  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.
