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The probability of benefit is a valuable and important measure of treatmenteffect, which has advantages over the average treatment effect. Particularlyfor an ordinal outcome, it has a better interpretation and can make apparentdifferent aspects of the treatment impact. Unfortunately, this measure, andvariations of it, are not identifiable even in randomized trials with perfectcompliance. There is, for this reason, a long literature on nonparametricbounds for unidentifiable measures of benefit. These have primarily focused onperfect randomized trial settings and one or two specific estimands. We expandthese bounds to observational settings with unmeasured confounders andimperfect randomized trials for all three estimands considered in theliterature: the probability of benefit, the probability of no harm, and therelative treatment effect.

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