Chris Spencer , Gavin Band and Matti Pirinen

where Y is the observed data, θ is the vector of the model parameters, p(Y |θ) is the likelihood function for the parameters and p(θ|M) is the prior distribution of the parameters under the model M . We use a logistic regression likelihood and treat the case-control status as the data Y . The parameters θ include the intercept, coe cients of any covariates and the allelic SNP e ect β, all measured on the log-odds scale. To simplify the computations, we follow the approach of Wake eld [9, 10]. Within each case-control collection, we approximate the logistic regression likelihood (up to a multiplicative constant) by a multivariate normal density function f(θ; θ̂, V̂θ), centered at the logistic regression maximum likelihood (ML) estimate θ̂, and having the covariance matrix V̂θ which is the inverse of the observed information matrix at θ̂. We use at priors for other parameters than β. Then the approximate Bayes factor (ABF) for association in the single study reduces to a ratio of marginal likelihood involving only β: ́ f(β; β̂, SE β)p(β|M1) dβ ́ f(β; β̂, SE2 β)p(β|M0) dβ , (2)