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The two-period crossover trial has the evident advantage that, by use of within-patient comparisons, the usual large between-patient variability is not used as a measure to compare treatments. A prerequisite, however, is that the order of the treatments does not substantially influence the outcome of the treatment. Crossover studies with a binary response (such as yes/no or present/absent), although widely used for initial screening of new compounds, have not previously been studied for such order effects. This study uses a mathematical model based on standard statistical tests to study to what extent such order effects, here identical to carryover effects, may reduce the power of detecting a treatment effect. It is concluded that, despite large carryover effects, the crossover study with a binary response remains a powerful method and that testing for carryover effects makes sense only if the null hypothesis of no treatment effect cannot be rejected.