User manual SPSS EXACT TESTS 7.0

[. . . ] SPSS Exact Tests 7. 0 for Windows TM Cyrus R. Patel/SPSS Inc. For more information about SPSS software products, please visit our WWW site at http://www. spss. com or contact Marketing Department SPSS Inc. 233 South Wacker Drive, 11th Floor Chicago, IL 60606-6307 Tel: (312) 651-3000 Fax: (312) 651-3668 SPSS is a registered trademark and the other product names are the trademarks of SPSS Inc. No material describing such software may be produced or distributed without the written permission of the owners of the trademark and license rights in the software and the copyrights in the published materials. [. . . ] Between rater 4 and rater 11, it is 0. 2857. Finally, between rater 21 and rater 11, it is 0. 4286. Therefore, the average of the three Spearman correlation coefficients is ( 0. 7381 + 0. 2857 + 0. 4286 ) / 3 = 0. 4841 . Substituting N = 3 and W = 0. 6561 into Equation 7. 20, you also get 0. 4841. 108 Chapter 7 Cochran's Q Test Suppose that the u ij values in the two-way layout shown in Table 7. 2 were all binary, with a 1 denoting success and a 0 denoting failure. A popular mathematical model for generating such binary data in the context of the two-way layout is the logistic regression model ij log -------------- = + i + j 1 ij Equation 7. 21 where, for all i = 1, 2, . . . N , and j = 1, 2, . . . K , ij = Pr ( U ij = 1 ), is the background log-odds of response, i is the block effect, and j is the treatment effect. All of these parameters are unknown, but for identifiability you can assume that N K i=1 i = j=1 j = 0 Friedman's test applied to such data is known as Cochran's Q test. As before, the null hypothesis that there is no treatment effect can be formally stated as H0 : ( 1 = 2 = . . . = K ) Equation 7. 22 Cochran's Q test is used to test H 0 against unordered alternatives of the form H 1 : j j for at least one ( j 1, j 2 ) pair 1 2 Equation 7. 23 Like Friedman's test, Cochran's Q is an omnibus test. The alternative hypothesis is simply that the treatments are different, not that one specific treatment is more effective than another. You can use the same test statistic as for Friedman's test. Because of the binary observations, the test statistic reduces to K 2 K( K 1) (B B) j=1 j Q = ------------------------------------------------------------------ Equation 7. 24 K Li L i l=1 i=1 N N 2 where B j is the total number of successes in the jth treatment, L i is the total number of successes in the ith block, and B denotes the average ( B 1 + B 2 + . . . The asymptotic distribution of Q is chi-square with ( K 1 ) degrees of freedom. The exact K-Sample Inference: Related Samples 109 and Monte Carlo results are calculated using the same permutational arguments used for Friedman's test. The exact, Monte Carlo and asymptotic two-sided p values are thus obtained by Equation 7. 7, Equation 7. 9, and Equation 7. 14, respectively. Example: Crossover Clinical Trial of Analgesic Efficacy This data set is taken from a three-treatment, three-period crossover clinical trial published by Snapinn and Small (1986). Twelve subjects each received, in random order, three treatments for pain relief: a placebo, an aspirin, and an experimental drug. The outcome of treatment j on subject i is denoted as either a success ( u ij = 1 ) or a failure ( u ij = 0 ) . Figure 7. 7 shows the data displayed in the Data Editor. Figure 7. 7 Crossover clinical trial of analgesic efficacy 110 Chapter 7 The Cochran's Q test can be used to determine if the response rates for the three treatments differ. The results are displayed in Figure 7. 8. Figure 7. 8 Cochran's Q results for study of analgesic efficacy Frequencies Value 0 Placebo Aspirin New Drug 10 8 3 1 2 4 9 Test Statistics Test Statistics1 N 12 Cochran's Q 7. 8001 df 2 Asymp. 0 is treated as a success. The exact p value is 0. 026 and indicates that the three treatments are indeed significantly different at the 5% level. The asymptotic p value, 0. 020, confirms this result. In this data set, there was very little difference between the exact and the asymptotic inference. However, the data set is fairly small, and a slightly different data configuration could have resulted in an important difference between the exact and asymptotic p values. [. . . ] Estimate of deviation between empirical distribution functions in two independent samples. Tests of significance using regression models for ordered categorical data. Applied nonparametric statistical methods. On a test whether two samples are from the same population. [. . . ]