  We investigate the bounding problem of causal effects in experimental studies
in which the outcome is truncated by death, meaning that the subject dies
before the outcome can be measured. Causal effects cannot be point identified
without instruments and/or tight parametric assumptions but can be bounded
under mild restrictions. Previous work on partial identification under the
principal stratification framework has primarily focused on the
`always-survivor' subpopulation. In this paper, we present a novel
nonparametric unified framework to provide sharp bounds on causal effects on
discrete and continuous square-integrable outcomes. These bounds are derived on
the `always-survivor', `protected', and `harmed' subpopulations and on the
entire population with/without assumptions of monotonicity and stochastic
dominance. The main idea depends on rewriting the optimization problem in terms
of the integrated tail probability expectation formula using a set of
conditional probability distributions. The proposed procedure allows for
settings with any type and number of covariates, and can be extended to
incorporate average causal effects and complier average causal effects.
Furthermore, we present several simulation studies conducted under various
assumptions as well as the application of the proposed approach to a real
dataset from the National Supported Work Demonstration.
