Graphical Tests for the Assumption of Gamma and Inverse Gaussian Frailty Distributions

The common choices of frailty distribution in lifetime data models include the Gamma and Inverse Gaussian distributions. We present diagnostic plots for these distributions when frailty operates in a proportional hazards framework. Firstly, we present plots based on the form of the unconditional survival function when the baseline hazard is assumed to be Weibull. Secondly, we base a plot on a closure property that applies for any baseline hazard, namely, that the frailty distribution among survivors at time t has the same form as the original distribution, with the same shape parameter but different scale parameter. We estimate the shape parameter at different values of t and examine whether it is constant, that is, whether plotted values form a straight line parallel to the time axis. We provide simulation results assuming Weibull baseline hazard and an example to illustrate the methods.     

Hierarchical likelihood approach for the Weibull frailty model

In this article, the proportional hazard model with Weibull frailty, which is outside the range of the exponential family, is used for analysing the right-censored longitudinal survival data. Complex multidimensional integrals are avoided by using hierarchical likelihood to estimate the regression parameters and to predict the realizations of random effects. The adjusted profile hierarchical likelihood is adopted to estimate the parameters in frailty distribution, during which the first- and second-order methods are used. The simulation studies indicate that the regression-parameter estimates in the Weibull frailty model are accurate, which is similar to the gamma frailty and lognormal frailty models. Two published data sets are used for illustration.     

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.     

Tests for Paired Lifetime Data with Frailty Models

We consider the problem of hypothesis testing of the equality of marginal survival distributions observed from paired lifetime data. Usual procedures include the paired t-test, which may perform poor for certain types of data. We propose asymptotic tests based on gamma frailty models with Weibull conditional distributions, and investigate their theoretical properties using large sample theory. For finite samples, we conduct simulations to evaluate the powers of the associated tests. For moderate and less skewed data, the proposed tests are the most powerful among the commonly applied testing procedures. A data example is illustrated to demonstrate the methods.     

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.     

A covariance-based test for shared frailty in multivariate lifetime data

We decompose the score statistic for testing for shared finite variance frailty in multivariate lifetime data into marginal and covariance-based terms. The null properties of the covariance-based statistic are derived in the context of parametric lifetime models. Its non-null properties are estimated using simulation and compared with those of the score test and two likelihood ratio tests when the underlying lifetime distribution is Weibull. Some examples are used to illustrate the covariance-based test. A case is made for using the covariance-based statistic as a simple diagnostic procedure for shared frailty in a parametric exploratory analysis of multivariate lifetime data and a link to the bivariate Clayton–Oakes copula model is shown.     

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.     

Fisher Information for Two Gamma Frailty Bivariate Weibull Models

The asymptotic properties of frailty models for multivariate survival data are not well understood. To study this aspect, the Fisher information is derived in the standard bivariate gamma frailty model, where the survival distribution is of Weibull form conditional on the frailty. For comparison, the Fisher information is also derived in the bivariate gamma frailty model, where the marginal distribution is of Weibull form.     

Frailty regression models in mixture distributions

We propose frailty regression models in mixture distributions and assume the distribution of frailty as gamma or positive stable or power variance function distribution. We consider Weibull mixture as an example. There are some interesting situations like survival times in genetic epidemiology, dental implants of patients and twin births (both monozygotic and dizygotic) where genetic behavior (which is unknown and random) of patients follows a known frailty distribution. These are the situations which motivate to study this particular model.     

