Average causal effect estimation via instrumental variables: the no simultaneous heterogeneity assumption

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 the 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 (NEM).
Methods: We describe the NO Simultaneous Heterogeneity (NOSH) 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 NOSH using simulations and by re-examining selected published studies.
Results: When NOSH holds, the Wald estimand equals the ACE even if both homogeneity assumptions and NEM (which we demonstrate to be special cases of – and therefore stronger than – NOSH) are violated.
Conclusions: NOSH is sufficient for identifying the ACE using IVs. Since NOSH is weaker than existing assumptions for ACE identification, doing so may be more plausible than previously anticipated.