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

TL;DR

This paper demonstrates that additive homogeneity in instrument-exposure relationships is not enough to identify the average derivative effect when using a binary instrument and a continuous, non-linear exposure.

Contribution

It clarifies the limitations of additive homogeneity assumptions for causal effect identification with binary instruments and continuous exposures.

Findings

Additive homogeneity is insufficient for ADE identification with non-linear effects.

Homogeneity suffices only for binary exposures and instruments.

Additional assumptions are necessary for continuous exposures with binary instruments.

Abstract

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.