Sparse Estimation in a Correlated Probit Model

Linear Mixed Models (LMMs) are important tools in statistical genetics. When used for feature selection, they allow practitioners to find a sparse set of genetic traits that best predict a continuous phenotype of interest, while simultaneously correcting for various confounding factors such as age, ethnicity and population structure. Formulated as models for linear regression, LMMs have been restricted to continuous phenotypes. Based on the idea of sparse probit regression with correlated noise (correlated probit regression), we generalize the LMM modeling paradigm to binary phenotypes. We thereby face the problem that marginalizing over the noise leads to an intractable, high-dimensional integral, but present a scalable approximate inference algorithm that lets us fit the model to high-dimensional data sets. We show on three real-world examples from different domains that in the setup of binary labels, our algorithm leads to better prediction accuracies and also selects features which show less correlation with the confounding factors.