A unified framework for bounding causal effects on the always-survivor and other populations

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.