A Weibull Regression Model with Gamma Frailties for Multivariate Survival Data

Frequently in the analysis of survival data, survival times within the same group are correlated due to unobserved co-variates. One way these co-variates can be included in the model is as frailties. These frailty random block effects generate dependency between the survival times of the individuals which are conditionally independent given the frailty. Using a conditional proportional hazards model, in conjunction with the frailty, a whole new family of models is introduced. By considering a gamma frailty model, often the issue is to find an appropriate model for the baseline hazard function. In this paper a flexible baseline hazard model based on a correlated prior process is proposed and is compared with a standard Weibull model. Several model diagnostics methods are developed and model comparison is made using recently developed Bayesian model selection criteria. The above methodologies are applied to the McGilchrist and Aisbett (1991) kidney infection data and the analysis is performed using Markov Chain Monte Carlo methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

Survival functions for the frailty models based on the discrete compound Poisson process

Frailty models are often used to model heterogeneity in survival analysis. The distribution of the frailty is generally assumed to be continuous. In some circumstances, it is appropriate to consider discrete frailty distributions. Having zero frailty can be interpreted as being immune, and population heterogeneity may be analysed using discrete frailty models. In this paper, survival functions are derived for the frailty models based on the discrete compound Poisson process. Maximum likelihood estimation procedures for the parameters are studied. We examine the fit of the models to earthquake and the traffic accidents’ data sets from Turkey.     

Modeling bivariate survival data using shared inverse Gaussian frailty model

ABSTRACT

The shared frailty models are often used to model heterogeneity in survival analysis. The most common shared frailty model is a model in which hazard function is a product of a random factor (frailty) and the baseline hazard function which is common to all individuals. There are certain assumptions about the baseline distribution and the distribution of frailty. In this paper, we consider inverse Gaussian distribution as frailty distribution and three different baseline distributions, namely the generalized Rayleigh, the weighted exponential, and the extended Weibull distributions. With these three baseline distributions, we propose three different inverse Gaussian shared frailty models. We also compare these models with the models where the above-mentioned distributions are considered without frailty. We develop 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. A search of the literature suggests that currently no work has been done for these three baseline distributions with a shared inverse Gaussian frailty so far. We also apply these three models by using a real-life bivariate survival data set of McGilchrist and Aisbett (1991 McGilchrist, C.A., Aisbett, C.W. (1991). Regression with frailty in survival analysis. Biometrics 47:461–466.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) related to the kidney infection data and a better model is suggested for the data using the Bayesian model selection criteria.     

Bayes Analysis and Comparison of Accelerated Weibull and Accelerated Birnbaum–Saunders Models

Several models are proposed in the literature for modeling fatigue data resulting from materials subject to cyclic stress and strain. Accelerated Weibull and accelerated Birnbaum–Saunders distributions are most commonly used models. Whereas the accelerated Weibull model is easier compared to accelerated Birnbaum–Saunders, it fails to represent the situation equally well. The present article focuses on Bayes analysis of the two models and provides a comparison based on some important Bayesian tools. Model compatibility study using predictive simulation ideas is preceded by the said comparison. Throughout, the posterior simulations are carried out by Markov chain Monte Carlo procedure.     

Exponentiated Weibull regression for time-to-event data

The Weibull, log-logistic and log-normal distributions are extensively used to model time-to-event data. The Weibull family accommodates only monotone hazard rates, whereas the log-logistic and log-normal are widely used to model unimodal hazard functions. The increasing availability of lifetime data with a wide range of characteristics motivate us to develop more flexible models that accommodate both monotone and nonmonotone hazard functions. One such model is the exponentiated Weibull distribution which not only accommodates monotone hazard functions but also allows for unimodal and bathtub shape hazard rates. This distribution has demonstrated considerable potential in univariate analysis of time-to-event data. However, the primary focus of many studies is rather on understanding the relationship between the time to the occurrence of an event and one or more covariates. This leads to a consideration of regression models that can be formulated in different ways in survival analysis. One such strategy involves formulating models for the accelerated failure time family of distributions. The most commonly used distributions serving this purpose are the Weibull, log-logistic and log-normal distributions. In this study, we show that the exponentiated Weibull distribution is closed under the accelerated failure time family. We then formulate a regression model based on the exponentiated Weibull distribution, and develop large sample theory for statistical inference. We also describe a Bayesian approach for inference. Two comparative studies based on real and simulated data sets reveal that the exponentiated Weibull regression can be valuable in adequately describing different types of time-to-event data.     

