Adjusting for ordinal covariates by inducing a partial ordering

In the context of randomized clinical trials, multiplicity arises in many forms. One prominent example is when a key endpoint is measured and analyzed both at baseline and after treatment. It is common to analyze each separately, but more efficient to adjust the post-treatment comparisons for the baseline values. Adjustment techniques generally treat the covariate (baseline value, in this case) as either nominal or continuous. Either is problematic when applied to an ordinal covariate, the former because it fails to exploit the natural ordering and the latter because it relies on an artifical notion of linear prediction and differences between values. We propose new methods for adjusting for ordinal covariates without having to treat them as nominal or continuous. Specifically, the information-preserving composite endpoint consists of the pair of values for each patient, one at baseline and one after treatment. Some of these patterns will indicate more improvement than others, yet some pairs of patterns are not comparable. Hence, the ordering is only partial. We develop an approach to testing and deriving estimators of magnitudes of the treatment effect based on comparing each observation in one group to each observation in the other group to which it is comparable.