Maximum Penalized Likelihood Estimation in a Gamma-Frailty Model

The shared frailty models allow for unobserved heterogeneity or for statistical dependence between observed survival data. The most commonly used estimation procedure in frailty models is the EM algorithm, but this approach yields a discrete estimator of the distribution and consequently does not allow direct estimation of the hazard function. We show how maximum penalized likelihood estimation can be applied to nonparametric estimation of a continuous hazard function in a shared gamma-frailty model with right-censored and left-truncated data. We examine the problem of obtaining variance estimators for regression coefficients, the frailty parameter and baseline hazard functions. Some simulations for the proposed estimation procedure are presented. A prospective cohort (Paquid) with grouped survival data serves to illustrate the method which was used to analyze the relationship between environmental factors and the risk of dementia.     

Shared frailty models with baseline generalized Pareto distribution

Abstract

In this article, we have considered three different shared frailty models under the assumption of generalized Pareto Distribution as baseline distribution. Frailty models have been used in the survival analysis to account for the unobserved heterogeneity in an individual risks to disease and death. These three frailty models are with gamma frailty, inverse Gaussian frailty and positive stable frailty. Then we introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters. We applied these three models to a kidney infection data and find the best fitted model for kidney infection data. We present a simulation study to compare true value of the parameters with the estimated values. Model comparison is made using Bayesian model selection criterion and a well-fitted model is suggested for the kidney infection data.     

Semiparametric Global Cross-ratio Models for Bivariate Censored Data

Abstract.  Multivariate correlated failure time data arise in many medical and scientific settings. In the analysis of such data, it is important to use models where the parameters have simple interpretations. In this paper, we formulate a model for bivariate survival data based on the Plackett distribution. The model is an alternative to the Gamma frailty model proposed by Clayton and Oakes. The parameter in this distribution has a very appealing odds ratio interpretation for dependence between the two failure times; in addition, it allows for negative dependence. We develop novel semiparametric estimation and inference procedures for the model. The asymptotic results of the estimator are developed. The performance of the proposed techniques in finite samples is examined using simulation studies; in addition, the proposed methods are applied to data from an observational study in cancer.     

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.     

Shared frailty models based on reversed hazard rate for modified inverse Weibull distribution as baseline distribution

The unknown or unobservable risk factors in the survival analysis cause heterogeneity between individuals. Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times, the shared frailty models were suggested. The most common shared frailty model is a model in which frailty act multiplicatively on the hazard function. In this paper, we introduce the shared gamma frailty model and the inverse Gaussian frailty model with the reversed hazard rate. We introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters involved in the model. We present a simulation study to compare the true values of the parameters with the estimated values. We also apply the proposed models to the Australian twin data set and a better model is suggested.     

Inference Methods for Correlated Left Truncated Lifetimes: Parent and Offspring Relations in an Adoption Study

The associations in mortality of adult adoptees and their biological or adoptive parents have been studied in order to separate genetic and environmental influences. The 1003 Danish adoptees born 1924–26 have previously been analysed in a Cox regression model, using dichotomised versions of the parents’ lifetimes as covariates. This model will be referred to as the conditional Cox model, as it analyses lifetimes of adoptees conditional on parental lifetimes. Shared frailty models may be more satisfactory by using the entire observed lifetime of the parents. In a simulation study, sample size, distribution of lifetimes, truncation- and censoring patterns were chosen to illustrate aspects of the adoption dataset, and were generated from the conditional Cox model or a shared frailty model with gamma distributed frailties. First, efficiency was compared in the conditional Cox model and a shared frailty model, based on the conditional approach. For data with type 1 censoring the models showed no differences, whereas in data with random or no censoring, the models had different power in favour of the one from which data were generated. Secondly, estimation in the shared frailty model by a conditional approach or a two-stage copula approach was compared. Both approaches worked well, with no sign of dependence upon the truncation pattern, but some sign of bias depending on the censoring. For frailty parameters close to zero, we found bias when the estimation procedure used did not allow negative estimates. Based on this evaluation, we prefer to use frailty models allowing for negative frailty parameter estimates. The conclusions from earlier analyses of the adoption study were confirmed, though without greater precision than using the conditional Cox model. Analyses of associations between parental lifetimes are also presented.     

