Partial Ordering of Populations

The two-way classification design structure involving ranks has the practical limitation that it may not be possible or practical to rank all p of the objects in ascending (or descending) order of preference. However, in many such instances it may be quite reasonable to expect each observer to identify a subset, say k < p, of the objects as being “most preferred” and exhibit the order of preference among the objects in this subset. In this paper a test statistic is developed for testing the hypothesis of no difference in preference among the p objects. Since the null distribution of this statistic is laborious to tabulate, a chi-square approximation and a beta approximation to this distribution are given with a discussion of their relative merits based on a limited tabulation of the statistic.