# Wilcoxon Test – Definition

### Wilcoxon Test Definition

The Wilcoxon test is a nonparametric test used in statistics for comparing two paired sets. This test is either called the Signed Rank test or the Rank Sum test. The main objective of this test is to ascertain and analyze the difference between both pair sets

The Wilcoxon Rank Sum test enables in testing the null hypothesis that the similar continuous distribution lies between two populations. This test is based on the assumption that the data should be from the similar population, and is paired, it should be quantified on minimum of one interval scale, and it should be selected on a random and independent basis.

The Wilcoxon Signed Rank test lies on the assumption that data lies in the degrees and hints of the differences between two paired groups. In case, the population information does not follow the pattern of a normal distribution, one can use this test instead of the t-test.

### A Little More on What is the Wilcoxon Test

In 1945, an American statistician named Frank Wilcoxon devised The Rank Sum and Signed Rank tests, and further published them in his research paper. Wilcoxon test created premises for hypothesis testing involving nonparametric statistics, which were applied to population data. This data was given ranks, but was not assigned any numbers, for example, movie reviews or customer satisfaction. Equations can easily define parametric distributions, but not nonparametric distribution. Nonparametric distribution, as the name suggests, doesn’t have any parameters.

Here are the questions that the Wilcoxon test provides answers for:

1. Do test scores vary from 5th grade to 5th grade considering the same set of students?
2. Will a specific drug affect health if given to the same individuals?

The model is based on the assumption that the information is derived from two dependent populations which follow the similar individual or stock every time and everywhere. This data is said to be continuous in nature. Due to the non-parametric nature of the test, the analysis doesn’t ask for a specific probability distribution of the variable that is dependent in nature.

Key points

1. The Wilcoxon test is either referred to as the Signed Rank test or the Rank Sum test. This nonparametric test involves comparison of two paired groups.
2. Due to the nonparametric nature of the t-test of paired student, the Signed Rank can be considered as a substitute for the t-test where there is no normal distribution in the population data.
3. The model revolves around the belief that the data originates from two similar or dependent populations, pursuing the similar person or share in terms of location or time.

### How to calculate a Wilcoxon test statistic?

Here are the steps that need to be followed for calculating the Wilcoxon test statistics:

1. Considering every item for a sample set of n items, calculate the difference score Di by subtracting two quantities.
2. No need to worry if the difference comes to be in positives or negatives. Ascertain a set of n perfect differences.
3. Exclude the values with a difference of zero. Consider a set of n absolute difference value without any zero values, where n’ ≤ n, and n’ is the actual size of the sample.
4. Now, give ranks Ri from 1 to n to the differences calculated in such a way that the smallest difference value receives rank 1, and the biggest one receives rank n. In case, there are at least two values with equivalent differences, they will receive the average rank based on the ranks they would have got if there were no ties present.
5. Based on the previous calculation if the difference was positive or negative, reallocated the positive or negative sign to each of the n ranks.
6. The summation of the positive ranks will provide the value of W, that is the Wilcoxon test statistic.

### References for “Wilcoxon Test”

https://www.investopedia.com/terms/w/wilcoxon-test.asp

https://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test

https://www.statisticssolutions.com/what-is-the-wilcoxon-sign-test/

### Academic research for “Wilcoxon Test”

A generalized Wilcoxon test for comparing arbitrarily singly-censored samples, Gehan, E. A. (1965). A generalized Wilcoxon test for comparing arbitrarily singly-censored samples. Biometrika, 52(1-2), 203-224.

A generalized two-sample Wilcoxon test for doubly censored data, Gehan, E. A. (1965). A generalized two-sample Wilcoxon test for doubly censored data. Biometrika, 52(3/4), 650-653.

Historical notes on the Wilcoxon unpaired two-sample test, Kruskal, W. H. (1957). Historical notes on the Wilcoxon unpaired two-sample test. Journal of the American Statistical Association, 52(279), 356-360.

Significance probabilities of the Wilcoxon test, Fix, E., & Hodges Jr, J. L. (1955). Significance probabilities of the Wilcoxon test. The Annals of Mathematical Statistics, 301-312.

Relative power of the Wilcoxon test, the Friedman test, and repeated-measures ANOVA on ranks, Zimmerman, D. W., & Zumbo, B. D. (1993). Relative power of the Wilcoxon test, the Friedman test, and repeated-measures ANOVA on ranks. The Journal of Experimental Education, 62(1), 75-86.

Early decision in the Wilcoxon two-sample test, Alling, D. W. (1963). Early decision in the Wilcoxon two-sample test. Journal of the American Statistical Association, 58(303), 713-720.

Estimating the power of the two-sample Wilcoxon test for location shift., Collings, B. J., & Hamilton, M. A. (1988). Estimating the power of the two-sample Wilcoxon test for location shift. Biometrics, 44(3), 847-860.

Robustness of the Wilcoxon test to a certain dependency between samples, Hollander, M., Pledger, G., & Lin, P. E. (1974). Robustness of the Wilcoxon test to a certain dependency between samples. The Annals of Statistics, 177-181.

Early decision in a censored Wilcoxon two-sample test for accumulating survival data, Halperin, M., & Ware, J. (1974). Early decision in a censored Wilcoxon two-sample test for accumulating survival data. Journal of the American Statistical Association, 69(346), 414-422.

The Wilcoxon test and non‐null hypotheses, Wetherill, G. B. (1960). The Wilcoxon test and non‐null hypotheses. Journal of the Royal Statistical Society: Series B (Methodological), 22(2), 402-418.