Approximation of bias and mean-squared error in two-sample Mendelian randomization analyses

Mendelian randomization (MR) is a type of instrumental variable (IV) analysis that uses genetic variants as IVs for a risk factor to study its causal effect on an outcome. Extensive investigations on the performance of IV analysis procedures, such as the one based on the two-stage least squares (2SLS) procedure, have been conducted under the one-sample scenario, where measures on IVs, the risk factor, and the outcome are assumed to be available for each study participant. Recent MR analysis usually is performed with data from two independent or partially overlapping genetic association studies (two-sample setting), with one providing information on the association between the IVs and the outcome, and the other on the association between the IVs and the risk factor. We investigate the performance of 2SLS in the two-sample-based MR when the IVs are weakly associated with the risk factor. We derive closed form formulas for the bias and mean squared error of the 2SLS estimate and verify them with numeric simulations under realistic circumstances. Using these analytic formulas, we can study the pros and cons of conducting MR analysis under one-sample and two-sample settings and assess the impact of having overlapping samples. We also propose and validate a bias-corrected estimator for the causal effect.