---

Titolo: Causal Bounds for Outcome-Dependent Sampling in Observational Studies

Abstract: Outcome-dependent sampling designs are common in many different
scientific fields, including epidemiology, ecology, and economics. As
with all observational studies, such designs often suffer from
unmeasured confounding, which generally precludes the nonparametric
identification of causal effects. Nonparametric bounds can provide a
way to narrow the range of possible values for a nonidentifiable
causal effect without making additional untestable assumptions. The
nonparametric bounds literature has almost exclusively focused on
settings with random sampling, and the bounds have often been derived
with a particular linear programming method. We derive novel bounds
for the causal risk difference, often referred to as the average
treatment effect, in six settings with outcome-dependent sampling and
unmeasured confounding for a binary outcome and exposure. Our
derivations of the bounds illustrate two approaches that may be
applicable in other settings where the bounding problem cannot be
directly stated as a system of linear constraints.