 we evaluate the properties of the estimators through simulations and compare the performance with that from a bivariate survival model that allows correlation between ages at onset only we apply the methodology to breast cancer using the kinship data from the washington ashkenazi study we also discuss potential applications of the proposed method in the area of cure modeling

semiparametric regression analysis of interval censored data with informative dropout interval censored data arise when the event time of interest can only be ascertained through periodic examinations in medical studies subjects may not complete the examination schedule for reasons related to the event of interest in this article we develop a semiparametric approach to adjust for such informative dropout in regression analysis of interval censored data specifically we propose a broad class of joint models under which the event time of interest follows a transformation model with a random effect and the dropout time follows a different transformation model but with the same random effect we consider nonparametric maximum likelihood estimation and develop an em algorithm that involves simple and stable calculations we prove that the resulting estimators of the regression parameters are consistent asymptotically normal and asymptotically efficient with a covariance matrix that can be consistently estimated through profile likelihood in addition we show how to consistently estimate the survival function when dropout represents voluntary withdrawal and the cumulative incidence function when dropout is an unavoidable terminal event furthermore we assess the performance of the proposed numerical and inferential procedures through extensive simulation studies finally we provide an application to data on the incidence of diabetes from a major epidemiological cohort study

modeling the association of bivariate interval censored data using the copula approach in a survival study it may not be possible to record the exact event time but only that the event has occurred between two time points or still has to occur leading to interval censored survival times recently sun et al scand j stat 2006 33 4 637 649 suggested to fit a clayton copula with nonparametric marginal distributions to estimate the association for bivariate interval censored failure data we propose here to model the marginal distributions with an accelerated failure time model with a flexible error term as suggested by komarek et al j comput graph stat 2005 14 3 726 745 in combination with a one parameter copula in addition we allow the association parameter of the copula to depend on covariates the performance of our method is illustrated by an extensive simulation study and is applied to tooth emergence data of permanent teeth measured on 4468 children from a longitudinal dental study

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

gamma frailty transformation models for multivariate survival times we propose a class of transformation models for multivariate failure times the class of transformation models generalize the usual gamma frailty model and yields a marginally linear transformation model for each failure time nonparametric maximum likelihood estimation is used for inference the maximum likelihood estimators for the regression coefficients are shown to be consistent and asymptotically normal and their asymptotic variances attain the semiparametric efficiency bound simulation studies show that the proposed estimation procedure provides asymptotically efficient estimates and yields good inferential properties for small sample sizes the method is illustrated using data from a cardiovascular study