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.     

Defective models induced by gamma frailty term for survival data with cured fraction

In this paper, we propose a defective model induced by a frailty term for modeling the proportion of cured. Unlike most of the cure rate models, defective models have advantage of modeling the cure rate without adding any extra parameter in model. The introduction of an unobserved heterogeneity among individuals has bring advantages for the estimated model. The influence of unobserved covariates is incorporated using a proportional hazard model. The frailty term assumed to follow a gamma distribution is introduced on the hazard rate to control the unobservable heterogeneity of the patients. We assume that the baseline distribution follows a Gompertz and inverse Gaussian defective distributions. Thus we propose and discuss two defective distributions: the defective gamma-Gompertz and gamma-inverse Gaussian regression models. Simulation studies are performed to verify the asymptotic properties of the maximum likelihood estimator. Lastly, in order to illustrate the proposed model, we present three applications in real data sets, in which one of them we are using for the first time, related to a study about breast cancer in the A.C.Camargo Cancer Center, São Paulo, Brazil.     

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.     

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.     

Comparison of Shared Frailty Models for Kidney Infection Data under Exponential Power Baseline Distribution

Shared frailty models are often used to model heterogeneity in survival analysis. There are certain assumptions about the baseline distribution and distribution of frailty. In this paper, four shared frailty models with frailty distribution gamma, inverse Gaussian, compound Poisson, and compound negative binomial with exponential power as baseline distribution are proposed. These models are fitted using Markov Chain Monte Carlo methods. These models are illustrated with a real life bivariate survival data set of McGilchrist and Aisbett (1991) related to kidney infection, and the best model is suggested for the data using different model comparison criteria.     

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.     

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.     

The Identifiability of Dependent Competing Risks Models Induced by Bivariate Frailty Models

In this paper, we propose to use a special class of bivariate frailty models to study dependent censored data. The proposed models are closely linked to Archimedean copula models. We give sufficient conditions for the identifiability of this type of competing risks models. The proposed conditions are derived based on a property shared by Archimedean copula models and satisfied by several well‐known bivariate frailty models. Compared with the models studied by Heckman and Honoré and Abbring and van den Berg, our models are more restrictive but can be identified with a discrete (even finite) covariate. Under our identifiability conditions, expectation–maximization (EM) algorithm provides us with consistent estimates of the unknown parameters. Simulation studies have shown that our estimation procedure works quite well. We fit a dependent censored leukaemia data set using the Clayton copula model and end our paper with some discussions. © 2014 Board of the Foundation of the Scandinavian Journal of Statistics     

Effect of frailty on marginal regression estimates in survival analysis

Unexplained heterogeneity in univariate survival data and association in multivariate survival can both be modelled by the inclusion of frailty effects. This paper investigates the consequences of ignoring frailty in analysis, fitting misspecified Cox proportional hazards models to the marginal distributions. Regression coefficients are biased towards 0 by an amount which depends in magnitude on the variability of the frailty terms and the form of frailty distribution. The bias is reduced when censoring is present. Fitted marginal survival curves can also differ substantially from the true marginals.     

Some results on discrete mixture failure rates

Some properties of the discrete mixture failure rates are studied. Specifically, similar to the continuous case, it is shown that the population mixture failure rate is always smaller than the unconditional expectation in the family of subpopulations failure rates. The analog of the multiplicative and the additive frailty models is introduced via the corresponding survival function. Another approach via the alternative discrete failure rate is also discussed. Stochastic comparisons for two mixed distributions with equal and different mixing distributions are studied.     

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.     

Robust Poisson likelihood estimation for frailty Cox models: A simulation study

Frailty models can be fit as mixed-effects Poisson models after transforming time-to-event data to the Poisson model framework. We assess, through simulations, the robustness of Poisson likelihood estimation for Cox proportional hazards models with log-normal frailties under misspecified frailty distribution. The log-gamma and Laplace distributions were used as true distributions for frailties on a natural log scale. Factors such as the magnitude of heterogeneity, censoring rate, number and sizes of groups were explored. In the simulations, the Poisson modeling approach that assumes log-normally distributed frailties provided accurate estimates of within- and between-group fixed effects even under a misspecified frailty distribution. Non-robust estimation of variance components was observed in the situations of substantial heterogeneity, large event rates, or high data dimensions.     

Modelling survival data using mixtures of frailties

Frailty models are often used to describe the extra heterogeneity in survival data by introducing an individual random, unobserved effect. The frailty term is usually assumed to act multiplicatively on a baseline hazard function common to all individuals. In order to apply the frailty model, a specific frailty distribution has to be assumed. If at least one of the latent variables is continuous, the frailty must follow a continuous distribution. In this paper, a finite mixture of continuous frailty distributions is used in order to describe situations in which one (or more) of the latent variables separates the population in study into two (or more) subpopulations. Closure properties of the unobserved quantity are given along with the maximum-likelihood estimates under the most common choices of frailty distributions. The model is illustrated on a set of lifetime 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.     

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.     

Effects of unmeasured heterogeneity in the linear transformation model for censored data

We investigate the effect of unobserved heterogeneity in the context of the linear transformation model for censored survival data in the clinical trials setting. The unobserved heterogeneity is represented by a frailty term, with unknown distribution, in the linear transformation model. The bias of the estimate under the assumption of no unobserved heterogeneity when it truly is present is obtained. We also derive the asymptotic relative efficiency of the estimate of treatment effect under the incorrect assumption of no unobserved heterogeneity. Additionally we investigate the loss of power for clinical trials that are designed assuming the model without frailty when, in fact, the model with frailty is true. Numerical studies under a proportional odds model show that the loss of efficiency and the loss of power can be substantial when the heterogeneity, as embodied by a frailty, is ignored. An erratum to this article can be found at     

Testing Heterogeneity for Frailty Distribution in Shared Frailty Model

Abstract

The frailties, representing extra variations due to unobserved measurements, are often assumed to be iid in shared frailty models. In medical applications, however, a speculation can arise that a data set might violate the iid assumption. In this paper we investigate this conjecture through an analysis of the kidney infection data in McGilchrist and Aisbett (McGilchrist, C. A., Aisbett, C. W. (1991). Regression with frailty in survival analysis. Biometrics 47:461–466). As a test procedure, we consider the cusum of squares test which is frequently used for monitoring a variance change in statistical models. Our result strongly sustains the heterogeneity of the frailty distribution.     

Multilevel Mixed Linear Models for Survival Data

For the analysis of correlated survival data mixed linear models are useful alternatives to frailty models. By their use the survival times can be directly modelled, so that the interpretation of the fixed and random effects is straightforward. However, because of intractable integration involved with the use of marginal likelihood the class of models in use has been severely restricted. Such a difficulty can be avoided by using hierarchical-likelihood, which provides a statistically efficient and fast fitting algorithm for multilevel models. The proposed method is illustrated using the chronic granulomatous disease data. A simulation study is carried out to evaluate the performance.     

Variable selection strategies in survival models with multiple imputations

In this paper, the variable selection strategies (criteria) are thoroughly discussed and their use in various survival models is investigated. The asymptotic efficiency property, in the sense of Shibata Ann Stat 8: 147-164, 1980, of a class of variable selection strategies which includes the AIC and all criteria equivalent to it, is established for a general class of survival models, such as parametric frailty or transformation models and accelerated failure time models, under minimum conditions. Furthermore, a multiple imputations method is proposed which is found to successfully handle censored observations and constitutes a competitor to existing methods in the literature. A number of real and simulated data are used for illustrative purposes.     

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.