User manual SPSS EXACT TESTS 7.0

[. . . ] 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. Use, duplication, or disclosure by the Government is subject to restrictions as set forth in subdivision (c)(1)(ii) of The Rights in Technical Data and Computer Software clause at 52. 227-7013. [. . . ] 174 in Chapter 14, you can compute Spearman's correlation coefficient for all possible pairs of raters. The Spearman correlation coefficient between rater 4 and rater 21 is 0. 7381. 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. 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. 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. To illustrate this point, ignore the data provided by the 12th subject. Running Cochran's Q test once more, this time on only the first 11 subjects, yields the results shown in Figure 7. 9. Figure 7. 9 Cochran's Q results for reduced analgesic efficacy data Frequencies Value 0 Placebo Aspirin New Drug 9 7 3 1 2 4 8 Test Statistics Test Statistics1 N 11 Cochran's Q 6. 2221 df 2 Asymp. [. . . ] FEXACT: A FORTRAN subroutine for Fisher's exact test on unordered r c contingency tables. Importance sampling for estimating exact probabilities in permutational inference. Theoretical epidemiology: Principles of occurrence research in medicine. On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. [. . . ]