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Kruskal-Wallis ANOVA with SPSS Dr. Hussam El-Gayar of the Urology & Nephrology Center at Mansoura University in Egypt found my SPSS lessons and asked me a question about the analysis of some of his data. He wanted to compare six independent groups. His criterion variable is continuously distributed. He had already conducted a one-way parametric ANOVA with pairwise comparisons. I took a peek at the results of that analysis and at the raw data and became concerned that both the homogeneity of variance and the normality assumptions of that analysis were tenable. I split the file by the grouping variable and obtained the stats necessary to determine if the assumptions of the parametric ANOVA were tenable. Click Analyze, Descriptive Statistics. Scoot the criterion variable into the Variable(s) box. Click Options and ask for means, variances, skewness, and kurtosis. Copyright 2005, Karl L. Wuensch - All rights reserved. 2 De scri ptive Statistics METHOD 1.00 2.00 3.00 4.00 5.00 6.00 DATA Valid N DATA Valid N DATA Valid N DATA Valid N DATA Valid N DATA Valid N (lis twis e) (lis twis e) (lis twis e) (lis twis e) (lis twis e) (lis twis e) N St atist ic 97 97 97 97 97 97 97 97 97 97 97 97 Mean St atist ic 39.8828 Variance St atist ic 95.739 Sk ewness St atist ic St d. Error -.448 .245 Kurtos is St atist ic St d. Error 2.212 .485 20.5936 33.483 -.207 .245 1.539 .485 28.0316 98.510 2.390 .245 16.615 .485 73.0409 4266.446 1.128 .245 1.625 .485 .8536 .484 2.827 .245 10.195 .485 129.4064 1606.246 .005 .245 -1. 178 .485 As you can see, there is both great heterogeneity of variance and, in some but not all groups, great skewness. In this case, a nonparametric analysis is indicated. An appropriate omnibus analysis is the Kruskal-Wallis ANOVA. To conduct it, click Analyze, Nonparametrics, K Independent Samples. Scoot the criterion variable in the Test Variable List and the grouping variable into the Grouping Variable box. Click Define Range and identify the lowest and highest numeric values for the grouping variable: 3 Click OK. The output includes mean ranks for each group: Ranks DATA METHOD 1.00 2.00 3.00 4.00 5.00 6.00 Total N 97 97 97 97 97 97 582 Mean Rank 358.90 198.29 269.71 359.47 49.87 512.77 Since our p value is less than the .05 criterion of statistical significance, we conclude that there are significant differences among the groups. Test Statisticsa,b Chi-Square df As ymp. Sig. DATA 431.651 5 .000 a. Kruskal Wallis Test b. Grouping Variable: METHOD We still do not know which groups differ significantly from which other groups, so we continue on, comparing each group with each other group. If we make all possible pairwise comparisons, we make 15 comparisons: Group 1 with 2, 3, 4, 5, and 6; Group 2 with 3, 4, 5, and 6; Group 3 with 4, 5, and 6; Group 4 with 5 and 6; and Group 5 with 6. 4 The Mann-Whitney U test is appropriate for comparing two independent groups. Click Analyze, Nonparametric Tests, 2 Independent Samples. Scoot the criterion variable into the Test Variable List and the grouping variable into the Grouping Variable box. Click Define Groups and give the group numbers for the two groups to be compared. Here I have asked to compare group 1 with group 6: Click OK. The output includes for each group the mean rank and sum of ranks. Both the Mann-Whitney and the Wilcoxon Rank Sum statistics are given. These two statistics are equivalent, yielding the same p value. Since our p value is less than the .05 criterion of statistical significance, we conclude that methods 1 and 6 differ from each other significantly. Ranks DATA METHOD 1.00 6.00 Total N 97 97 194 Test Statisticsa Mann-Whitney U Wilcoxon W Z As ymp. Sig. (2-tailed) DATA 9.000 4762.000 -12.089 .000 a. Grouping Variable: METHOD Mean Rank 49.09 145.91 Sum of Ranks 4762.00 14153.00 5 Having illustrated how to compare two groups, I left to Dr. El-Gayar the task of making the remaining 14 pairwise comparisons. One final comment: Since some of the groups are greatly skewed, it is a good idea to report medians rather than, or in addition to, means. SPSS Descriptives does not compute medians, but SPSS Frequencies does. Here is how to do it: First, split the file by the grouping method. Then click Analyze, Descriptive Statistics, Frequencies. Select the criterion variable and click Statistics. Check Median and click Continue. Click Format and check Suppress Tables With More Than ___ Categories. Enter the number 2 in the categories box and click Continue. 6 Click OK. The output includes the median for each group: Statistics DA TA 1.00 N 2.00 Median N 3.00 Median N 4.00 Median N 5.00 Median N 6.00 Median N Valid Missing Median Valid Missing Valid Missing Valid Missing Valid Missing Valid Missing 97 0 39.8800 97 0 22.0000 97 0 30.0062 97 0 65.2500 97 0 .6704 97 0 131.8400 Copyright 2005, Karl L. Wuensch - All rights reserved.