This thesis considers three problems in causal inference. First, for the censoring by death problem, we propose a set of ranked average score assumptions making use of survival information both before and after the measurement of a non-mortality outcome to tighten the bounds on the survivor average causal effect (SACE) obtained in the previous literature that utilized survival information only before the measurement. We apply our method to a randomized trial study of the effect of mechanical ventilation with lower tidal volume vs. traditional tidal volume for acute lung injury patients. Our bounds on the SACE are much shorter than the bounds obtained using only the survival information before the measurement of the non-mortality outcome. Second, for the IV method with nonignorable missing covariates problem, we develop a method to estimate the causal effect of a treatment in observational studies using an IV when there are nonignorable missing covariates, i.e., missingness depending on the partially observed compliance class besides the fully observed outcome, covariates and IV. We apply our method to a motivating study in neonatal care to study the effectiveness of high level compared to low level NICUs. Third, besides the association with the treatment, there are two key assumptions for the IV to be valid: (i) the IV is essentially random conditioning on observed covariates, (ii) the IV affects outcomes only by altering the treatment, the so-called ``exclusion restriction". These two assumptions are often said to be untestable; however, that is untrue if testable means checking the compatibility of assumptions with other things we think we know. A test of this sort may result in an aporia. We discuss this subject in the context of our on-going study of the effects of delivery by cesarean section on the survival of extremely premature infants of 23-24 weeks gestational age.
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