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{{Short description|Non-parametric statistical test}}

The '''Friedman test''' is a non-parametric statistical test developed by [[Biography:Milton Friedman|Milton Friedman]].<ref>{{cite journal
 | last = Friedman
 | first = Milton
 |date=December 1937
 | 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 = 10.1080/01621459.1937.10503522
 | jstor = 2279372
 }}</ref><ref>{{cite journal
 | last = Friedman
 | first = Milton
 |date=March 1939
 | 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 = 10.1080/01621459.1939.10502372
 | jstor = 2279169
 }}</ref><ref>{{cite journal
 | last = Friedman
 | first = Milton
 |date=March 1940
 | 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 = 10.1214/aoms/1177731944
 | jstor=2235971
 | doi-access = free
 }}</ref> Similar to the [[Parametric statistics|parametric]] repeated measures ANOVA, it is used to detect differences in treatments across multiple test attempts. The procedure involves [[Ranking|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:
* <math display="inline">n</math> wine judges each rate <math display="inline">k</math> different wines. Are any of the <math display="inline">k</math> wines ranked consistently higher or lower than the others?
* <math display="inline">n</math> welders each use <math display="inline">k</math> welding torches, and the ensuing welds were rated on quality. Do any of the <math display="inline">k</math> torches produce consistently better or worse welds?

The Friedman test is used for one-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.

The Friedman test is widely supported by many [[Software:Comparison of statistical packages|statistical software packages]].

== Method ==
# Given data <math>\{x_{ij}\}_{n\times k}</math>, that is, a [[Matrix (mathematics)|matrix]] with <math>n</math> rows (the ''blocks''), <math>k</math> 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 matrix <math>\{r_{ij}\}_{n \times k}</math> where the entry <math>r_{ij}</math> is the rank of <math>x_{ij}</math> within block <math>i</math>.
# Find the values <math>\bar{r}_{\cdot j} = \frac{1}{n} \sum_{i=1}^n {r_{ij}}</math>
# The test statistic is given by <math>Q = \frac{12n}{k(k+1)} \sum_{j=1}^k \left(\bar{r}_{\cdot j}-\frac{k+1}{2}\right)^2</math>. Note that the value of <math display="inline">Q</math> does need to be adjusted for tied values in the data.<ref>{{cite web |title=FRIEDMAN TEST in NIST Dataplot |date=August 20, 2018 |url=https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/friedman.htm}}</ref>
# Finally, when <math display="inline">n</math> or <math display="inline">k</math> is large (i.e. <math display="inline">n>15</math> or <math display="inline">k> 4</math>), the [[Probability distribution|probability distribution]] of <math display="inline">Q</math> can be approximated by that of a [[Chi-squared distribution|chi-squared distribution]]. In this case the [[P-value|p-value]] is given by <math>\mathbf{P}(\chi^2_{k-1} \ge Q)</math>. If <math display="inline">n</math> or <math display="inline">k</math> is small, the approximation to chi-square becomes poor and the p-value should be obtained from tables of <math display="inline">Q</math> specially prepared for the Friedman test. If the p-value is [[Statistical significance|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's Q test]].
* The [[Sign test]] (with a two-sided alternative) is equivalent to a Friedman test on two groups. 
* [[Kendall's W]] is a normalization of the Friedman statistic between 0 and 1.
* The [[Wilcoxon signed-rank test]] is a nonparametric test of nonindependent data from only two groups.
* The Skillings–Mack test is a general Friedman-type statistic that can be used in almost any block design with an arbitrary missing-data structure.
* The Wittkowski test is a general Friedman-Type statistics similar to Skillings-Mack test. When the data do not contain any missing value, it gives the same result as Friedman test. But if the data contain missing values, it is both, more precise and sensitive than Skillings-Mack test.<ref>{{cite journal
 | last = Wittkowski
 | first = Knut M. 
 | title = Friedman-Type statistics and consistent multiple comparisons for unbalanced designs with missing data
 | journal = Journal of the American Statistical Association
 | volume = 83
 | issue = 404
 | pages = 1163–1170
 | doi = 10.1080/01621459.1988.10478715
 | jstor = 2290150
 | year = 1988 
 | citeseerx = 10.1.1.533.1948 
 }}</ref> An implementation of the test exists in [[R (programming language)|R]].<ref>{{cite web |title=muStat package (R code) |date=August 23, 2012 |url=https://cran.r-project.org/package=muStat/}}</ref>

