Recalibrating Mendelian randomization under winner's curse, sample structure and polygenicity

Recently, Hu et al. (2024) conducted a benchmarking study showing that most existing Mendelian randomization (MR) methods exhibit substantial bias and inflated type-I error rates in real data. They attributed these failures to two largely neglected sources of bias: winner's curse and polygenicity-induced bias. Although a few methods have been developed to address one or both of these issues, existing approaches either do not fully account for both biases or are restricted to the univariable setting. In this paper, we propose a multivariable Rao-Blackwellization that corrects winner's curse while accounting for polygenicity and sample structure in a unified framework. Unlike univariable Rao-Blackwellization, where instrument selection yields a truncated normal statistic amenable to a Mills-ratio correction, multivariable Rao-Blackwellization conditions on a noncentral $\chi^2$ statistic, for which no analogous correction is available. We derive closed-form conditional moments under this instrument selection model and use them to construct bias-corrected summary statistics that can be integrated into a wide range of existing MR methods. Simulations and real data analyses show that, when combined with methods such as MR-cML and MR-BEE, the proposed correction substantially improves type-I error control and yields more robust inference.