# Sparse estimation for case-control studies with multiple subtypes of cases

## TL;DR

This paper compares methods for analyzing case-control studies with multiple subtypes, proposing adaptations of stratified logistic regression and data shared lasso, and demonstrating their effectiveness through simulations and real data.

## Contribution

It introduces a novel approach combining stratified conditional logistic regression with data shared lasso for subtype analysis in case-control studies.

## Key findings

- Data shared lasso improves estimation when subtypes are homogeneous.
- Symmetric multinomial logistic regression reduces to data shared lasso under certain conditions.
- Proper modeling of subtype homogeneity enhances analysis accuracy.

## Abstract

The analysis of case-control studies with several subtypes of cases is increasingly common, e.g. in cancer epidemiology. For matched designs, we show that a natural strategy is based on a stratified conditional logistic regression model. Then, to account for the potential homogeneity among the subtypes of cases, we adapt the ideas of data shared lasso, which has been recently proposed for the estimation of regression models in a stratified setting. For unmatched designs, we compare two standard methods based on L1-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 the subtypes of cases: 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. We also present preliminary results from the analysis a case-control study nested within the EPIC cohort, where the objective is to identify metabolites associated with the occurrence of subtypes of breast cancer.