Randomization-based covariance adjustment of win ratios and win odds for randomized multi-visit studies with ordinal outcomes

In many randomized multi-visit studies, some response variables have an ordinal scale for outcomes. The win odds (accounts for ties) and the win ratio (ignores ties) are useful for treatment comparisons for ordinal outcomes. This paper discusses the application of randomization-based covariance and stratification adjustment of the win odds (and win ratio) to enable their more convenient use. Adjustment for strata is through the weighted average of within stratum two-sample U statistics for numerators and denominators for the stratified win odds (or win ratio). As randomization-based, invocation of covariance adjustment is through constraints to zeros for baseline covariate differences in the joint vector with logarithms of stratified win odds (or win ratios) for the respective visits. Such adjustment has no formal assumptions about the distributions of response variables or co-variates or the relationships of covariates to response variables; but the resulting adjusted stratified win odds (or win ratios) have narrower confidence intervals than their unadjusted counterparts when covariates have at least moderately strong associations with response variables. There is illustration of such results for the stratified win odds (and win ratio) for a randomized multi-visit clinical trial with an ordinal outcome for a chronic respiratory disorder.