== Post hoc analysis ==

Post-hoc tests were proposed by Schaich and Hamerle (1984)<ref>Schaich, E. & Hamerle, A. (1984). Verteilungsfreie statistische Prüfverfahren. Berlin: Springer. {{ISBN|3-540-13776-9}}.</ref> as well as Conover (1971, 1980)<ref>Conover, W. J. (1971, 1980). Practical nonparametric statistics. New York: Wiley. {{ISBN|0-471-16851-3}}.</ref> in order to decide which groups are significantly different from each other, based upon the mean rank differences of the groups. These procedures are detailed in Bortz, Lienert and Boehnke (2000, p.&nbsp;275).<ref>Bortz, J., Lienert, G. & Boehnke, K. (2000). Verteilungsfreie Methoden in der Biostatistik. Berlin: Springer. {{ISBN|3-540-67590-6}}.</ref> Eisinga, Heskes, Pelzer and Te Grotenhuis (2017)<ref>{{cite journal | last1 = Eisinga | first1 = R. | last2 = Heskes | first2 = T. | last3 = Pelzer | first3 = B. | last4 = Te Grotenhuis | first4 = M. | year = 2017 | title = Exact ''p''-values for pairwise comparison of Friedman rank sums, with application to comparing classifiers | doi = 10.1186/s12859-017-1486-2 | journal = BMC Bioinformatics | volume = 18 | issue = 1 | pages = 68 | pmc = 5267387 | pmid=28122501 | url=http://rdcu.be/oOf9 | doi-access = free }}</ref> provide an exact test for pairwise comparison of Friedman rank sums, implemented in [[R (programming language)|R]]. The Eisinga c.s. exact test offers a substantial improvement over available approximate tests, especially if the number of groups (<math>k</math>) is large and the number of blocks (<math>n</math>) is small.

Not all statistical packages support post-hoc analysis for Friedman's test, but user-contributed code exists that provides these facilities (for example in [[Software:SPSS|SPSS]],<ref>{{cite web |title=Post-hoc comparisons for Friedman test |url=http://timo.gnambs.at/en/scripts/friedmanposthoc |access-date=2010-02-22 |archive-url=https://web.archive.org/web/20121103040410/http://timo.gnambs.at/en/scripts/friedmanposthoc |archive-date=2012-11-03 |url-status=dead }}</ref> and in [[R (programming language)|R]].<ref>{{cite web |title=Post hoc analysis for Friedman's Test (R code) |date=February 22, 2010 |url=https://www.r-statistics.com/2010/02/post-hoc-analysis-for-friedmans-test-r-code/ }}</ref>). Also, there is a specialized package available in [[R (programming language)|R]] containing numerous non-parametric methods for post-hoc analysis after Friedman.<ref>{{cite web |title=PMCMRplus: Calculate Pairwise Multiple Comparisons of Mean Rank Sums Extended |date=17 August 2022 |url=https://cran.r-project.org/web/packages/PMCMRplus/index.html }}</ref>

== References ==
{{Reflist|30em}}

== Further reading ==
* {{cite book |last=Daniel |first=Wayne W. |chapter=Friedman two-way analysis of variance by ranks |title=Applied Nonparametric Statistics |location=Boston |publisher=PWS-Kent |edition=2nd |year=1990 |isbn=978-0-534-91976-4 |pages=262–74 |chapter-url=https://books.google.com/books?id=0hPvAAAAMAAJ&pg=PA262 }}
* {{cite book |last=Kendall |first=M. G. |title=Rank Correlation Methods |year=1970 |edition=4th |location=London |publisher=Charles Griffin |isbn=978-0-85264-199-6 }}
* {{cite book |last1=Hollander |first1=M. |last2=Wolfe |first2=D. A. |title=Nonparametric Statistics |year=1973 |location=New York |publisher=J. Wiley |isbn=978-0-471-40635-8 |url=https://archive.org/details/nonparametricsta00holl |url-access=registration }}
* {{cite book |last1=Siegel |first1=Sidney |last2=Castellan |first2=N. John Jr. |title=Nonparametric Statistics for the Behavioral Sciences |year=1988 |edition=2nd |location=New York |publisher=McGraw-Hill |isbn=978-0-07-100326-1 |url=https://books.google.com/books?id=ha3AQgAACAAJ }}

{{statistics|inference|collapsed}}

{{DEFAULTSORT:Friedman Test}}
[[Category:Analysis of variance]]
[[Category:Statistical tests]]
[[Category:Nonparametric statistics]]

{{Sourceattribution|Friedman test}}