Transfer learning-based two-sample Mendelian randomization method for heterogeneous population

Population heterogeneity presents a significant challenge for two-sample Mendelian randomization (MR), often leading to biased estimates of causal effects. This heterogeneity arises when covariate distributions differ across populations, especially when these covariates function both as confounders and effect modifiers. To address this issue, we propose a new method, transfer learning-based Mendelian randomization (TLMR) that leverages observable effect modifiers to transfer predicted exposures from a source population to a target population. This transfer enables causal effect estimation in the target population while properly accounting for population differences. TLMR is developed under minimal modeling assumptions, allowing flexible exposure modeling, and supporting both continuous and binary outcomes. We further extend TLMR to accommodate reverse transfer in the outcome model that broadens its applicability in practical settings. Through extensive simulations, we demonstrate that TLMR yields robust and consistent estimates in heterogeneous populations, outperforming eight widely used MR methods that exhibit substantial estimation bias. Even in homogeneous populations or in the absence of effect modification, TLMR performs comparably to existing approaches. Finally, we systematically evaluate the causal relationship between body mass index and pulmonary function, demonstrating the practical utility and improved accuracy of TLMR in real-world analysis.