Gamma shared frailty model based on reversed hazard rate

Abstract

Frailty models are used in survival analysis to account for unobserved heterogeneity in individual risks to disease and death. To analyze bivariate data on related survival times (e.g., matched pairs experiments, twin, or family data), shared frailty models were suggested. Shared frailty models are frequently used to model heterogeneity in survival analysis. The most common shared frailty model is a model in which hazard function is a product of random factor(frailty) and baseline hazard function which is common to all individuals. There are certain assumptions about the baseline distribution and distribution of frailty. In this paper, we introduce shared gamma frailty models with reversed hazard rate. We introduce Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in the model. We present a simulation study to compare the true values of the parameters with the estimated values. Also, we apply the proposed model to the Australian twin data set.     

Gamma frailty models for bivariate survival data

In this paper, we introduce the shared gamma frailty models with two different baseline distributions namely, the generalized log-logistic and the generalized Weibull. We introduce the Bayesian estimation procedure to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. We apply these models to a real-life bivariate survival data set of McGilchrist and Aisbett related to the kidney infection data and a better model is suggested for the data.     

Estimation of the cumulative baseline hazard function for dependently right-censored failure time data

In this paper, we study the properties of a special class of frailty models when the frailty is common to several failure times. The models are closely linked to Archimedean copula models. We establish a useful formula for cumulative baseline hazard functions and develop a new estimator for cumulative baseline hazard functions in bivariate frailty regression models. Based on our proposed estimator, we present a graphical model checking procedure. We fit a leukemia data set using our model and end our paper with some discussions.     

Correlated gamma frailty models for bivariate survival data

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g., matched pairs experiments, twin or family data) the shared frailty models were suggested. Shared frailty models are used despite their limitations. To overcome their disadvantages correlated frailty models may be used. In this article, we introduce the gamma correlated frailty models with two different baseline distributions namely, the generalized log logistic, and the generalized Weibull. We introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. Also we apply these models to a real life bivariate survival dataset related to the kidney infection data and a better model is suggested for the data.     

Shared frailty model based on reversed hazard rate for left censored data

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in the individual risks to disease and death. To analyze the bivariate data on related survival times (e.g., matched pairs experiments, twin or family data), the shared frailty models were suggested. In this article, we introduce the shared gamma frailty models with the reversed hazard rate. We develop the Bayesian estimation procedure using the Markov chain Monte Carlo (MCMC) technique to estimate the parameters involved in the model. We present a simulation study to compare the true values of the parameters with the estimated values. We apply the model to a real life bivariate survival dataset.     

A Composite Likelihood Approach to Multivariate Survival Data

This paper is about the statistical analysis of multivariate survival data. We discuss the additive and multiplicative frailty models which have been the most popular models for multivariate survival data. As an alternative to the additive and multiplicative frailty models, we propose basing inference on a composite likelihood function that only requires modelling of the marginal distribution of pairs of failure times. Each marginal distribution of a pair of failure times is here assumed to follow a shared frailty model. The method is illustrated with a real-life example.     

Semiparametric mixed-effects models for clustered doubly censored data

The Cox proportional frailty model with a random effect has been proposed for the analysis of right-censored data which consist of a large number of small clusters of correlated failure time observations. For right-censored data, Cai et al. [3] proposed a class of semiparametric mixed-effects models which provides useful alternatives to the Cox model. We demonstrate that the approach of Cai et al. [3] can be used to analyze clustered doubly censored data when both left- and right-censoring variables are always observed. The asymptotic properties of the proposed estimator are derived. A simulation study is conducted to investigate the performance of the proposed estimator.     

