Tests for balanced incomplete block ranked data with ties

We consider balanced incomplete block data when ties occur and propose new statistics for testing (a) differences in mean ranks, (b) differences in distributions of ranks, (c) differences in nonlinear effects of ranks and (d) linear contrasts. A sensory evaluation example where the data are ranks is given. Preference ranking to compare products is well known, and statistical procedures are commonly used to verify that apparent differences between product rankings are due to other than chance effects. Sometimes in taste-testing experiments there are too many products for one judge or consumer to reliably compare at one sitting. This loss of reliability is often associated with sensory fatigue. In such cases balanced incomplete block designs can be employed whereby each judge or consumer tastes only some of the products. Let Nij be the count in the (i, j)th cell of the t × k table of counts based on the ranks of t products ranked k at a time by b consumers or judges. The number b is chosen so that if k < t then each product is equally replicated r times, where r = bk/t. This Nij is the number of times that product i receives rank j. If there are no product differences the expectation of Nij is r/k. If X2 is the usual Pearson chi-squared statistic for testing homogeneity of the distributions of the ranks for the t products, then Schach (1979) showed that asymptotically