Effects attributable to treatment: Inference in experiments and observational studies with a discrete pivot

In randomisation and permutation inference, pivotal arguments remove the hypothesised treatment effect, thereby basing inferences on the null distribution in which the treatment has no effect. This is common, for instance, with additive treatment effects. The current paper uses ‘attributable effects’ to expand substantially the scope of pivotal arguments. Attributable effects are defined for three cases, namely the 2×2 contingency table, displacement effects and the Mann–Whitney–Wilcoxon statistic, and in each case removing an appropriate attributable effect restores the familiar null randomisation distribution of the associated statistic, yielding exact inferences. The procedure extends immediately for use in sensitivity analysis in nonrandomised observational studies.