Monotonicity assumptions in estimating the treatment effect for a principal stratum

In addition to the treatment effect for all randomized patients, sometimes it is of interest to understand the treatment effect for a principal stratum, a subset of patients defined by one or more post-baseline variables. For example, what is the treatment effect for those patients who could be compliant with the experimental treatment? One commonly used assumption for estimating such a treatment effect is deterministic monotonicity, which assumes that a patient with a stratum-defining event under one treatment would also have that event under the alternative treatment. Alternatively, a less widely used stochastic monotonicity condition assumes the probability of a patient in a stratum with a stratum-defining event under one treatment is no smaller (or no larger) than that under the alternative treatment. In this article, we discuss the lack of plausibility of the deterministic monotonicity assumption and the advantages of using the principal score for estimating principal strata effects in clinical trials through theoretic argument and a real data example from a 2x2 cross-over study. As we illustrate, in some cases, methods based on modeling the probability of strata membership using baseline covariates (the principal score) may lead to reliable inferences without the need for making monotonicity assumptions.