Applications of Extensions of Bivariate Rank Sum Statistics to the Crossover Design to Compare Two Treatments Through Four Sequence Groups

Summary This article describes applications of extensions of bivariate rank sum statistics to the crossover design with four sequence groups for two treatments. A randomized clinical trial in ophthalmology provides motivating background for the discussion. The bilateral design for this study has four sequence groups T:T, T:P, P:T, and P:P, respectively, for T as test treatment or P as placebo in the corresponding order for the left and right eyes. This article describes how to use the average of the separate Wilcoxon rank sum statistics for the left and right eyes for the overall comparison between T and P with the correlation between the two eyes taken into account. An extension of this criterion with better sensitivity to potential differences between T and P through reduction of the applicable variance has discussion in terms of a conceptual model with constraints for within‐side homogeneity of groups with the same treatment and between‐side homogeneity of the differences between T and P. Goodness of fit for this model can have assessment with test statistics for its corresponding constraints. Simulation studies for the conceptual model confirm better power for the extended test statistic with its full invocation than other criteria without this property. The methods summarized here are illustrated for the motivating clinical trial in ophthalmology, but they are applicable to other situations with the crossover design with four sequence groups for either two locations for two treatments at the same time for a patient or two successive periods for the assigned treatments for a recurrent disorder. This article also notes that the methods based on its conceptual model can have unsatisfactory power for departures from that model where the difference between T and P via the T:T and P:P groups is not similar to that via the T:P and P:T groups, as might occur when T has a systemic effect in a bilateral trial. For this situation, more robust test statistics have identification, but there is recognition that the parallel groups design with only the T:T and P:P groups may be more useful than the bilateral design with four sequence groups.