Homogeneity in the instrument-treatment association is not sufficient for the Wald estimand to equal the average causal effect when the exposure is continuous

Background: Interpreting results of instrumental variable (IV) analysis as well-defined causal estimands requires further assumptions in addition to the core IV assumptions of relevance, independence, and the exclusion restriction. One such assumption is additive homogeneity in the instrument-exposure association, which is postulated to render the conventional Wald estimand as equal to the average causal effect (ACE). Methods: We used theoretical arguments and an illustrative example to assess whether instrument-exposure additive homogeneity identifies the ACE when the exposure is continuous and the instrument is either binary or continuous. Results: Instrument-exposure additive homogeneity is insufficient to identify the ACE when the instrument is binary, the exposure is continuous and the effect of the exposure on the outcome is non-linear on the additive scale. If the exposure is binary, the exposure-outcome effect is necessarily additive linear, so the homogeneity condition is sufficient. For a continuous instrument, the instrument-exposure additive homogeneity is sufficient regardless of the exposure-outcome effect being linear or not. Conclusions: For binary instruments, additive homogeneity in the instrument-exposure association identifies the ACE if the exposure is also binary. Otherwise, additional assumptions (such as additive linearity of the exposure-outcome effect) are required.