Wilcoxon rank-sum test

Wilcoxon rank-sum test, also known as the Mann-Whitney U test, is a non-parametric test used to compare two independent samples, such as those from University of California, Berkeley and Massachusetts Institute of Technology, to determine if one sample has consistently larger values than the other, as studied by Frank Wilcoxon and Henry Mann. This test is widely used in various fields, including medicine at Johns Hopkins University and Harvard Medical School, biology at Stanford University and University of Oxford, and psychology at Yale University and University of Cambridge. The test is named after Frank Wilcoxon, who developed it, and Henry Mann, who later developed a similar test, as recognized by the American Statistical Association and Royal Statistical Society.

Introduction

The Wilcoxon rank-sum test is used to compare the distribution of two independent samples, such as those from University of Michigan and University of California, Los Angeles, to determine if one sample has consistently larger values than the other, as applied in clinical trials at National Institutes of Health and World Health Organization. This test is a non-parametric alternative to the two-sample t-test, which assumes that the data follows a normal distribution, as discussed by Ronald Fisher and Karl Pearson. The Wilcoxon rank-sum test is more robust and can handle non-normal data, making it a popular choice in many fields, including engineering at California Institute of Technology and Georgia Institute of Technology, and economics at University of Chicago and London School of Economics. The test has been widely used in various studies, including those published in Journal of the American Medical Association and The Lancet, and has been recognized by National Academy of Sciences and American Academy of Arts and Sciences.

Assumptions

The Wilcoxon rank-sum test assumes that the data is independent and identically distributed, as stated by Andrey Markov and Emile Borel. The test also assumes that the data is continuous, as discussed by Henri Lebesgue and David Hilbert. Additionally, the test assumes that the samples are randomly selected from the population, as emphasized by Jerzy Neyman and Egon Pearson. If these assumptions are not met, the results of the test may not be valid, as noted by John Tukey and Frederick Mosteller. The test is sensitive to outliers, as studied by John Wilder Tukey and William Gosset, and may not perform well with heavily skewed data, as discussed by Karl Pearson and Ronald Fisher.

Method

The Wilcoxon rank-sum test involves ranking the data from both samples, as described by Frank Wilcoxon and Henry Mann. The test then calculates the sum of the ranks for each sample, as computed by Blaise Pascal and Pierre-Simon Laplace. The test statistic is calculated as the smaller of the two sums, as derived by Abraham de Moivre and Carl Friedrich Gauss. The test then uses a permutation test or a normal approximation to determine the p-value, as developed by Ronald Fisher and Jerzy Neyman. The p-value represents the probability of observing a test statistic as extreme or more extreme than the one observed, assuming that the null hypothesis is true, as explained by Karl Pearson and John Wilder Tukey.

Interpretation

The results of the Wilcoxon rank-sum test are interpreted in terms of the p-value, as discussed by Ronald Fisher and Jerzy Neyman. If the p-value is below a certain significance level, such as 0.05, the null hypothesis is rejected, and it is concluded that one sample has consistently larger values than the other, as applied in medical research at National Institutes of Health and World Health Organization. The test can also be used to estimate the probability that a value from one sample is larger than a value from the other sample, as studied by Frank Wilcoxon and Henry Mann. This probability is known as the probability of superiority, as recognized by American Statistical Association and Royal Statistical Society.

Examples

The Wilcoxon rank-sum test has been used in a variety of studies, including those in medicine at Johns Hopkins University and Harvard Medical School, biology at Stanford University and University of Oxford, and psychology at Yale University and University of Cambridge. For example, a study published in Journal of the American Medical Association used the Wilcoxon rank-sum test to compare the effectiveness of two different treatments for cancer patients at Memorial Sloan Kettering Cancer Center and MD Anderson Cancer Center. Another study published in The Lancet used the test to compare the outcomes of patients who received surgery at Massachusetts General Hospital and University of California, San Francisco versus those who did not. The test has also been used in engineering at California Institute of Technology and Georgia Institute of Technology, and economics at University of Chicago and London School of Economics.

Related_tests

The Wilcoxon rank-sum test is related to other non-parametric tests, such as the Kruskal-Wallis test and the Friedman test, as discussed by William Kruskal and Milton Friedman. The test is also related to the two-sample t-test, which assumes that the data follows a normal distribution, as compared by Ronald Fisher and Karl Pearson. The Wilcoxon rank-sum test is a more robust alternative to the two-sample t-test, as noted by John Tukey and Frederick Mosteller. The test has been recognized by National Academy of Sciences and American Academy of Arts and Sciences, and has been widely used in various fields, including medicine at National Institutes of Health and World Health Organization, and biology at Stanford University and University of Oxford. Category:Statistical tests