Correlated Frailty Models Research Articles

Convergent stochastic algorithm for estimation in general multivariate correlated frailty models using integrated partial likelihood

The Cox model with unspecified baseline hazard is often used to model survival data. In the case of correlated event times, this model can be extended by introducing random effects, also called frailty terms, leading to the frailty model. Few methods have been put forward to estimate parameters of such frailty models, and they often consider only a particular distribution for the frailty terms and specific correlation structures. In this paper, a new efficient method is introduced to perform parameter estimation by maximizing the integrated partial likelihood. The proposed stochastic estimation procedure can deal with frailty models with a broad choice of distributions for the frailty terms and with any kind of correlation structure between the frailty components, also allowing random interaction terms between the covariates and the frailty components. The almost sure convergence of the stochastic estimation algorithm towards a critical point of the integrated partial likelihood is proved. Numerical convergence properties are evaluated through simulation studies and comparison with existing methods is performed. In particular, the robustness of the proposed method with respect to different parametric baseline hazards and misspecified frailty distributions is demonstrated through simulation. Finally, the method is applied to a mastitis and a bladder cancer dataset.

Modeling Negatively Skewed Survival Data in Accelerated Failure Time and Correlated Frailty Models

Modeling Negatively Skewed Survival Data in Accelerated Failure Time and Correlated Frailty Models

Compound negative binomial multivariate correlated frailty model for long-term survivors

In the correlated frailty models, it is believed that all the individuals in the population will eventually experience the event of interest. This may not always be the situation in reality. There may be a certain fraction of the population which is immune to the event and hence may not experience the event under study. For such individuals, frailty may not be active i.e., zero. Hence, frailty distribution should provide a positive mass at zero. In this article, a new correlated frailty model with compound negative binomial distribution as frailty distribution, is introduced to incorporate the non-susceptibility of individuals. The inferential problem is solved in a Bayesian framework using Markov Chain Monte Carlo methods. The proposed model is then applied to a real life data set.

Comparison of correlated frailty models

In this article, we propose inverse Gaussian correlated frailty model based on logistic exponential as baseline distribution. The Bayesian approach of Markov Chain Monte Carlo (MCMC) technique was employed to estimate the parameters involved in the models. A simulation study was performed to compare the true values and estimated value of the parameters. Comparison of the proposed model with correlated gamma frailty model and without frailty model was done using Bayesian comparison techniques. The better model for the infectious disease data related to kidney infection is also suggested.

The Modelling of a Cure Fraction in Bivariate Time-to-Event Data

Three correlated frailty models are used to analyze bivariate timeto- event data by assuming gamma, log-normal and compound Poisson distributed frailty. All approaches allow to deal with right censored lifetime data and account for heterogeneity as well as for a non-susceptible (cure) fraction in the study population. In the gamma and compound Poisson model traditional ML estimation methods are used, whereas in the log-normal model MCMC methods are applied. Breast cancer incidence data of Swedish twin pairs illustrate the practical relevance of the models, which are used to estimate the size of the susceptible fraction and the correlation between the frailties of the twin partners. We discuss future directions of development of the methods and additional thoughts concerning their advantages and use.

A comparison of different bivariate correlated frailty models and estimation strategies

Frailty models are becoming increasingly popular in multivariate survival analysis. Shared frailty models in particular are often used despite their limitations. To overcome their disadvantages numerous correlated frailty models were established during the last decade. In the present study, we examine bivariate correlated frailty models, and especially the behavior of the parameter estimates when using different estimation strategies. We consider three different bivariate frailty models: the gamma model and two versions of the log-normal model. The traditional maximum likelihood procedure of parameter estimation in the gamma case with an explicit available likelihood function is compared with maximum likelihood methods based on numerical integration and a Bayesian approach using MCMC methods with the help of a comprehensive simulation study. We detected a strong dependence between the two parameter estimates (variance and correlation of frailties) in the bivariate correlated frailty model and analyzed this dependence in detail.

Random‐effects Cox proportional hazards model: General variance components methods for time‐to‐event data

Proportional hazards regression models are commonly used to study factors associated with time-to-event data. Because many complex genetic diseases exhibit variation in age at onset, it is important to have the capability to perform survival analyses on data collected from individuals whose observations are correlated due to shared genes or environment. While there are widely accepted methods for variance components analysis for simple quantitative traits, a parallel methodology for survival data has not been available. This manuscript outlines a method to perform variance component analyses under general random effects proportional hazards models. This method is based on a Laplace approximation, and makes computation for correlated time-to-event data feasible. The correlated frailty models described here can be used to perform genetic analyses, and other analyses with structured random effects, on age-at-onset data in a manner analogous to standard variance components methods for quantitative traits. We illustrate the use of the method by examining the heritability of breast cancer in a large familial cohort study. We also perform variance components linkage analyses on data simulated for the Twelfth Genetic Analysis Workshop (GAW12), and further examine the performance of this method for linkage analysis in a simulation study. The breast cancer analyses support significant heritability of disease age-at-onset that is of moderate size. The variance component linkage analyses successfully identify the location of the disease genes that were simulated to have a direct impact on age-at-onset. The methods outlined here make it possible to perform general variance components analyses on time-to-event endpoints, even on large data sets, in a computationally efficient manner.

On Use of Bivariate Survival Models with Cure Fraction

Correlated Gamma Frailty Model In our original article, we proposed the methodology in the very general setting of copula model (Genest and Mackay, 1986; Marshall and Olkin, 1988). We illustrated the application of the methodology using three specific distribution of the general class, namely Clayton's (1978), Frank's (1979), and Stable family (Hougaard, 1986). It is true that all of these three models correspond to an underlying shared frailty model. The class of Copula models, however, are much more general and can include correlated frailty models. In fact, the correlated frailty model the authors proposed to use as an extension of our methodology is a special case of the copula model with the copula function given by

Cancer incidence for Swedish twins studied by means of bivariate frailty models.

Cancer incidence rates for Swedish twins born between 1928 and 1965 and who both were alive at age 30 are studied by means of bivariate frailty models. Altogether, 7,280 fraternal (DZ) and 4,699 identical (MZ) twin pairs were followed up through December 31, 1995, for cancer status. The association between cancer incidence rates was statistically greater among the MZ than among the DZ pairs and stronger between women than between men; however, the magnitude of this association is relatively small and decreases over time. The relative decrease in dependency (association) is most easily detected using shared frailty models but may also be demonstrated, at least for women, using correlated frailty models. We also demonstrate that estimates of the correlation coefficient are similar when using any correlated frailty models derived from the power variance family but that these estimates disagree regarding the age at which the dependence is most important. The relative importance of dependence across age may sometimes be more interesting than the correlation coefficient itself. The latter may usually be estimated using alternative methods. Furthermore, when estimating correlation coefficients close to the boundary of the parameter space, simulation studies indicate that the correlated inverse Gaussian frailty model is more robust than the gamma frailty model.