Selecting invalid instruments to improve Mendelian randomization with two-sample summary data

Mendelian randomization (MR) is a widely-used method to estimate the causal relationship between a risk factor and disease. A fundamental part of any MR analysis is to choose appropriate genetic variants as instrumental variables. Genome-wide association studies often reveal that hundreds of genetic variants may be robustly associated with a risk factor, but in some situations investigators may believe that only a smaller subset of these variants are valid instruments. Nevertheless, using the full set of instruments could lead to biased but more precise estimates, and therefore in terms of mean squared error it may be unclear which set of instruments is optimal. For this purpose, we consider a method for "focused" instrument selection whereby genetic variants are selected to minimise the estimated asymptotic mean squared error of causal effect estimates. In a setting of many weak and locally invalid instruments, we consider a novel strategy to construct confidence intervals for post-selection focused estimators which guards against the worst case loss in asymptotic coverage. In empirical applications to: (i) validate lipid drug targets; and (ii) investigate vitamin D effects on a wide range of outcomes, our findings suggest that the optimal selection of instruments does not involve only a small number of biologically-justified valid instruments, but also many potentially invalid instruments.