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# Wilcoxon-Signed Rank and Wilcoxon Rank-Sum Tests for Testing Hypotheses on Dependent and Independent Populations

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The Wilcoxon test, which can refer to either the Rank Sum test or the Signed Rank test version, is a nonparametric statistical test that compares two paired groups

The tests essentially calculate the difference between sets of pairs and analyzes these differences to establish if they are statistically significantly different from one another.

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The Wilcoxon signed rank test is the non-parametric of the dependent samples t-test. Because the dependent samples t-test analyzes if the average difference of two repeated measures is zero, it requires metric (interval or ratio) and normally distributed data; the Wilcoxon sign test uses ranked or ordinal data; thus, it is a common alternative to the dependent samples t-test when its assumptions are not met. The Wilcoxon signed rank test relies on the W-statistic. For large samples with n>10 paired observations the W-statistic approximates a normal distribution. The W statistic is a non-parametric test, thus it does not need multivariate normality in the data. A research team wants to test whether a new teaching method increases the literacy of children. Therefore, the researchers take measure the literacy of 20 children before and after the teaching method has been applied. The literacy is measured on a scale from 0 to 10, with 10 indicating high literacy. The initial baseline shows an average literacy score of 5.9 and after the method has been used the average increases to 7.6. A dependent samples t-test cannot be used, as the distribution does not approximate a normal distribution. Also both measurements are not independent from each other; therefore, the Mann-Whitney U-test cannot be used

The first step of the Wilcoxon sign test is to calculate the differences of the repeated measurements and to calculate the absolute differences.

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The Wilcoxon signed-rank test refers to a non-parametric statistical theory that is very significant in carrying out tests of two related models as well as repeated dimensions on individual samples to establish whether there are variations in their populace mean ranks (Gravetter & Wallnau, 2009). Further, the test is also useful in assessing the differences existing between the population mean ranks of matched samples

Moreover, the analysis plays an alternative means of assessment to the paired Student’s test for corresponding pairs as well as the t-test for independent samples in the event that the populace is not normally distributed. In carrying out the Wilcoxon test, the statistics from the corresponding population are paired off. The test also applies random sampling of the independent pairs (Gravetter & Wallnau, 2009). Moreover, an ordinal scale is vital in measuring the statistics following a normal distribution. In essence, the hypothesis testing of non-parametric data is essential in assessing records that can be placed in a given order but lack the statistical figures. In fact, the test is invaluable in analyzing clientele fulfillment (Gravetter & Wallnau, 2009).

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Thus, as a future work, we consider extending this test to fuzzy interval-valued data which would lead to a fuzzy interval of p-values. A very straightforward and exact solution may be obtained by using Algorithm 1, Algorithm 2 for each level-cut of the fuzzy interval-valued data, and building the level cut of the p-value fuzzy interval. This solution is tractable only if the membership values are quantized (i.e

can take only a finite number of values). Otherwise, it would lead to a too computationally expensive algorithm. Finding a low computational cost algorithm for extending the generalized Wilcoxon rank-sum test remains an open problem.

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Cleves, M. A. (2008). An introduction to survival analysis using stata. New York, NY: Stata Press.

Gravetter, F. J. & Wallnau, L. B. (2009). Statistics for the behavioral sciences. Belmont, CA: Cengage Learning. 