Bayesian Variable Selection in Multinomial Probit Models to Identify Molecular Signatures of Disease Stage

Abstract

Here we focus on discrimination problems where the number of predictors substantially exceeds the sample size and we propose a Bayesian variable selection approach to multinomial probit models. Our method makes use of mixture priors and Markov chain Monte Carlo techniques to select sets of variables that differ among the classes. We apply our methodology to a problem in functional genomics using gene expression profiling data. The aim of the analysis is to identify molecular signatures that characterize two different stages of rheumatoid arthritis.

Cross-validation prior choice in Bayesian probit regression with many covariates

“…This covers values of c inducing a lot of regularisation as well as values inducing very little and significantly extends the guideline range of Sha et al (2004) for these data. Applying their guidelines leads to a range of (0.1, 2.27) for the Arthritis dataset and (0.1, 2.26) for the Colon Tumour dataset.…”

“…The latter is typically linked to successful variable selection, which is our main concern in the type of applications considered here. Interestingly, the guideline range for choosing c proposed in Sha et al (2004) covers our preferred value in one of the datasets we examine here, but remains very far from this optimal value in the other 4 .…”

“…In both cases, Bayesian variable selection for the extremes of c (and thus of regularisation) is associated with poorer predictive performance. The guideline range for c suggested by Sha et al (2004) includes the optimal value of c in the case of the Arthritis dataset, but the optimal value of c is well outside this range for the Colon Tumour dataset.…”

“…For example, we may want to find genes that discriminate between two disease states using samples taken from patients in the first disease state (y i = 1) or the second one (y i = 0). Typically, the number of measured gene expressions (covariates), say p, will be much larger than the number of samples, say n. A popular approach to this problem is variable selection in a probit regression model (Sha et al, 2004;Lee et al, 2003) 1 . Usually, it is assumed that the response y can be modelled in terms of a (small) subset of the p covariates.…”

“…in Fernández et al (2001). However, we will follow Sha et al (2004) and Brown and Vannucci (1998) by assuming that α ∼ N(0, h). The prior distribution for the regression coefficients β γ is the so-called ridge prior…”