  Background: Interpreting instrumental variable results often requires further
assumptions in addition to the core assumptions of relevance, independence, and
the exclusion restriction. Methods: We assess whether instrument-exposure
additive homogeneity renders the Wald estimand equal to the average derivative
effect (ADE) in the case of a binary instrument and a continuous exposure.
Results: Instrument-exposure additive homogeneity is insufficient for ADE
identification 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.
For a binary exposure, the exposure-outcome effect is necessarily additive
linear, so the homogeneity condition is sufficient. Conclusions: For binary
instruments, instrument-exposure additive homogeneity identifies the ADE if the
exposure is also binary. Otherwise, additional assumptions (such as additive
linearity of the exposure-outcome effect) are required.
