We investigate  identification of causal parameters in case-control and related studies. The odds ratio in the sample is our main estimand of interest and we articulate its relationship with causal parameters under various scenarios. It turns out that the odds ratio is generally a sharp upper bound for counterfactual relative risk under some monotonicity assumptions, without resorting to strong ignorability, nor to the rare-disease assumption. Further , we propose semparametrically efficient , easy-to-implement, machine-learning-friendly estimators of the aggregated (log) odds ratio by exploiting an explicit form of the efficient influence function. Using our new estimators , we develop methods for causal inference and illustrate the usefulness of our methods by a real-data example .
We investigate partial identification of causal relative and attributable risk---the ratio of two counterfactual proportions and the difference between them---in case-control and case-population studies. The odds ratio is shown to be a sharp upper bound on causal relative risk under the monotone treatment response and monotone treatment selection assumptions, without resorting to strong ignorability, nor to the rare-disease assumption. Sharp bounds on causal attributable risk are also obtained under the same assumptions. Paying special attention to the (conditional) odds ratio , we propose a semiparametrically efficient estimator of the aggregated (log) odds ratio . Further , we develop easy-to-implement causal inference procedures for relative and attributable risk. Finally, we showcase our methodology by applying it to two unique datasets in the literature. We find that attending private school may have little effect on entering a very selective university in Pakistan and that dropping out of school could substantially increase relative and attributable risk of joining a criminal gang in Brazil .
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semparametrically efficient , easy-to-implement, machine-learning-friendly estimators
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by exploiting an explicit form of the efficient influence function. Using our new estimators
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methods for causal inference and illustrate the usefulness of our methods by a real-data example
easy-to-implement causal inference procedures for relative and attributable risk. Finally, we showcase our methodology by applying it to two unique datasets in the literature. We find that attending private school may have little effect on entering a very selective university in Pakistan and that dropping out of school could substantially increase relative and attributable risk of joining a criminal gang in Brazil