Causal Inference under Outcome-Based Sampling with Monotonicity Assumptions

We study causal inference under case-control and case-population sampling. For this purpose, we focus on the binary-outcome and binary-treatment case, where the parameters of interest are causal relative and attributable risk defined via the potential outcome framework. It is shown that strong ignorability is not always as powerful as it is under random sampling and that certain monotonicity assumptions yield comparable results in terms of sharp identified intervals. Specifically, the usual odds ratio is shown to be a sharp identified upper bound on causal relative risk under the monotone treatment response and monotone treatment selection assumptions. We then discuss averaging the conditional (log) odds ratio and propose an algorithm for semiparametrically efficient estimation when averaging is based only on the (conditional) distributions of the covariates that are identified in the data. We also offer algorithms for causal inference if the true population distribution of the covariates is desirable for aggregation. We show the usefulness of our approach by studying two empirical examples from social sciences: the benefit of attending private school for entering a prestigious university in Pakistan and the causal relationship between staying in school and getting involved with drug-trafficking gangs in Brazil.