Wilcoxon signed-rank test

Wilcoxon signed-rank test
Name	Wilcoxon signed-rank test
Type	nonparametric statistical test
Developer	Frank Wilcoxon
Introduced	1945
Purpose	compare paired samples or matched observations

Wilcoxon signed-rank test The Wilcoxon signed-rank test is a nonparametric method for comparing paired or matched observations to assess median differences without assuming normality. It was introduced by Frank Wilcoxon and is widely used in analyses alongside methods from Karl Pearson, Ronald Fisher, Jerzy Neyman, and Egon Pearson in the development of modern inferential statistics. Practitioners in fields from Harvard University and Johns Hopkins University to the World Health Organization have employed it when data violate assumptions of parametric alternatives such as the paired t-test used at institutions like Massachusetts Institute of Technology and Stanford University.

Introduction

The test is applied to paired measurements, matched designs, or repeated measures collected in studies conducted by organizations like National Institutes of Health, Centers for Disease Control and Prevention, and Food and Drug Administration. It ranks absolute differences and uses signed ranks to form a test statistic, an approach informed by earlier rank-based ideas from statisticians associated with University of Cambridge, University of Oxford, and research programs at Bell Labs and Princeton University. Use cases span clinical trials at Mayo Clinic, behavioral research at University of California, Berkeley, and econometric studies at London School of Economics.

Statistical definition and assumptions

The Wilcoxon signed-rank test assumes that paired differences are independent and come from a continuous distribution symmetric about a common median; foundational concepts trace to work by Abraham de Moivre and later formalization by figures at University of Chicago. It does not require normality like the paired t-test advocated by William Sealy Gosset (who published as "Student") and is robust in small samples considered by researchers at Columbia University and Yale University. The primary assumptions recognized in guidelines from agencies such as European Medicines Agency and National Institute for Health and Care Excellence include: matched pairs, continuous or ordinal scale, independence of pairs, and distributional symmetry of differences.

Test procedure and computation

Computation begins by forming differences for each paired observation as in clinical measurement protocols from Cleveland Clinic studies, discarding zeros as done in analyses at Johns Hopkins Hospital, ranking absolute differences, reassigning the original signs, and summing positive and negative signed ranks to obtain the test statistic. For small n, exact tables originally disseminated through outlets like Biometrika and publishers associated with Wiley-Blackwell are used; for larger n, a normal approximation with continuity correction similar to methods in texts from Cambridge University Press or Springer is applied. Implementation is routine in software maintained by groups at R Project, Python Software Foundation, and statistical packages produced by SAS Institute and IBM.

Variants and related tests

Variants include procedures handling tied ranks and zero differences used in meta-analyses published by Cochrane Collaboration and adjustments proposed by scholars at University of Pennsylvania and University of Michigan. Related nonparametric tests include the sign test discussed in literature by Ronald Fisher, the Mann–Whitney U test associated with work at University College London, and procedures for censored data developed in collaboration with investigators at Fred Hutchinson Cancer Research Center and Memorial Sloan Kettering Cancer Center. Multivariate extensions and exact permutation alternatives have been developed by researchers affiliated with Carnegie Mellon University and Stanford University.

Exact and asymptotic distributions

For small samples the exact null distribution of the signed-rank statistic is tabulated as in early reports in Biometrika and computed via enumeration methods from computing groups at Los Alamos National Laboratory. For moderate to large samples asymptotic normality applies under conditions discussed by Andrey Kolmogorov and Aleksandr Lyapunov; continuity corrections and variance adjustments are standard, with edgeworth expansions appearing in theoretical work from Institute for Advanced Study and publications linked to Princeton University Press.

Applications and examples

Applications span clinical endpoints in trials overseen by World Health Organization and Food and Drug Administration, psychological measurement studies at University of Cambridge and Yale University, and environmental monitoring projects run by United Nations Environment Programme. Example analyses include pre-post intervention pain scores in trials run by Mayo Clinic, matched economic indicators in policy work at International Monetary Fund, and paired gene expression assays in research at Broad Institute and Cold Spring Harbor Laboratory.

Power, effect size, and sample size considerations

Power and sample size planning for the signed-rank test are discussed in methodological guidance from National Institutes of Health and textbooks published by Wiley-Blackwell and Cambridge University Press. Effect size measures compatible with rank-based methods are used in meta-analyses by the Cochrane Collaboration and in reporting standards endorsed by agencies such as European Medicines Agency. Comparative power to the paired t-test depends on departure from normality, with seminal simulations performed by researchers at University of Toronto and Michigan State University informing modern practice.

Category:Statistical tests