The two-period crossover trial has the evident advantage that by the use of within-patients comparisons, the usually larger between-patient variability is not used as a measuring stick to compare treatments. However, a prerequisite 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. In the present paper we use 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. We come to the conclusion that in spite of 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.