Mann-Whitney-Wilcoxon test, statistical

It has been advocated that the area under the ROC curve is a relative measure of a tesfs performance. A Wilcoxon statistic (or equivalently the Mann-Whitney U-Test) statists cally determines which ROC curve has more area under it. Less computationally intensive alternatives, which are no longer necessary, have been described. These methods are particularly helpful when the curves do not intersect. When the ROC curves of two laboratory tests for the same disease intersect, they may offer quite different performances even though the areas under their curves are identical. The performance depends on the region of the curve (i.e., high sensitivity versus high specificity) chosen. Details on how to compare statistically individual points on two curves have been developed elsewhere. ... 

Rank test. A statistical test, such as the Mann-Whitney-Wilcoxon test, carried out on the ranks of the data rather than on the original data. Thus, the statistic used in the test, the rank statistic, is calculated from ranks of the data. Usually such tests are associated with randomization. Given knowledge of the randomization procedure, the distribution of the rank statistic is perfectly general under the null h5q)othesis. The rank is but the guinea stamp. The Mann s the gowd for a that (Burns). 

The Wilcoxon rank-sum test is a nonparametric test for assessing whether two samples of measurements come from the same distribution. That is, as an alternative to the two-sample f-test, this test can be used to discover differentially expressed candidates under two conditions. For example, again consider the measurements of the probe set used for the two-sample t-test. The gene expression values are 12.79, 12.53, and 12.46 for the naive condition and 11.12, 10.77, and 11.38 for the 48-h activated condition. Measurement 12.79 has rank 6, measurement 12.53 has 5, and measurement 12.46 has rank 4. The rank sum of the naive condition is 6 -I- 5 -I-4=15. Then after the sum is subtracted by ni(ni-I-l)/2 = 3 x 4/2 = 6, the Wilcoxon rank-sum test statistic becomes 9. Considering all of the combinations of the three measurements, we can compute the probability that the rank sum happens more extremely than 9. The probability becomes its p-value. This is the most extreme among the 20 combinations thus the p-value is 2 x Pr( W > 9) = 2 x = 0.1 for the two-sided test. It is hard to say that the probe set is differentially expressed since the p-value 0.1 > 0.05. This test is also called the Mann - Whitney- Wilcoxon test because this test was proposed initially by Wilcoxon for equal sample sizes and extended to arbitrary sample sizes by Mann and Whitney. As a nonparametric alternative to the paired t-test for the two related samples, the Wilcoxon signed-rank test can be used. The statistic is computed by ordering absolute values of differences of paired samples. For example, consider a peptide in the platelet study data. Their differences for each... 

The analysis of rank data, what is generally called nonparametric statistical analysis, is an exact parallel of the more traditional (and familiar) parametric methods. There are methods for the single comparison case (just as Student s t-test is used) and for the multiple comparison case (just as analysis of variance is used) with appropriate post hoc tests for exact identification of the significance with a set of groups. Four tests are presented for evaluating statistical significance in rank data the Wilcoxon Rank Sum Test, distribution-free multiple comparisons, Mann-Whitney U Test, and the Kruskall-Wallis nonparametric analysis of variance. For each of these tests, tables of distribution values for the evaluations of results can be found in any of a number of reference volumes (Gad, 1998). 

The choice of an appropriate statistical method is important, and a method suitable for the comparison of two groups in terms of an ordinal outcome measurement is the Mann-Whitney/Wilcoxon rank-sum test (not to be confused with the Wilcoxon matched-pairs signed ranks test, which is appropriate for paired data - see later). It is both inefficient and inappropriate to use a qualitative data test (such as a simple chi-square) for such a measurement, and the application of quantitative data tests (such as one of the f-tests) is also invalid. 

The test statistics from a Mann-Whitney are linearly related to those of Wilcoxon. The two tests will always yield the same result. The Mann-Whitney is presented here for historical completeness, as it has been much favored in reproductive and developmental toxicology studies. However, it should be noted that the author does not include it in the decision tree for method selection (Figure 22.2). 

As the results were not distributed normally, median values were used for the descriptive statistics, while parameter-free test procedures were used for the analytical statistics (Wilcoxon test for paired differences by Wilcoxon/Mann and Whitney) (ClauC and Ebner 1992). 

A "comparison between groups," performed by the U test of Wilcoxon-Mann-Whitney. Only P values lower than 1% were considered statistically significant. 

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