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www.downloadslide.com 15.7 The Kruskal–Wallis Test for the One-Way Layout 765 a b Do the data present sufficient evidence to indicate that the distributions of wing stroke frequencies differ for the two species? Use the test based on the Mann–Whitney U statistic **with** α as close to, but not exceeding, .10. Give the approximate p-value associated **with** the test. 15.28 Cancer treatment using chemotherapy employs chemicals that kill both cancer cells and normal cells. In some instances, the toxicity of the cancer drug—that is, its effect on normal cells— can be reduced by the simultaneous injection of a second drug. A study was conducted to determine whether a particular drug injection was beneficial in reducing the harmful effects of a chemotherapy treatment on the survival time for rats. Two randomly selected groups of rats, 12 rats in each group, were used for the experiment. Both groups, call them A and B, received the toxic drug in a dosage large enough to cause death, but group B also received the antitoxin that was intended to reduce the toxic effect of the chemotherapy on normal cells. The test was terminated at the end of 20 days, or 480 hours. The lengths of survival time for the two groups of rats, to the nearest 4 hours, are shown in the following table. Do the data provide sufficient evidence to indicate that rats receiving the antitoxin tended to survive longer after chemotherapy than those not receiving the antitoxin? Use the Mann–Whitney U test **with** a value of α near .05. Only Chemotherapy (A) Chemotherapy plus Drug (B) 84 140 128 184 168 368 92 96 184 480 92 188 76 480 104 244 72 440 180 380 144 480 120 196 15.7 The Kruskal–Wallis Test for the One-Way Layout In Section 13.3, we presented an analysis of variance (ANOVA) procedure to compare the means of k populations. The resultant F test was based on the assumption that independent random samples were taken from normal populations **with** equal variances. That is, as discussed in Section 15.2, we were interested in testing whether all the populations had the same distribution versus the alternative that the populations differed in location. A key element in the development of the procedure was the quantity identified as the sum of squares for treatments, SST. As we pointed out in the discussion in Section 13.3, the larger the value of SST, the greater will be the weight of evidence favoring rejection of the null hypothesis that the means are all equal. In this section, we present a nonparametric technique to test whether the populations differ in location. Like the other nonparametric techniques discussed in this chapter,

766 Chapter 15 Nonparametric **Statistics** www.downloadslide.com the Kruskal–Wallis procedure requires no assumptions about the actual form of the probability distributions. As in Section 13.3, we assume that independent random samples have been drawn from k populations that differ only in location. However, we need not assume that these populations possess normal distributions. For complete generality, we permit the sample sizes to be unequal, and we let n i , for i = 1, 2,...,k, represent the size of the sample drawn from the ith population. Analogously to the procedure of Section 15.5, combine all the n 1 +n 2 +···+n k = n observations and rank them from 1 (the smallest) to n (the largest). Ties are treated as in previous sections. That is, if two or more observations are tied for the same rank, then the average of the ranks that would have been assigned to these observations is assigned to each member of the tied group. Let R i denote the sum of the ranks of the observations from population i and let R i = R i /n i denote the corresponding average of the ranks. If R equals the overall average of all of the ranks, consider the rank analogue of SST, which is computed by using the ranks rather than the actual values of the measurements: k∑ V = n i (R i − R) 2 . i=1 If the null hypothesis is true and the populations do not differ in location, we would expect the R i values to be approximately equal and the resulting value of V to be relatively small. If the alternative hypothesis is true, we would expect this to be exhibited in differences among the values of the R i values, leading to a large value for V . Notice that R = (sum of the first n integers)/n = [n(n + 1)/2]/n = (n + 1)/2 and thus that k∑ ( V = n i R i − n + 1 ) 2 . 2 i=1 Instead of focusing on V , Kruskal and Wallis (1952) considered the statistic H = 12V/[n(n + 1)], which may be rewritten (see Exercise 15.35) as H = 12 n(n + 1) k∑ i=1 R 2 i n i − 3(n + 1). As previously noted, the null hypothesis of equal locations is rejected in favor of the alternative that the populations differ in location if the value of H is large. Thus, the corresponding α-level test calls for rejection of the null hypothesis in favor of the alternative if H > h(α), where h(α) is such that, when H 0 is true, P[H > h(α)] = α. If the underlying distributions are continuous and if there are no ties among the n observations, the null distribution of H can (tediously) be found by using the methods of Chapter 2. We can find the distribution of H for any values of k and n 1 , n 2 ,...,n k by calculating the value of H for each of the n! equally likely permutations of the ranks of the n observations (see Exercise 15.36). These calculations have been performed and tables developed for some relatively small values of k and for n 1 , n 2 ,...,n k [see, for example, Table A.12 of Hollander and Wolfe (1999)]. Kruskal and Wallis showed that if the n i values are “large” the null distribution of H can be approximated by a χ 2 distribution **with** k − 1 degrees of freedom (df). This approximation is generally accepted to be adequate if each of the n i values is greater than or equal to 5. Our examples and exercises are all such that this large sample

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