  In many empirical settings, directly observing a treatment variable may be
infeasible although an error-prone surrogate measurement of the latter will
often be available. Causal inference based solely on the observed surrogate
measurement of the hidden treatment may be particularly challenging without an
additional assumption or auxiliary data. To address this issue, we propose a
method that carefully incorporates the surrogate measurement together with a
proxy of the hidden treatment to identify its causal effect on any scale for
which identification would in principle be feasible had contrary to fact the
treatment been observed error-free. Beyond identification, we provide general
semiparametric theory for causal effects identified using our approach, and we
derive a large class of semiparametric estimators with an appealing multiple
robustness property. A significant obstacle to our approach is the estimation
of nuisance functions involving the hidden treatment, which prevents the direct
application of standard machine learning algorithms. To resolve this, we
introduce a novel semiparametric EM algorithm, thus adding a practical
dimension to our theoretical contributions. This methodology can be adapted to
analyze a large class of causal parameters in the proposed hidden treatment
model, including the population average treatment effect, the effect of
treatment on the treated, quantile treatment effects, and causal effects under
marginal structural models. We examine the finite-sample performance of our
method using simulations and an application which aims to estimate the causal
effect of Alzheimer's disease on hippocampal volume using data from the
Alzheimer's Disease Neuroimaging Initiative.
