  In clinical trials, it is often of interest to understand the principal
causal effect (PCE), the average treatment effect for a principal stratum (a
subset of patients defined by the potential outcomes of one or more
post-baseline variables). Commonly used assumptions include monotonicity,
principal ignorability, and cross-world assumptions of principal ignorability
and principal strata independence. In this article, we evaluate these
assumptions through a 2$\times$2 cross-over study in which the potential
outcomes under both treatments can be observed, provided there are no
carry-over and study period effects. From this example, it seemed the
monotonicity assumption and the within-treatment principal ignorability
assumptions did not hold well. On the other hand, the assumptions of
cross-world principal ignorability and cross-world principal stratum
independence conditional on baseline covariates seemed reasonable. With the
latter assumptions, we estimated the PCEs, defined by whether the blood glucose
standard deviation increased in each treatment period, without relying on the
cross-over feature, producing estimates close to the results when exploiting
the cross-over feature. To the best of our knowledge, this article is the first
attempt to evaluate the plausibility of commonly used assumptions for
estimating PCEs using a cross-over trial.
