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For comparison, we use the Mann-Whitney test to compare the two samples of 10th graders discussed earlier in this chapter. The sum of ranks for the “before” group is 69.55, and for the “one year later group,” 86.57. The test statistic is significant at p = .019, yielding the same conclusion as the independent-samples t-test, p = .011. This comparison also shows that nonparametric tests do have higher levels of significance. As mentioned earlier, the Mann-Whitney test (as a nonparametric test) does not calculate the group means; separate, descriptive analysis needs to be undertaken for that information. A nonparametric alternative to the paired-samples t-test is the Wilcoxon signed rank test. This test assigns ranks based on the absolute values of these differences (Table 12.5). The signs of the differences are retained (thus, some values are positive and others are negative). For the data in Table 12.5, there are seven positive ranks (with mean rank = 6.57) and three negative ranks (with mean rank = 3.00). The Wilcoxon signed rank test statistic is normally distributed. The Wilcoxon signed rank test statistic, Z, for a difference between these values is 1.89 (p = .059 > .05). Hence, according to this test, the differences between the before and after scores are not significant. Getting Started Calculate a t-test and a Mann-Whitney test on data of your choice. Again, nonparametric tests result in larger p-values. The paired-samples t-test finds that p = .038 < .05, providing sufficient statistical evidence to conclude that the differences are significant. It might also be noted that a doubling of the data in Table 12.5 results in finding a significant difference between the before and after scores with the Wilcoxon signed rank test, Z = 2.694, p = .007. Table 12.5 Wilcoxon Signed Rank Test The Wilcoxon signed rank test can also be adapted as a nonparametric alternative to the one-sample t-test. In that case, analysts create a second variable that, for each observation, is the test value. For example, if in Table 12.5 we wish to test whether the mean of variable “before” is different from, say, 4.0, we create a second variable with 10 observations for which each value is, say, 4.0. Then using the Wilcoxon signed rank test for the “before” variable and this new,
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Evan M. Berman (Essential Statistics for Public Managers and Policy Analysts)