  Noncompliance and missing data often occur in randomized trials, which
complicate the inference of causal effects. When both noncompliance and missing
data are present, previous papers proposed moment and maximum likelihood
estimators for binary and normally distributed continuous outcomes under the
latent ignorable missing data mechanism. However, the latent ignorable missing
data mechanism may be violated in practice, because the missing data mechanism
may depend directly on the missing outcome itself. Under noncompliance and an
outcome-dependent nonignorable missing data mechanism, previous studies showed
the identifiability of complier average causal effect for discrete outcomes. In
this paper, we study the semiparametric identifiability and estimation of
complier average causal effect in randomized clinical trials with both
all-or-none noncompliance and the outcome-dependent nonignorable missing
continuous outcomes, and propose a two-step maximum likelihood estimator in
order to eliminate the infinite dimensional nuisance parameter. Our method does
not need to specify a parametric form for the missing data mechanism. We also
evaluate the finite sample property of our method via extensive simulation
studies and sensitivity analysis, with an application to a double-blinded
psychiatric clinical trial.
