title: Exact p-values for pairwise comparison of Friedman rank sums, with application to comparing classifiers.
abstract: BACKGROUND
  The Friedman rank sum test is a widely-used nonparametric method in computational biology. In addition to examining the overall null hypothesis of no significant difference among any of the rank sums, it is typically of interest to conduct pairwise comparison tests. Current approaches to such tests rely on large-sample approximations, due to the numerical complexity of computing the exact distribution. These approximate methods lead to inaccurate estimates in the tail of the distribution, which is most relevant for p-value calculation.
  
  
  RESULTS
  We propose an efficient, combinatorial exact approach for calculating the probability mass distribution of the rank sum difference statistic for pairwise comparison of Friedman rank sums, and compare exact results with recommended asymptotic approximations. Whereas the chi-squared approximation performs inferiorly to exact computation overall, others, particularly the normal, perform well, except for the extreme tail. Hence exact calculation offers an improvement when small p-values occur following multiple testing correction. Exact inference also enhances the identification of significant differences whenever the observed values are close to the approximate critical value. We illustrate the proposed method in the context of biological machine learning, were Friedman rank sum difference tests are commonly used for the comparison of classifiers over multiple datasets.
  
  
  CONCLUSIONS
  We provide a computationally fast method to determine the exact p-value of the absolute rank sum difference of a pair of Friedman rank sums, making asymptotic tests obsolete. Calculation of exact p-values is easy to implement in statistical software and the implementation in R is provided in one of the Additional files and is also available at http://www.ru.nl/publish/pages/726696/friedmanrsd.zip .
authors.affiliation: Department of Social Science Research Methods, Radboud University Nijmegen, PO Box 9104, , 6500 HE, Nijmegen, The Netherlands. r.eisinga@maw.ru.nl.
authors.forename: Rob
authors.identifier: http://orcid.org/0000-0002-8349-5226
authors.initials: R
authors.lastname: Eisinga
authors.affiliation: Institute for Computing and Information Sciences, Radboud University Nijmegen, Nijmegen, The Netherlands.
authors.forename: Tom
authors.initials: T
authors.lastname: Heskes
authors.affiliation: Department of Social Science Research Methods, Radboud University Nijmegen, PO Box 9104, , 6500 HE, Nijmegen, The Netherlands.
authors.forename: Ben
authors.initials: B
authors.lastname: Pelzer
authors.affiliation: Department of Social Science Research Methods, Radboud University Nijmegen, PO Box 9104, , 6500 HE, Nijmegen, The Netherlands.
authors.forename: Manfred
authors.initials: M
authors.lastname: Te Grotenhuis
mesh_terms: D019295:Computational Biology / Q000379:methods; D006801:Humans; D020407:Internet; D018709:Statistics, Nonparametric; D014584:User-Computer Interface
keywords: Classifier comparison; Exact p-value; Friedman test; Machine learning; Multiple comparison; Nonparametric statistics; Rank sum difference