Modeling heterogeneity for bivariate survival data by shared gamma frailty regression model

In the analysis of survival data with parametric models, it is well known that the Weibull model is not suitable for modeling survival data where the hazard rate is non-monotonic. For such cases, where hazard rates are bathtub-shaped or unimodal (or hump-shaped), log-logistic, lognormal, Birnbaun-Saunders, and inverse Gaussian models are used for the computational simplicity and popularity among users. When models are inadequate and inappropriate, compound Rayleigh, arctangent, generalized Weibull, and Weibull-Pareto composite models are also used. Out of these models log-logistic (LL) model is frequently used. The log-logistic distribution (LLD) has the advantage of having simple algebraic expressions for its survivor and hazard functions and a closed form for its distribution function. In this paper, we consider gamma distribution as frailty distribution and LLD as baseline distribution for bivariate survival times. The problem of analyzing and estimating parameters of bivariate LLD with shared gamma frailty is of interest and the focus of this paper. We introduce Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in the proposed model. We present a simulation study and two real data examples to compute Bayesian estimates of the parameters and their standard errors and then compare the true values of the parameters with the estimated values for different sample sizes. A search of the literature suggests there is currently no work has been done for bivariate log-logistic regression model with shared gamma frailty using Bayesian approach.