Multivariate non-parametric methods for Mann-Whitney statistics to analysecross-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( greater than or equal to 2) per iods is examined through Mann-Whitney rank measures of association. For eac h period, these statistics estimate the probability of larger response for a randomly selected patient in one group relative to a randomly selected pa tient 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 m atrix for the (d + u) Mann-Whitney estimates. The effects of periods and tr eatments on the respective Mann-Whitney estimates are evaluated through lin ear (or log-linear) models. For estimation of the parameters in these model s, a modified weighted least squares method is applied through a (2d - 1) l ess than or equal to (d + u) dimensional basis which effectively addresses potentially near singularities in the estimated covariance matrix of the Ma nn-Whitney estimates. The proposed methods are applicable to response varia bles with an interval or an ordered categorical scale. Their scope addition ally has capabilities for controlling strata in the design of a cross-over study or concomitant variables for which covariance adjustment is of intere st for reduction of variance. Applications of the methods are illustrated t hrough three cross-over studies with different specifications for the two s equences of two treatments during two to four periods. Copyright (C) 1999 J ohn Wiley & Sons, Ltd.