Testing and correcting for weak and pleiotropic instruments in two-sample multivariable Mendelian randomization

Multivariable Mendelian randomization (MVMR) is a form of instrumental variable analysis which estimates the direct effect of multiple exposures on an outcome using genetic variants as instruments. Mendelian randomization and MVMR are frequently conducted using two-sample summary data where the association of the genetic variants with the exposures and outcome are obtained from separate samples. If the genetic variants are only weakly associated with the exposures either individually or conditionally, given the other exposures in the model, then standard inverse variance weighting will yield biased estimates for the effect of each exposure. Here, we develop a two-sample conditional F-statistic to test whether the genetic variants strongly predict each exposure conditional on the other exposures included in a MVMR model. We show formally that this test is equivalent to the individual level data conditional F-statistic, indicating that conventional rule-of-thumb critical values of F> 10, can be used to test for weak instruments. We then demonstrate how reliable estimates of the causal effect of each exposure on the outcome can be obtained in the presence of weak instruments and pleiotropy, by repurposing a commonly used heterogeneity Q-statistic as an estimating equation. Furthermore, the minimized value of this Q-statistic yields an exact test for heterogeneity due to pleiotropy. We illustrate our methods with an application to estimate the causal effect of blood lipid fractions on age-related macular degeneration.