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

TL;DR

This paper investigates the theoretical properties of L1 penalized likelihood estimators in case-control studies, demonstrating their consistency and subset selection capabilities under the sampling scheme, with a novel tuning parameter approach.

Contribution

It extends the theoretical understanding of L1 penalized likelihood estimators in case-control studies, including their equivalence and consistency properties, with a new data-driven tuning method.

Findings

L1 penalized estimators coincide for retrospective and prospective likelihoods.

Establishes sup-norm consistency of the odds ratio after model selection.

Proposes a computationally efficient tuning parameter selection method.

Abstract

Dimension reduction and variable selection are performed routinely in case-control studies, but the literature on the theoretical aspects of the resulting estimates is scarce. We bring our contribution to this literature by studying estimators obtained via L1 penalized likelihood optimization. We show that the optimizers of the L1 penalized retrospective likelihood coincide with the optimizers of the L1 penalized prospective likelihood. This extends the results of Prentice and Pyke (1979), obtained for non-regularized likelihoods. We establish both the sup-norm consistency of the odds ratio, after model selection, and the consistency of subset selection of our estimators. The novelty of our theoretical results consists in the study of these properties under the case-control sampling scheme. Our results hold for selection performed over a large collection of candidate variables, with cardinality allowed to depend and be greater than the sample size. We complement our theoretical results with a novel approach of determining data driven tuning parameters, based on the bisection method. The resulting procedure offers significant computational savings when compared with grid search based methods. All our numerical experiments support strongly our theoretical findings.