International Journal of Statistical Distributions and Applications 2019; 5(2): 38-45 http://www.sciencepublishinggroup.com/j/ijsda doi: 10.11648/j.ijsd.20190502.13 ISSN: 2472-3487 (Print); ISSN: 2472-3509 (Online) On Bayesian Estimation of Dirichlet Process Lognormal Mixture Models and Comparison of Treatments in Censoring Henry Ondicho Nyambega 1, * , George Otieno Orwa 2 1 Department of Mathematics, Kisii University, Kisii, Kenya 2 Department of Statistics and Actuarial Science, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya Email address: * Corresponding author To cite this article: Henry Ondicho Nyambega, George Otieno Orwa. On Bayesian Estimation of Dirichlet Process Lognormal Mixture Models and Comparison of Treatments in Censoring. International Journal of Statistical Distributions and Applications. Vol. 5, No. 2, 2019, pp. 38-45. doi: 10.11648/j.ijsd.20190502.13 Received: June 13, 2019; Accepted: July 12, 2019; Published: July 30, 2019 Abstract: One current interest in medical research is the comparison of treatments in the analysis of survival times of patients. This is particularly problematic, especially for censored data, and when these data consists of several groups, where each group has distinct properties and characteristics but belong to the same distribution. There are various modeling schemes that have been contemplated to overcome these complexities inherent in the data. One such possibility is the Bayesian approach which integrates prior knowledge in analysis. In this paper, we focus on the use of Bayesian lognormal mixture model (MLNM) with related Dirichlet process (DP) prior distribution for estimating patient survival. The advances in the Bayesian paradigm have considerably bolstered the development and application of mixture modelling methodology in the field of survival analysis. The proposed MLN model is compared with the conventional parametric lognormal and the nonparametric Kaplan Meier (K-M) models used to estimate survival to establish model robustness. A simulation study that investigates the impact of censoring on these models is also described. Real data from past research is used to show the resulting Dirichlet process mixture model’s robustness in the comparison of censored treatment. The results indicate that the proposed lognormal mixtures provide a better fit to complex data. Further, the MLN models are able to estimate various survival distributions and therefore appropriate to compare treatments. Clinicians will find these models useful especially when confronted with the obstacle of choosing a suitable therapy for a disease. Keywords: Bayesian, Lognormal, Mixture Models, Treatment Comparison, Win BUGS 1. Introduction Most medical data are censored and/or arise from several homogenous subgroups relating to one or several characteristics, for example when different treatments are administered to patients. There is therefore an increasing need for efficient estimation of patient survival through comparison of treatments. Several modeling procedures have been postulated in literature. One such scheme is the Bayesian framework that incorporates prior information regarding the data without compromising the accuracy of estimates [2]. And as reported by [5], models based on Bayesian nonparametrics present more flexibility leading to ease of computation. Bayesian nonparametric models however suffer from the possible impediments of inference, particularly due to the challenges associated with prior choice. The Dirichlet Process (DP) is the prior of choice for Bayesian non-parametric models. However the DP is inadequate in these settings since the posterior is not DP but a mixture of DP. Mixture DP distributions offer the options of discreteness and flexibility especially since they consider data as represented by weighted sum of distributions, with each distribution characterized by a unique parameter set representing a subspace of the population. Given the