Abstract
In this paper, we study the performance of Bayesian computational methods to estimate the parameters of a bivariate survival model based on the Ali–Mikhail–Haq copula with marginal distributions given by Weibull distributions. The estimation procedure was based on Monte Carlo Markov Chain (MCMC) algorithms. We present three version of the Metropolis–Hastings algorithm: Independent Metropolis–Hastings (IMH), Random Walk Metropolis (RWM) and Metropolis–Hastings with a natural-candidate generating density (MH). Since the creation of a good candidate generating density in IMH and RWM may be difficult, we also describe how to update a parameter of interest using the slice sampling (SS) method. A simulation study was carried out to compare the performances of the IMH, RWM and SS. A comparison was made using the sample root mean square error as an indicator of performance. Results obtained from the simulations show that the SS algorithm is an effective alternative to the IMH and RWM methods when simulating values from the posterior distribution, especially for small sample sizes. We also applied these methods to a real data set.
Highlights
In survival studies, it is common to observe two or more lifetimes for the same client, patient or equipment
The copula model is useful for modeling this kind of bivariate data. It has been used in several articles, including the following: [1] describes a comparison between bivariate frailty models, and models based on bivariate exponential and Weibull distributions; [2] proposes a copula model to study the association between survival time of individuals infected with HIV and persistence time of infection; [3] models the association of bivariate failure times by copula functions, and investigates two-stage parametric and semi-parametric procedures; and [4] considers a Gaussian copula model and estimates the copula association parameter using a two-stage estimation procedure
We compared the performances of the three algorithms using the effective sample size and the integrated autocorrelation time [14]. Results obtained from these simulations show that the algorithm that applied the slice sampling (SS) algorithm is an effective alternative for standard Monte Carlo Markov Chain (MCMC) methods (IMH and Random Walk Metropolis (RWM))
Summary
It is common to observe two or more lifetimes for the same client, patient or equipment. We apply the Ali–Mikhail–Haq (AMH) copula to model bivariate survival data with random right-censored observations. We compared the performances of the three algorithms using the effective sample size and the integrated autocorrelation time [14] Results obtained from these simulations show that the algorithm that applied the SS algorithm is an effective alternative for standard MCMC methods (IMH and RWM). This data set is related to diabetic retinopathy, described in The Diabetic Retinopathy Study Research Group [15], and is available in the ‘survival’ package [16] of the R software [17] For this case, we compared the performance of the algorithms.
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