Consider data from a Mendelian randomization study on J genetic variants G1,...,GJ, a continuous exposure X and a continuous outcome Y. All confounding variables are subsumed into a single variable U. We initially assume that all genetic variants are valid IVs, and further assume that all the relationships between variables in Figure 1 are linear without heterogeneity or effect modification: X|Gj=γ0+γjGj+εXjY|Gj=Γ0+ΓjGj+εYj.Assumption IV1 tells us that all variants are associated with the exposure, so γj≠0 for all j. Assumptions IV2 and IV3 tell us that the genetic associations with the outcome Γj are equal to the genetic associations with the exposure γj multiplied by the causal effect of the exposure on the outcome β: so Γj=βγj. The error terms εXj and εYj are assumed to be normally distributed and contain contributions from the confounder U and all genetic variants except Gj. In a one‐sample setting, the exposure and outcome data are collected on the same individuals, in which case εXj and εYj are correlated. If exposure and outcome data are collected on different sets of individuals (known as two‐sample Mendelian randomization
