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We consider the problem of variable selection in high-dimensional settings with missing observations among the covariates. To address this relatively understudied problem, we propose a new synergistic procedure—adaptive Bayesian SLOPE with missing values—which effectively combines SLOPE (sorted  l1  regularization) with the spike-and-slab LASSO (SSL) and is accompanied by an efficient stochastic approximation of expected maximization (SAEM) algorithm to handle missing data. Similarly as in SSL, the regression coefficients are regarded as arising from a hierarchical model consisting of two groups: the spike for the inactive and the slab for the active. However, instead of assigning independent spike and slab Laplace priors for each covariate, here we deploy a joint SLOPE “spike-and-slab” prior which takes into account the ordering of coefficient magnitudes in order to control for false discoveries. We position our approach within a Bayesian framework which allows for simultaneous variable selection and parameter estimation while handling missing data. Through extensive simulations, we demonstrate satisfactory performance in terms of power, false discovery rate (FDR) and estimation bias under a wide range of scenarios including complete data and existence of missingness. Finally, we analyze a real dataset consisting of patients from Paris hospitals who underwent severe trauma, where we show competitive performance in predicting platelet levels. Our methodology has been implemented in C++ and wrapped into open source R programs for public use. Supplemental files for this article are available online.

Adaptive Bayesian SLOPE: Model Selection With Incomplete Data