Accommodating stochastic departures from percentile invariance in causal models

Consider a clinical trial in which participants are randomized to a single-dose treatment or a placebo control and assume that the adherence level is accurately recorded. If the treatment is effective, then good adherers in the treatment group should do better than poor ad- herers because they received more drug; the treatment group data follow a dose–response curve. But, good adherers to the placebo often do better than poor adherers, so the observed adherence–response in the treatment group cannot be completely attributed to the treatment. Efron and Feldman proposed an adjustment to the observed adherence–response in the treatment group by using the adherence–response in the control group. It relies on a percentile invariance assumption under which each participantʹs adherence percentile within their assigned treatment group does not depend on the assigned group (active drug or placebo). The Efron and Feldman approach is valid under percentile invariance, but not necessarily under departures from it. We propose an analysis based on a generalization of percentile invariance that allows adherence percentiles to be stochastically permuted across treatment groups, using a broad class of stochastic permutation models. We show that approximate maximum likelihood estimates of the underlying doseresponse curve perform well when the stochastic permutation process is correctly specified and are quite robust to model misspecification.