Multivariate non-parametric methods for Mann-Whitney statistics to analyse cross-over studies with two treatment sequences

A non-parametric strategy for the analysis of ordinal data from cross-over studies with two treatment sequences and d(> or = 2) periods is examined through Mann-Whitney rank measures of association. For each period, these statistics estimate the probability of larger response for a randomly selected patient in one group relative to a randomly selected patient in the other group. Such estimates are as well formed for comparisons between groups for u pairs of periods with the same treatment. Methods for U-statistics are used to produce a consistent estimate of the covariance matrix for the (d + u) Mann-Whitney estimates. The effects of periods and treatments on the respective Mann-Whitney estimates are evaluated through linear (or log-linear) models. For estimation of the parameters in these models, a modified weighted least squares method is applied through a (2d - 1) < or = (d + u) dimensional basis which effectively addresses potentially near singularities in the estimated covariance matrix of the Mann-Whitney estimates. The proposed methods are applicable to response variables with an interval or an ordered categorical scale. Their scope additionally has capabilities for controlling strata in the design of a cross-over study or concomitant variables for which covariance adjustment is of interest for reduction of variance. Applications of the methods are illustrated through three cross-over studies with different specifications for the two sequences of two treatments during two to four periods.