Friedman test

The Friedman test is a non-parametric statistical test developed by the U.S. economist Milton Friedman. Similar to the parametric repeated measures ANOVA, it is used to detect differences in treatments across multiple test attempts. The procedure involves ranking each row (or block) together, then considering the values of ranks by columns. Applicable to complete block designs, it is thus a special case of the Durbin test.

Classic examples of use are:

• n wine judges rate k different wines. Are any wines ranked consistently higher or lower than the others?
• n wines are rated by k different judges. Are the judges ratings consistent with each other?
• n welders use k welding torches, and the ensuing welds were rated on quality. Do any of the torches produce consistently better or worse welds?

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The Friedman test is used for two-way repeated measures analysis of variance by ranks. In its use of ranks it is similar to the Kruskal-Wallis one-way analysis of variance by ranks.

Method

1. Given data , that is, a tableau with n rows (the blocks), k columns (the treatments) and a single observation at the intersection of each block and treatment, calculate the ranks within each block. If there are tied values, assign to each tied value the average of the ranks that would have been assigned without ties. Replace the data with a new tableau where the entry r_ij is the rank of x_ij within block i.
2. Find the values:
   • 
   • 
   • ,
   • 
3. The test statistic is given by . Note that the value of Q as computed above does not need to be adjusted for tied values in the data.
4. Finally, when n or k is large (i.e. n > 15 or k > 4), the probability distribution of Q can be approximated by that of a chi-square distribution. In this case the p-value is given by . If n or k is small, the approximation to chi-square becomes poor and the p-value should be obtained from tables of Q specially prepared for the Friedman test. If the p-value is significant, appropriate post-hoc multiple comparisons tests would be performed.

Related tests

• When using this kind of design for a binary response, one instead uses the Cochran test.

References

Primary sources

• Friedman, Milton (December 1937). "The use of ranks to avoid the assumption of normality implicit in the analysis of variance". Journal of the American Statistical Association 32 (200): 675–701. doi:10.2307/2279372. http://links.jstor.org/sici?sici=0162-1459%28193712%2932%3A200%3C675%3ATUORTA%3E2.0.CO%3B2-3. 
• Friedman, Milton (March 1939). "A correction: The use of ranks to avoid the assumption of normality implicit in the analysis of variance". Journal of the American Statistical Association 34 (205): 109. doi:10.2307/2279169. http://links.jstor.org/sici?sici=0162-1459%28193903%2934%3A205%3C109%3AACTUOR%3E2.0.CO%3B2-N. 
• Friedman, Milton (March 1940). "A comparison of alternative tests of significance for the problem of m rankings". The Annals of Mathematical Statistics 11 (1): 86–92. doi:10.1214/aoms/1177731944. http://links.jstor.org/sici?sici=0003-4851%28194003%2911%3A1%3C86%3AACOATO%3E2.0.CO%3B2-J. 

Secondary sources

• Corder, G. W. and Foreman, D. I, Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach. (2009). New York: J. Wiley.
• Friedman test at Institute of Phonetic Sciences (IFA)
• Kendall, M. G. Rank Correlation Methods. (1970, 4th ed.) London: Charles Griffin.
• Hollander, M., and Wolfe, D. A. Nonparametric Statistics. (1973). New York: J. Wiley.
• Siegel, Sidney, and Castellan, N. John Jr. Nonparametric Statistics for the Behavioral Sciences. (1988, 2nd ed.) New York: McGraw-Hill.