Wilcoxon signed-rank test

Wilcoxon signed-rank test is a non-parametric statistical test used to compare two related samples or repeated measurements on a single sample, developed by Frank Wilcoxon in the 1940s, a contemporary of Ronald Fisher and Jerzy Neyman. The test is often used in clinical trials and medical research, such as studies published in the Journal of the American Medical Association and the New England Journal of Medicine. It is also widely used in social sciences research, including studies published in Nature and Science (journal), and has been applied in various fields, including psychology research published in Psychological Bulletin and Journal of Personality and Social Psychology.

Introduction

The Wilcoxon signed-rank test is used to determine if there is a significant difference between two related samples or repeated measurements on a single sample, and has been used in various studies, including those conducted by University of California, Berkeley and Harvard University. This test is a non-parametric alternative to the paired t-test, which assumes that the data follows a normal distribution, a concept discussed by Karl Pearson and R.A. Fisher. The Wilcoxon signed-rank test is more robust and can handle non-normal data, making it a popular choice in fields such as biostatistics and epidemiology, where researchers like Bradford Hill and Richard Doll have applied it. The test has been implemented in various statistical software packages, including R (programming language), SAS, and SPSS, developed by companies like IBM and SAS Institute.

Assumptions

The Wilcoxon signed-rank test assumes that the data is paired, meaning that each observation in one sample has a corresponding observation in the other sample, a concept discussed by John Tukey and Frederick Mosteller. The test also assumes that the data is measured on an ordinal scale, which means that the data can be ranked in order, but the differences between consecutive ranks may not be equal, a concept explored by Louis Guttman and Paul Lazarsfeld. Additionally, the test assumes that the data is symmetric around the median, a concept discussed by Andrey Markov and Emil Julius Gumbel. If these assumptions are not met, alternative tests such as the sign test or the Friedman test may be used, which have been developed by researchers like Frank Anscombe and Milton Friedman.

Method

The Wilcoxon signed-rank test involves ranking the absolute differences between the paired observations, a method discussed by Henry Mann and Donald Whitney. The test statistic is calculated as the sum of the ranks of the positive differences, a concept explored by E.J.G. Pitman and G.A. Barnard. The test statistic is then compared to a critical value from the Wilcoxon signed-rank distribution, which is a discrete distribution that depends on the sample size, a concept discussed by Maurice Kendall and Bernard Babington Smith. The test can be performed using statistical software packages like MATLAB, Python (programming language), and Julia (programming language), developed by companies like MathWorks and Google.

Interpretation

The Wilcoxon signed-rank test produces a test statistic and a p-value, which can be used to determine if there is a significant difference between the two related samples or repeated measurements on a single sample, a concept discussed by Jerzy Neyman and Egon Pearson. If the p-value is below a certain significance level, typically 0.05, the null hypothesis that the two samples are equal is rejected, a concept explored by R.A. Fisher and Karl Pearson. The test can also be used to estimate the median difference between the two samples, a concept discussed by John Tukey and Frederick Mosteller. The results of the test can be published in academic journals like Journal of the Royal Statistical Society and Biometrika, and have been applied in various fields, including research conducted by National Institutes of Health and World Health Organization.

Examples

The Wilcoxon signed-rank test has been used in various studies, including a study on the effect of caffeine on athletic performance published in the Journal of Applied Physiology, and a study on the effect of meditation on anxiety published in the Journal of Consulting and Clinical Psychology. The test has also been used in quality control studies, such as a study on the effect of temperature on manufacturing processes published in the Journal of Quality Technology. Additionally, the test has been used in finance research, such as a study on the effect of interest rates on stock prices published in the Journal of Finance, and has been applied by researchers like Eugene Fama and Robert Shiller.

Related_tests

The Wilcoxon signed-rank test is related to other non-parametric tests, such as the Mann-Whitney U test and the Kruskal-Wallis test, which have been developed by researchers like Henry Mann and William Kruskal. The test is also related to parametric tests, such as the paired t-test and the ANOVA test, which have been developed by researchers like R.A. Fisher and Karl Pearson. The Wilcoxon signed-rank test can be used as an alternative to these tests when the assumptions of normality are not met, a concept discussed by John Tukey and Frederick Mosteller. The test has been implemented in various statistical software packages, including R (programming language), SAS, and SPSS, developed by companies like IBM and SAS Institute, and has been applied in various fields, including research conducted by University of Oxford and University of Cambridge. Category:Statistical tests