  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.
