Two-sample Mendelian randomization (MR) is a widely used methodology in epidemiology. In two-sample MR, summary data (typically regression coefficients and standard errors) quantifying the association between multiple genetic variants and the exposure and the outcome are used in an instrumental variable framework to estimate the causal effect of the exposure on the outcome. Most two-sample MR methods assume that the association between each genetic variant and the exposure, as well as the causal effect of the exposure on the outcome, are constant on the additive scale. These assumptions are useful because they imply that, if all genetic variants were valid instruments, they would all estimate the same causal parameter - namely, the constant causal effect. We call this condition summary-level homogeneity. In this paper, we show that summary-level homogeneity follows from the following conditions: the causal effect is linear on the additive scale within population strata, but is not necessarily constant across strata; and heterogeneity in the causal effect is uncorrelated with heterogeneity in the association between each genetic variant and the exposure. Under these conditions, typical two-sample MR methods can be interpreted as estimators of the average causal effect. These results clarify that point-identifying assumptions required for two-sample MR methods are weaker than previously anticipated, making them more plausible and clarifying the interpretation of the resulting estimates if those assumptions hold.

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