- Friedman test
The Friedman test is a non-parametric
statistical test developed by the U.S. economistMilton Friedman . Similar to the parametricrepeated measures ANOVA , it is used to detect differences in treatments across multiple test attempts. The procedure involvesranking each row (or "block") together, then considering the values of ranks by columns. Applicable tocomplete block design s, it is thus a special case of theDurbin test .Classic examples of use are:
* "n" wine judges rate "k" different wines. Are the ratings consistent?
* "n" welders use "k" welding torches, and the ensuing welds were rated on quality. Is there one torch that produced better welds than the others? [http://www.texasoft.com/winkfrie.html]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
#Given data x_{ij}}_{n imes k}, 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 r_{ij}}_{n imes k} where the entry r_{ij} is the rank of x_{ij} within block i.
#Find the values:
#*ar{r}_{cdot j} = frac{1}{n} sum_{i=1}^n {r_{ij
#*ar{r} = frac{1}{nk}sum_{i=1}^n sum_{j=1}^k r_{ij}
#*SS_t = nsum_{j=1}^k (ar{r}_{cdot j} - ar{r})^2,
#*SS_e = frac{1}{n(k-1)} sum_{i=1}^n sum_{j=1}^k (r_{ij} - ar{r})^2
#The test statistic is given by Q = frac{SS_t}{SS_e}. Note that the value of Q as computed above does not need to be adjusted for tied values in the data.
#Finally, when n or k is large (i.e. n > 15 or k > 4), theprobability distribution of Q can be approximated by that of achi-square distribution. In this case thep-value is given by mathbf{P}(chi^2_{k-1} ge Q). 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-hocmultiple 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
*cite journal
last = Friedman
first = Milton
authorlink = Milton Friedman
year = 1937
month = December
title = The use of ranks to avoid the assumption of normality implicit in the analysis of variance
journal = Journal of the American Statistical Association
volume = 32
issue = 200
pages = 675–701
doi = 10.2307/2279372
url = http://links.jstor.org/sici?sici=0162-1459%28193712%2932%3A200%3C675%3ATUORTA%3E2.0.CO%3B2-3
*cite journal
last = Friedman
first = Milton
authorlink = Milton Friedman
year = 1939
month = March
title = A correction: The use of ranks to avoid the assumption of normality implicit in the analysis of variance
journal = Journal of the American Statistical Association
volume = 34
issue = 205
pages = 109
doi = 10.2307/2279169
url = http://links.jstor.org/sici?sici=0162-1459%28193903%2934%3A205%3C109%3AACTUOR%3E2.0.CO%3B2-N
*cite journal
last = Friedman
first = Milton
authorlink = Milton Friedman
year = 1940
month = March
title = A comparison of alternative tests of significance for the problem of "m" rankings
journal = The Annals of Mathematical Statistics
volume = 11
issue = 1
pages = 86–92
doi =
url = http://links.jstor.org/sici?sici=0003-4851%28194003%2911%3A1%3C86%3AACOATO%3E2.0.CO%3B2-JSecondary sources
* [http://www.fon.hum.uva.nl/Service/Statistics/Friedman.html Friedman test at Institute of Phonetic Sciences (IFA)]
* [http://www.texasoft.com/winkfrie.html Texasoft statistics tutorial]
*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.
Wikimedia Foundation. 2010.