  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.
