Supplementary Materials for “A Composite Likelihood Approach to Latent Multivariate Gaussian Modeling of SNP Data with Application to Genetic Association Testing”

We first introduce a multivariate probit model for ordinal responses as implemented in R package mprobit. As before, we have n independent observations (Yi, X ′ i) with Yi = 0 or 1 as disease status andX ′ i = (Xi1, ..., Xik) as genotypes at k SNPs for subject i, i = 1, ..., n. In addition, we have some possible covariates A. We assume that there is a Latent Gaussian variable Z ′ i = (Zi1, ..., Zik) for each Xi: Zi ∼ MV N(Aα,R) with R = (rjl) as a k × k correlation matrix. There exists two constants c1 and c2 such that Pr(Xij = 0) = Pr(Zij ≤ c1) and Pr(Xij = 1) = Pr(c1 < Zij ≤ c2). Note that the cut-off parameters C = (c1, c2) ′ are common for and independent of the SNPs. For simplicity of notation, we define intervals Tij = (−∞, c1], (c1, c2] and (c2,∞) for Xij = 0, 1 and 2, respectively. The (full) likelihood is