Semiparametric Likelihood Estimation in the Clayton–Oakes Failure Time Model

Multivariate failure time data arise when the sample consists of clusters and each cluster contains several possibly dependent failure times. The Clayton–Oakes model (Clayton, 1978; Oakes, 1982) for multivariate failure times characterizes the intracluster dependence parametrically but allows arbitrary specification of the marginal distributions. In this paper, we discuss estimation in the Clayton–Oakes model when the marginal distributions are modeled to follow the Cox (1972) proportional hazards regression model. Parameter estimation is based on an approximate generalized maximum likelihood estimator. We illustrate the model's application with example datasets.     

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.     

Conditional and marginal estimates in case-control family data – extensions and sensitivity analyses

This work considers two specific estimation techniques for the family-specific proportional hazards model and for the population-averaged proportional hazards model. So far, these two estimation procedures were presented and studied under the gamma frailty distribution mainly because of its simple interpretation and mathematical tractability. Modifications of both procedures for other frailty distributions, such as the inverse Gaussian, positive stable and a specific case of discrete distribution, are presented. By extensive simulations, it is shown that under the family-specific proportional hazards model, the gamma frailty model appears to be robust to frailty distribution mis-specification in both bias and efficiency loss in the marginal parameters. The population-averaged proportional hazards model, is found to be robust under the gamma frailty model mis-specification only under moderate or weak dependency within cluster members.     

Shared gamma frailty models based on reversed hazard rate for modeling Australian twin data

In this paper, we consider shared gamma frailty model with the reversed hazard rate (RHR) with two different baseline distributions, namely the generalized inverse Rayleigh and the exponentiated Gumbel distributions. With these two baseline distributions we propose two different shared frailty models. We develop the Bayesian estimation procedure using Markov Chain Monte Carlo technique to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. A search of the literature suggests that currently no work has been done for these two baseline distributions with a shared gamma frailty with the RHR so far. We also apply these two models by using a real life bivariate survival data set of Australian twin data given by Duffy et a1. (1990) and a better model is suggested for the data.     

Small Sample Bias in the Gamma Frailty Model for Univariate Survival

The gamma frailty model is a natural extension of the Cox proportional hazards model in survival analysis. Because the frailties are unobserved, an E-M approach is often used for estimation. Such an approach is shown to lead to finite sample underestimation of the frailty variance, with the corresponding regression parameters also being underestimated as a result. For the univariate case, we investigate the source of the bias with simulation studies and a complete enumeration. The rank-based E-M approach, we note, only identifies frailty through the order in which failures occur; additional frailty which is evident in the survival times is ignored, and as a result the frailty variance is underestimated. An adaption of the standard E-M approach is suggested, whereby the non-parametric Breslow estimate is replaced by a local likelihood formulation for the baseline hazard which allows the survival times themselves to enter the model. Simulations demonstrate that this approach substantially reduces the bias, even at small sample sizes. The method developed is applied to survival data from the North West Regional Leukaemia Register.     

Frailty models and copulas: similarities and differences

Copulas and frailty models are important tools to model bivariate survival data. Equivalence between Archimedean copula models and shared frailty models, e.g. between the Clayton-Oakes copula model and the shared gamma frailty model, has often been claimed in the literature. In this note we show that, in both the models, there is indeed a well-known equivalence between the copula functions; the modeling of the marginal survival functions, however, is quite different. The latter fact leads to different joint survival functions.     

Diagnostic Plots for Assessing the Frailty Distribution in Multivariate Survival Data

In biomedical studies, frailty models arecommonly used in analyzing multivariate survival data, wherethe objective of the study is to estimate both the covariateeffect and the dependence between the multivariate survival times.However, inference based on these models are dependent on thedistributional assumption of frailty. We propose a diagnosticplot for assessing the frailty assumption. The proposed methodis based on the cross-ratio function and the diagnostic plotsuggested by Oakes (1989). We use kernel regression smoothingwith bandwidth choice by cross-validation, to obtain the proposedplot. The resulting plot is capable of differentiating betweenthe gamma and positive stable frailty models when strong associationis present. We illustrate the feasibility of our method usingsimulation studies under known frailty distributions. The approachis applied to data on blindness for each eye of diabetic patientswith adult onset diabetes and a reasonable fit to the gamma frailtymodel is found.