Validity and Interpretation of Two-Sample Mendelian Randomization with Binary Traits

Abstract

Two-sample Mendelian randomization (MR) is widely applied to binary exposures and outcomes. Yet standard MR models rely on linear effect assumptions that are difficult to interpret for binary traits. Although liability-based interpretations have been suggested, it remains unclear whether conventional summary-data MR is formally justified in this setting or what causal parameter it identifies. We develop a liability-threshold framework in which binary traits arise from underlying continuous liabilities. We derive explicit relationships between genome-wide association study (GWAS) coefficients obtained from logistic or linear regression on binary traits and marginal genetic associations on the liability scale. Under small genetic effects, typical for complex traits, observed-scale GWAS coefficients are approximately proportional to liability-scale associations. This proportionality implies that standard two-sample MR methods remain statistically coherent for binary traits. MR applied to binary exposures or outcomes estimates a scaled causal effect between underlying liabilities rather than an effect on the observed binary scale. The scaling factor depends primarily on trait prevalence and is directly computable. Simulations and UK Biobank analyses confirm that, after rescaling, MR using binary traits recovers liability-scale causal effects consistent with analyses based on continuous traits. We provide a formal statistical justification for summary-data MR with binary traits and clarify the causal parameter being estimated. These results support routine MR practice for binary exposures and outcomes while enabling coherent interpretation of effect sizes.

Zixuan Wu

Ph.D. student

Jingshu Wang

Assistant Professor in Statistics