Approximate marginal likelihoods for shrinkage parameter estimation in penalized logistic regression analysis of case-control data

Inference of associations between disease status and rare exposures is complicated by the finite-sample bias of the maximum likelihood estimator for logistic regression. Penalised likelihood methods are useful for reducing such bias. In this project, we studied penalisation by a family of log-F priors indexed by a shrinkage parameter m. We propose a method for estimating m based on an approximate marginal likelihood obtained by Laplace approximation. Derivatives of the approximate marginal likelihood for m are challenging to compute, and so we explore several derivative-free optimization approaches to obtaining the maximum marginal likelihood estimate. We conduct a simulation study to evaluate the performance of our method under a variety of data-generating scenarios, and applied the method to real data from a genetic association study of Alzheimer's disease.