Causal bounds for outcome-dependent sampling in observational studies
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Outcome-dependent sampling designs are common in many different scientific fields including ecology, economics, and medicine. 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 assumptions. The nonparametric bounds literature has almost exclusively focused on settings with random sampling and applications of the linear programming approach. We derive novel bounds for the causal risk difference in six settings with outcome-dependent sampling and unmeasured confounding. Our derivations of the bounds illustrate two general approaches that can be applied in other settings where the bounding problem cannot be directly stated as a system of linear constraints. We illustrate our derived bounds in a real data example involving the effect of vitamin D concentration on mortality.
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