  In clinical trials, principal stratification analysis is commonly employed to
address the issue of truncation by death, where a subject dies before the
outcome can be measured. However, in practice, many survivor outcomes may
remain uncollected or be missing not at random, posing a challenge to standard
principal stratification analyses. In this paper, we explore the
identification, estimation, and bounds of the average treatment effect within a
subpopulation of individuals who would potentially survive under both treatment
and control conditions. We show that the causal parameter of interest can be
identified by introducing a proxy variable that affects the outcome only
through the principal strata, while requiring that the treatment variable does
not directly affect the missingness mechanism. Subsequently, we propose an
approach for estimating causal parameters and derive nonparametric bounds in
cases where identification assumptions are violated. We illustrate the
performance of the proposed method through simulation studies and a real
dataset obtained from a Human Immunodeficiency Virus (HIV) study.
