Friedman test (statistics)

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The Friedman test is a statistical test for examining three or more paired samples for equality of the location parameter . Since it does not assume a normal distribution of the data in the samples, it is one of the non-parametric methods . It is an extension of the sign test to the application for more than two samples and a parameter-free alternative to the analysis of variance with repeated measurements . Was named the test after the American economist Milton Friedman , who developed it and in the 1937 journal Journal of the American Statistical Association published. An extension of the Friedman test for use on data sets with missing values ​​is the Durbin Skillings Mack test .

Test description

The Friedman test assumes that the values ​​are paired between samples and independent of one another within the samples. The analysis relies on sorting the values ​​in each paired set of data from smallest to largest, sorting each set of values ​​separately. The ranks in each sample are then added up. The p-value as a measure of statistical significance is lower, the greater the differences between the sums of ranks of the individual samples.

Assuming that the samples examined have a comparable frequency distribution, the test's null hypothesis is the assumption that there is no difference in position between the samples . A p-value less than 0.05 is therefore generally interpreted in such a way that the median value of at least one of the examined samples differs significantly from that of the other samples. However, significant differences in terms of distribution can also lead to a significant p-value if the situation is comparable.

Alternative procedures

The Friedman test is a parameter-free alternative to parametric analysis of variance with repeated measurements , which requires a normal distribution of the data. Instead of the Friedman test, the likewise nonparametric Quade test can be used. This usually has a higher test strength for the comparison of up to five samples , while the Friedman test for more than five samples is considered to be stronger in most cases. The Quade test is also clearly superior to the Friedman test for data with different ranges in the individual samples, since the Quade test, unlike the Friedman test, uses the range to weight the individual ranks. On the other hand, in contrast to the Quade test, the Friedman test can also be used for ordinally scaled data that were collected, for example, as rank data or are based on the rank transformation of cardinally scaled measured values, and for which therefore no ranges can be determined.

The likewise nonparametric Kruskal-Wallis test , which, like the Friedman test, is used to analyze the variance of three or more samples, is used in contrast to this test to compare unpaired data. A non-parametric test to compare two paired samples is the Wilcoxon signed rank test . Its use for multiple two-group comparisons between several samples should, however, either be limited to a few comparisons planned in advance or supplemented by a correction of the alpha error accumulation , which can be carried out, for example, with the Bonferroni method .

The Skillings Mack test developed by John Skillings and Gregory Mack, also known as the Durbin Skillings Mack test based on previous research by James Durbin, is a generalization of the Friedman test to data sets with missing values. In the case of complete data sets, it is equivalent to the Friedman test.

literature

  • Milton Friedman: The Use of Ranks to Avoid the Assumption of Normality Implicit in the Analysis of Variance. In: Journal of the American Statistical Association. 32 (200) / 1937, pp. 675-701, doi : 10.1080 / 01621459.1937.10503522 JSTOR 2279372 ; Correction in: A Correction: The Use of Ranks to Avoid the Assumption of Normality Implicit in the Analysis of Variance. 34 (205) / 1939, p. 109, doi : 10.1080 / 01621459.1939.10502372 .
  • The Friedman Two-Way Analysis of Variance by Ranks. In: David Sheskin: Handbook of Parametric and Nonparametric Statistical Procedures. Fourth edition. CRC Press, Boca Raton 2007, ISBN 1-58-488814-8 , pp. 1075-1088.
  • James Durbin: Incomplete Blocks in Ranking Experiments. In: British Journal of Psychology. 4/1951, pp. 85-90, doi : 10.1111 / j.2044-8317.1951.tb00310.x .
  • John H. Skillings, Gregory A. Mack: On the Use of a Friedman-Type Statistic in Balanced and Unbalanced Block Designs. In: Technometrics. 23 (2) / 1981, pp. 171-177, doi : 10.1080 / 00401706.1981.10486261 JSTOR 1268034 .
  • Knut M. Wittkowski: Friedman-Type Statistics and Consistent Multiple Comparisons for Unbalanced Designs with Missing Data. In: Journal of the American Statistical Association Vol. 83, No. 404 (Dec., 1988), pp. 1163-1170, doi : 10.2307 / 2290150 .