We provide a formal justification for the validity of the MR design by building a causal model which includes features such as assortative mating, linkage disequilibrium, population stratification and transmission ratio distortion. We then propose an “almost exact” randomization test for MR based on explicitly modelling the distribution of crossovers.

We develop an approach to statistical inference in stochastic optimization problems when both the function to minimized, and the set over which it is minimized, must be estimated empirically. We apply this inference procedure to the problem of selection bias in large population cohorts such as UK Biobank. We propose a sensitivity analysis which is able to flexibly incorporate a wide variety of population-level information, while providing valid statistical inference.

In MR studies, the exposure is often a coarsened approximation to some latent continuous trait. Genetically driven variation in the outcome can exist within categories of the exposure, violating the exclusion restriction. I derive a closed-form expression for the resulting bias and propose a simple correction that can be used with summary-level data to provide MR estimates with interpretable effect sizes.