In the survival analysis, when there is a cure fraction and the occurrence times of events are correlated, the cure frailty model is utilized. The main objective is to propose a method of analysis for two types of correlated frailty in the non-mixture cured model in order to separate the individual and shared heterogeneity between subjects. The cure models with correlated frailty and promotion time are considered. In both models, the likelihood function are based on piecewise exponential distribution for hazard function. To estimate the parameters, hierarchical Bayesian modeling is employed. Due to non-closed forms of the posteriors, they are estimated by MCMC algorithms. The Cox correlated frailty model is used as a benchmark and models are compared by DIC Criterion . The results show the superiority of cure models with correlated frailty.

Part of the recent literature on the evaluation of surrogate endpoints is started by a definition of validity in terms of both trial-level and individual-level association between a potential surrogate and a true endpoint. In another part, we review the main considerable statistical methods being proposed for the evaluation of a biomarker as surrogate endpoints, which have developed and consider how the validation process might be arranged within the regulatory and practical constraints evaluation. In the present work, we propose a new. Bayesian approach to evaluate individual level surrogacy. Deferent variations to prior distributions were implemented for responses with binomial distribution. Then these methods are compared in a simulation study. Finally, we apply and compare the previous and new methodology using a clinical study.

Sometimes it is impossible to directly measure the effect of intervention (medicine or therapeutic methods) in medical researches. That is because of high costs, long time, the aggressiveness of therapeutic methods, lack of clinical responses, and etc. In such cases, the effect of intervention on surrogate variables is measured. Many statistical studies have been accomplished for measuring the validity of surrogates and introducing a criterion for testing. The first criterion was established based on hypothesis testing. Other criterions were introduced over time. Then by using the classic methods, the Likelihood Ratio Factor was introduced. After that, the Bayesian Likelihood Ratio Factor developed and published. This article aims to introduce the Bayesian Likelihood Ratio Factor based on time dependent data. The illness under study is lung disease in victims of chemical weapons. The surrogate therapy method uses the forced expiratory volume at fist second.

Graphical mixture models provide a powerful tool to visually depict the conditional independence relationships between high-dimensional heterogeneous data. In the study of these models, the distribution of the mixture components is mostly considered multivariate normal with different covariance matrices. The resulting model is known as the Gaussian graphical mixture model. The nonparanormal graphical mixture model has been introduced by replacing the limiting normal assumption with a semiparametric Gaussian copula, which extends the nonparanormal graphical model and mixture models. This study proposes clustering based on the nonparanormal graphical mixture model with two forms of $ell_1$ penalty function (conventional and unconventional), and its performance is compared with the clustering method based on the Gaussian graphical mixture model. The results of the simulation study on normal and nonparanormal datasets in ideal and noisy settings, as well as the application to breast cancer data set, showed that the combination of the nonparanormal graphical mixture model and the penalty term depending on the mixing proportions, both in terms of cluster reconstruction and parameters estimation, is more accurate than the other model-based clustering methods.