Estimation of total mediation effect for a binary trait in a case-control study for high-dimensional omics mediators

Mediation analysis helps uncover how exposures impact outcomes through intermediate variables. Traditional mean-based total mediation effect measures may suffer from the cancellation of opposite component-wise effects, and existing methods often lack the power to capture weak effects in high-dimensional mediators. Additionally, most high-dimensional mediation analysis methods have focused on continuous outcomes, with limited attention to binary outcomes, particularly in case-control studies. To fill this gap, we propose an R2 total mediation effect measure within the liability framework that offers a clear and intuitive causal interpretation, provides additional insights beyond the mean-based measures, and is invariant to disease prevalence. We develop a cross-fitted, modified Haseman-Elston regression-based estimation procedure tailored for mediation analysis in case-control studies, which can also be applied to cohort studies. Our estimator remains consistent in the presence of non-mediators and weak effects, as demonstrated in extensive simulations. Theoretical justification for consistency is provided under mild conditions and without requiring exact mediator selection. In a case-control substudy of the Womens Health Initiative involving 2150 individuals, we found that many metabolites were mediators with weak effects in the path from BMI to coronary heart disease, and we estimated that 89% (95% CI: 57%-100%) of the BMI-explained variation in underlying CHD liability is mediated by the measured metabolomics. The proposed estimation procedure is implemented in the R package "r2MedCausal", available on GitHub.