Average Causal Effect Estimation Via Instrumental Variables: the No Simultaneous Heterogeneity Assumption

Abstract

Background: 

Instrumental variables (IVs) can be used to provide evidence as to whether a treatment $X$ has a causal effect on an outcome $Y$. Even if the instrument $Z$ satisfies the three core IV assumptions of relevance, independence, and exclusion restriction, further assumptions are required to identify the average causal effect (ACE) of $X$ on $Y$. Sufficient assumptions for this include homogeneity in the causal effect of $X$ on $Y$; homogeneity in the association of $Z$ with $X$; and no effect modification.

Methods: 

We describe the no simultaneous heterogeneity assumption, which requires the heterogeneity in the $X$-$Y$ causal effect to be mean independent of (i.e., uncorrelated with) both $Z$ and heterogeneity in the $Z$-$X$ association. This happens, for example, if there are no common modifiers of the $X$-$Y$ effect and the $Z$-$X$ association, and the $X$-$Y$ effect is additive linear. We illustrate the assumption of no simultaneous heterogeneity using simulations and by re-examining selected published studies.

Results: 

Under no simultaneous heterogeneity, the Wald estimand equals the ACE even if both homogeneity assumptions and no effect modification (which we demonstrate to be special cases of—and therefore stronger than—no simultaneous heterogeneity) are violated.

Conclusions: 

The assumption of no simultaneous heterogeneity is sufficient for identifying the ACE using IVs. Since this assumption is weaker than existing assumptions for ACE identification, doing so may be more plausible than previously anticipated.