Principal stratification designs to estimate input data missing due to death

We consider studies of cohorts of individuals after a critical event, such as an injury, with the following characteristics. First, the studies are designed to measure "input" variables, which describe the period before the critical event, and to characterize the distribution of the input variables in the cohort. Second, the studies are designed to measure "output" variables, primarily mortality after the critical event, and to characterize the predictive (conditional) distribution of mortality given the input variables in the cohort. Such studies often possess the complication that the input data are missing for those who die shortly after the critical event because the data collection takes place after the event. Standard methods of dealing with the missing inputs, such as imputation or weighting methods based on an assumption of ignorable missingness, are known to be generally invalid when the missingness of inputs is nonignorable, that is, when the distribution of the inputs is different between those who die and those who live. To address this issue, we propose a novel design that obtains and uses information on an additional key variable-a treatment or externally controlled variable, which if set at its "effective" level, could have prevented the death of those who died. We show that the new design can be used to draw valid inferences for the marginal distribution of inputs in the entire cohort, and for the conditional distribution of mortality given the inputs, also in the entire cohort, even under nonignorable missingness. The crucial framework that we use is principal stratification based on the potential outcomes, here mortality under both levels of treatment. We also show using illustrative preliminary injury data that our approach can reveal results that are more reasonable than the results of standard methods, in relatively dramatic ways. Thus, our approach suggests that the routine collection of data on variables that could be used as possible treatments in such studies of inputs and mortality should become common.