Causal Inference Under Outcome-Based Sampling with Monotonicity Assumptions

Abstract

We study causal inference under case-control and case-population sampling. Specifically, we focus on the binary-outcome and binary-treatment case, where the parameters of interest are causal relative and attributable risks 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 offer algorithms for inference on the causal parameters that are aggregated over the true population distribution of the covariates. We show the usefulness of our approach by studying three empirical examples: the benefit of attending private school for entering a prestigious university in Pakistan; the relationship between staying in school and getting involved with drug-trafficking gangs in Brazil; and the link between physicians’ hours and size of the group practice in the United States.

Keywords:

• Attributable risk
• Odds ratio
• Partial identification
• Relative risk

Acknowledgments

The authors would like to thank the editor, an associate editor, two anonymous referees, Guido Imbens, Chuck Manski, Francesca Molinari and seminar participants at Cemmap, Oxford, and Penn State for helpful comments and Leandro Carvalho and Rodrigo Soares for sharing their dataset and help.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported in part by the European Research Council (ERC-2014-CoG-646917-ROMIA) and by the UK Economic and Social Research Council (ESRC) through research grant (ES/P008909/1) to the CeMMAP.