We discuss two-armed clinical trials in which an ordinal response is observed on entry and at follow-up. In certain subject areas, such data are often reduced to the counts of patients who have deteriorated, remained the same, or improved. We develop asymptotic distribution theory to compare this approach to the use of nonparametric tests based on poststratification of the follow-up responses according to the entry levels. The theory for the latter approach is established by a general result for two-sample U statistics based on bivariate data. Application to data from clinical trials and evaluation of asymptotic relative efficiencies and powers for a variety of contexts illustrate the increased sensitivity of poststratified Wilcoxon tests.