Sharp symbolic nonparametric bounds for measures of benefit in observational and imperfect randomized studies with ordinal outcomes

The probability of benefit is a valuable and important measure of treatment effect, which has advantages over the average treatment effect. Particularly for an ordinal outcome, it has a better interpretation and can make apparent different aspects of the treatment impact. Unfortunately, this measure, and variations of it, are not identifiable even in randomized trials with perfect compliance. There is, for this reason, a long literature on nonparametric bounds for unidentifiable measures of benefit. These have primarily focused on perfect randomized trial settings and one or two specific estimands. We expand these bounds to observational settings with unmeasured confounders and imperfect randomized trials for all three estimands considered in the literature: the probability of benefit, the probability of no harm, and the relative treatment effect.