Friedman test

For US Army cryptologist William F. Friedman's cryptanalytic test, see Vigenère cipher § Friedman test.

For Friedman pregnancy test, see Rabbit test.

The Friedman test is a non-parametric statistical test developed by Milton Friedman.^[1]^[2]^[3] 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 each rate k different wines. Are any wines ranked consistently higher or lower than the others?
n welders each use k welding torches, and the ensuing welds were rated on quality. Do any of the 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.

Friedman test is widely supported by many statistical software packages.

Method

Given data $\{x_{ij}\}_{n\times k}$ , that is, a matrix 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 matrix $\{r_{ij}\}_{n \times k}$ where the entry $r_{ij}$ is the rank of $x_{ij}$ within block $i$ .
Find the values:
- $\bar{r}_{\cdot j} = \frac{1}{n} \sum_{i=1}^n {r_{ij}}$
- $\bar{r} = \frac{1}{nk}\sum_{i=1}^n \sum_{j=1}^k r_{ij}$
- $SS_t = n\sum_{j=1}^k (\bar{r}_{\cdot j} - \bar{r})^2$ ,
- $SS_e = \frac{1}{n(k-1)} \sum_{i=1}^n \sum_{j=1}^k (r_{ij} - \bar{r})^2$
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), the probability distribution of Q can be approximated by that of a chi-squared distribution. In this case the p-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-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.
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

Post hoc analysis

Post-hoc tests were proposed by Schaich and Hamerle (1984)^[4] as well as Conover (1971, 1980)^[5] 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, pp. 275).^[6]

Not all statistical packages support Post-hoc analysis for Friedman's test, but user-contributed code exists that provides these facilities (for example in SPSS,^[7] and in R.^[8])

References

↑ 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. American Statistical Association. 32 (200): 675–701. doi:10.2307/2279372. JSTOR 2279372.
↑ 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. American Statistical Association. 34 (205): 109. doi:10.2307/2279169. JSTOR 2279169.
↑ 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. JSTOR 2235971.
↑ Schaich, E. & Hamerle, A. (1984). Verteilungsfreie statistische Prüfverfahren. Berlin: Springer. ISBN 3-540-13776-9.
↑ Conover, W. J. (1971, 1980). Practical nonparametric statistics. New York: Wiley. ISBN 0-471-16851-3.
↑ Bortz, J., Lienert, G. & Boehnke, K. (2000). Verteilungsfreie Methoden in der Biostatistik. Berlin: Springer. ISBN 3-540-67590-6.
↑ "Post-hoc comparisons for Friedman test".
↑ "Post hoc analysis for Friedman's Test (R code)". February 22, 2010.

Daniel, Wayne W. (1990). "Friedman two-way analysis of variance by ranks". Applied Nonparametric Statistics (2nd ed.). Boston: PWS-Kent. pp. 262–274. ISBN 0-534-91976-6.
Kendall, M. G. (1970). Rank Correlation Methods (4th ed.). London: Charles Griffin. ISBN 0-85264-199-0.
Hollander, M.; Wolfe, D. A. (1973). Nonparametric Statistics. New York: J. Wiley. ISBN 0-471-40635-X.
Siegel, Sidney; Castellan, N. John Jr. (1988). Nonparametric Statistics for the Behavioral Sciences (2nd ed.). New York: McGraw-Hill. ISBN 0-07-100326-6.

Statistics

Descriptive statistics

Continuous data

Center	Mean arithmetic geometric harmonic Median Mode

Dispersion	Variance Standard deviation Coefficient of variation Percentile Range Interquartile range

Shape	Moments Skewness Kurtosis L-moments

Count data

Index of dispersion

Summary tables

Dependence

Graphics

Data collection

Study design	Population Statistic Effect size Statistical power Sample size determination Missing data

Survey methodology	Sampling Standard error stratified cluster Opinion poll Questionnaire

Controlled experiments	Design control optimal Controlled trial Randomized Random assignment Replication Blocking Interaction Factorial experiment

Uncontrolled studies	Observational study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Frequentist inference

Point estimation	Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance Rao–Blackwellization Lehmann–Scheffé theorem Median unbiased Plug-in

Interval estimation	Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife

Testing hypotheses	1- & 2-tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons

Parametric tests	Likelihood-ratio Wald Score

Specific tests

Z (normal) Student's t-test F

Goodness of fit	Chi-squared Kolmogorov–Smirnov Anderson–Darling Normality (Shapiro–Wilk) Likelihood-ratio test Model selection Cross validation AIC BIC

Rank statistics	Sign Sample median Signed rank (Wilcoxon) Hodges–Lehmann estimator Rank sum (Mann–Whitney) Nonparametric anova 1-way (Kruskal–Wallis) 2-way (Friedman) Ordered alternative (Jonckheere–Terpstra)

Bayesian inference

Correlation	Pearson product–moment Partial correlation Confounding variable Coefficient of determination

Regression analysis	Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)

Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression

Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity

Generalized linear model	Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions

Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

Categorical

Multivariate

Time-series

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality

Specific tests	Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey

Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR)

Frequency domain	Spectral density estimation Fourier analysis Wavelet

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time

Hazard function	Nelson–Aalen estimator

Test	Log-rank test

Applications

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics

Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification

Social statistics	Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population statistics Psychometrics

Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

Category
Portal
Commons
WikiProject

Milton Friedman

Academic career	Statistics Friedman test Decision theory Friedman–Savage utility function Economics Monetarism Friedman rule Friedman's k-percent rule Miracle of Chile Permanent income hypothesis

Works	Essays in Positive Economics (1953) Capitalism and Freedom (1962) A Monetary History of the United States (1963) Free to Choose (1980) Milton Friedman bibliography

Family	Rose Friedman (wife) David D. Friedman (son) Patri Friedman (grandson)

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Friedman test

Method

Related tests

Post hoc analysis

References

Further reading