Sparse Estimation in a Correlated Probit Model

Among the goals of statistical genetics is to find associations between genetic data and binary phenotypes, such as heritable diseases. Often, the data are obfuscated by confounders such as age, ethnicity, or population structure. Linear mixed models are linear regression models that correct for confounding by means of correlated label noise; they are widely appreciated in the field of statistical genetics. We generalize this modeling paradigm to binary classification, where we face the problem that marginalizing over the noise leads to an intractable, high-dimensional integral. We present a scalable, approximate inference algorithm that lets us fit the model to high-dimensional data sets. The algorithm selects features based on an $\ell_1$ norm regularizer which are up to 40% less confounded compared to the outcomes of uncorrected feature selection, as we show. The proposed method also outperforms Gaussian process classification and uncorrelated probit regression in terms of prediction performance.