In a two-period crossover trial where residual carryover is suspected, it is often advised that first-period data only be used in an analysis appropriate for a parallel design. However, it has been shown (Willan and Pater, 1986, Biometrics 42, 593-599) that the crossover analysis is more powerful than the parallel analysis if the residual carryover, expressed as a proportion of treatment effect, is less than 2- square root of 2(1 - rho), where rho is the intrasubject correlation coefficient. Choosing between the analyses based on the empirical evaluation of this condition is equivalent to choosing the analysis with the larger corresponding test statistic. Approximate nominal significance levels are presented that maintain the desired level when basing the analysis on the maximum test statistic. Furthermore, the power and precision of the analysis based on the maximum test statistic are compared to the crossover and parallel analyses.