  Instrumental variables (IV) are a commonly used tool to estimate causal
effects from non-randomized data. A prototype of an IV is a randomized trial
with non-compliance where the randomized treatment assignment serves as an IV
for the non-ignorable treatment received. Under a monotonicity assumption, a
valid IV non-parametrically identifies the average treatment effect among a
non-identifiable complier subgroup, whose generalizability is often under
debate. In many studies, there could exist multiple versions of an IV, for
instance, different nudges to take the same treatment in different study sites
in a multi-center clinical trial. These different versions of an IV may result
in different compliance rates and offer a unique opportunity to study IV
estimates' generalizability. In this article, we introduce a novel nested IV
assumption and study identification of the average treatment effect among two
latent subgroups: always-compliers and switchers, who are defined based on the
joint potential treatment received under two versions of a binary IV. We derive
the efficient influence function for the SWitcher Average Treatment Effect
(SWATE) and propose efficient estimators. We then propose formal statistical
tests of the generalizability of IV estimates based on comparing the
conditional average treatment effect among the always-compliers and that among
the switchers under the nested IV framework. We apply the proposed framework
and method to the Prostate, Lung, Colorectal and Ovarian (PLCO) Cancer
Screening Trial and study the causal effect of colorectal cancer screening and
its generalizability.
