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Dimension reduction and variable selection are performed routinely incase-control studies, but the literature on the theoretical aspects of theresulting estimates is scarce. We bring our contribution to this literature bystudying estimators obtained via L1 penalized likelihood optimization. We showthat the optimizers of the L1 penalized retrospective likelihood coincide withthe optimizers of the L1 penalized prospective likelihood. This extends theresults of Prentice and Pyke (1979), obtained for non-regularized likelihoods.We establish both the sup-norm consistency of the odds ratio, after modelselection, and the consistency of subset selection of our estimators. Thenovelty of our theoretical results consists in the study of these propertiesunder the case-control sampling scheme. Our results hold for selectionperformed over a large collection of candidate variables, with cardinalityallowed to depend and be greater than the sample size. We complement ourtheoretical results with a novel approach of determining data driven tuningparameters, based on the bisection method. The resulting procedure offerssignificant computational savings when compared with grid search based methods.All our numerical experiments support strongly our theoretical findings.

Dimension reduction and variable selection in case control studies via regularized likelihood optimization