Estimation and Inference Under Simple Order Restrictions: Hierarchical Bayesian Approach

Abstract

In Chap. 13, we focus on hierarchical Bayesian modeling of dose-response microarray data. The materials presented in the first part of the chapter are closely related to the classification procedure discussed in Chap. 10 in the sense that we fit several order-restricted models for each gene. However, in contrast with Chap. 10, we do not aim to select the model with the best goodness-of-fit but to test the null hypothesis of no dose effect. We formulate order-restricted hierarchical Bayesian model for dose-response data and present gene specific examples to illustrate the estimation procedures. Within the hierarchical Bayesian framework one of the major challenges is related to the question of how to perform Bayesian inference and in particular how to adjust for multiplicity. We discuss the Bayesian variable selection (BVS) method (O‘Hara et al., Bayesian Anal 4(1):85–118, 2009) which we use in order to calculate the posterior probability of a specific model given the data and the model parameters. Following Newton et al. (Biostatistics 5(2):155–176, 2004; Hierarchical mixture models for expression profiles. In Do KM, Muller P, Vannucci M (eds) Bayesian Inference for gene expression and proteomics. Cambridge University Press, Cambridge, 2007), we use the posterior probability of the null model to control for multiplicity using the direct posterior probability method for multiplicity adjustment.