A study of moments and likelihood estimators of the odd Weibull distribution

The odd Weibull distribution is a three-parameter generalization of the Weibull and the inverse Weibull distributions having rich density and hazard shapes for modeling lifetime data. This paper explored the odd Weibull parameter regions having finite moments and examined the relation to some well-known distributions based on skewness and kurtosis functions. The existence of maximum likelihood estimators have shown with complete data for any sample size. The proof for the uniqueness of these estimators is given only when the absolute value of the second shape parameter is between zero and one. Furthermore, elements of the Fisher information matrix are obtained based on complete data using a single integral representation which have shown to exist for any parameter values. The performance of the odd Weibull distribution over various density and hazard shapes is compared with generalized gamma distribution using two different test statistics. Finally, analysis of two data sets has been performed for illustrative purposes.     

Non-parametric Bayesian analysis of clustered survival data

Clustered survival data are often modelled with frailty models which incorporate frailties to model the cluster specific heterogeneity and the dependence between observations in the same cluster. For the analysis of the frailty models, we propose Bayesian modelling with beta process prior on the cumulative hazard function and describe the details of the posterior computation. We demonstrate the method with two data sets using three different frailty distributions: gamma, log-normal and log-logistic distributions. We also empirically demonstrate the difficulty in checking the assumed frailty distribution with the posterior sample of the frailties.     

A Distribution for Multivariate Frailty Based on the Compound Poisson Distribution with Random Scale

Frailty models are often used to model heterogeneity in survival analysis. The most common frailty model has an individual intensity which is a product of a random factor and a basic intensity common to all individuals. This paper uses the compound Poisson distribution as the random factor. It allows some individuals to be non-susceptible, which can be useful in many settings. In some diseases, one may suppose that a number of families have an increased susceptibility due to genetic circumstances. Then, it is logical to use a frailty model where the individuals within each family have some shared factor, while individuals between families have different factors. This can be attained by randomizing the Poisson parameter in the compound Poisson distribution. To our knowledge, this is a new distribution. The power variance function distributions are used for the Poisson parameter. The subsequent appearing distributions are studied in some detail, both regarding appearance and various statistical properties. An application to infant mortality data from the Medical Birth Registry of Norway is included, where the model is compared to more traditional shared frailty models.     

Discrete Linnik Weibull distribution

Abstract

We introduce a new family of distributions using truncated discrete Linnik distribution. This family is a rich family of distributions which includes many important families of distributions such as Marshall–Olkin family of distributions, family of distributions generated through truncated negative binomial distribution, family of distributions generated through truncated discrete Mittag–Leffler distribution etc. Some properties of the new family of distributions are derived. A particular case of the family, a five parameter generalization of Weibull distribution, namely discrete Linnik Weibull distribution is given special attention. This distribution is a generalization of many distributions, such as extended exponentiated Weibull, exponentiated Weibull, Weibull truncated negative binomial, generalized exponential truncated negative binomial, Marshall-Olkin extended Weibull, Marshall–Olkin generalized exponential, exponential truncated negative binomial, Marshall–Olkin exponential and generalized exponential. The shape properties, moments, median, distribution of order statistics, stochastic ordering and stress–strength properties of the new generalized Weibull distribution are derived. The unknown parameters of the distribution are estimated using maximum likelihood method. The discrete Linnik Weibull distribution is fitted to a survival time data set and it is shown that the distribution is more appropriate than other competitive models.     

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.     

Characterizations of distributions through coincidences of semiparametric families

Semiparametric families are families that have both a real parameter and a parameter that is itself a distribution. A number of semiparametric families suitable for lifetime data are introduced: scale, power, frailty (proportional hazards), age, moment, Laplace transform, and convolution parameter families. The coincidence of two families provides a characterization of the underlying distribution. Characterizations of the Weibull, gamma, lognormal, and Gompertz distributions are obtained.