Sparse estimation for case-control studies with multiple disease subtypes

The analysis of case-control studies with several disease subtypes is increasingly common, e.g. in cancer epidemiology. For matched designs, a natural strategy is based on a stratified conditional logistic regression model. Then, to account for the potential homogeneity among disease subtypes, we adapt the ideas of data shared lasso, which has been recently proposed for the estimation of stratified regression models. For unmatched designs, we compare two standard methods based on $L_1$-norm penalized multinomial logistic regression. We describe formal connections between these two approaches, from which practical guidance can be derived. We show that one of these approaches, which is based on a symmetric formulation of the multinomial logistic regression model, actually reduces to a data shared lasso version of the other. Consequently, the relative performance of the two approaches critically depends on the level of homogeneity that exists among disease subtypes: more precisely, when homogeneity is moderate to high, the non-symmetric formulation with controls as the reference is not recommended. Empirical results obtained from synthetic data are presented, which confirm the benefit of properly accounting for potential homogeneity under both matched and unmatched designs, in terms of estimation and prediction accuracy, variable selection and identification of heterogeneities. We also present preliminary results from the analysis of a case-control study nested within the EPIC (European Prospective Investigation into Cancer and nutrition) cohort, where the objective is to identify metabolites associated with the occurrence of subtypes of breast cancer.