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Friedman Test with Effect Size

Source: R/GLM_ANOVA.R
compute_friedman_test.Rd

Runs the Friedman rank-sum test for a complete block design (repeated measures) and returns the test statistic, p-value, and Kendall's W (kendall_w) as the effect size.

In simple terms, this tests whether the same subjects (blocks) respond differently across conditions (groups), and quantifies how consistently the subjects rank the conditions in the same order.

Usage

compute_friedman_test(formula, df, ci = FALSE, conf.level = 0.95, nboot = 1000)

Source

Q is computed with the same tie-corrected formula as stats::friedman.test (R Core Team, R package stats, licensed GPL-2 | GPL-3).

Arguments

formula

A formula in the form y ~ group | block, where group is the repeated condition and block identifies the subject.

df

A data frame in long format containing the variables in formula, with one row per block and group.

ci

If TRUE, adds a percentile bootstrap confidence interval for kendall_w.

conf.level

Confidence level of the interval.

nboot

Number of bootstrap resamples.

Value

A one-row data frame with:

  • formula: model formula used

  • method: test name

  • kendall_w: Kendall's coefficient of concordance, \(Q/(n(k-1))\)

  • kendall_w_lower, kendall_w_upper: confidence interval of kendall_w (only if ci = TRUE)

  • Q: Friedman chi-squared statistic, corrected for ties

  • df: degrees of freedom (\(k-1\))

  • n: number of complete blocks used

  • p: p-value

Details

The observations are ranked within each block. With \(n\) blocks, \(k\) groups, \(R_j\) the rank sum of group \(j\) and \(t\) the sizes of the groups of tied values within blocks, $$Q=\frac{12\sum_{j=1}^{k}\left(R_j-n(k+1)/2\right)^2}{nk(k+1)-\sum(t^3-t)/(k-1)}$$ Under the null hypothesis \(Q\) approximately follows a chi-squared distribution with \(k-1\) degrees of freedom.

kendall_w is in [0, 1]. 0 means the blocks rank the groups in no consistent order; 1 means every block ranks the groups in the same order.

As in rstatix::friedman_effsize, the rules of thumb for kendall_w are the Cohen (1988) benchmarks for correlations:

  • tiny: < 0.1

  • small: 0.1 to < 0.3

  • medium: 0.3 to < 0.5

  • large: >= 0.5

These are rough guides; what counts as a meaningful effect depends on the field.

Rows with a missing group or block are removed. Blocks with a missing value in any group are removed, as in the default method of stats::friedman.test; n reports how many blocks remain. A block with more than one observation for the same group is an error.

The confidence interval resamples blocks with replacement nboot times, recomputes kendall_w in each resample and takes the \((1-conf.level)/2\) and \(1-(1-conf.level)/2\) quantiles. Use set.seed for reproducible intervals.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.

Friedman, M. (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:10.1080/01621459.1937.10503522

Kendall, M. G., & Babington Smith, B. (1939). The problem of m rankings. The Annals of Mathematical Statistics, 10(3), 275-287. doi:10.1214/aoms/1177732186

R Core Team (2026). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. doi:10.32614/R.manuals

Examples

form <- formula(uptake ~ conc | Plant)
friedman.test(formula = form, data = df_co2)
#> 
#> 	Friedman rank sum test
#> 
#> data:  uptake and conc and Plant
#> Friedman chi-squared = 59.677, df = 6, p-value = 5.236e-11
#> 
rstatix::friedman_effsize(df_co2, form, ci = TRUE, conf.level = 0.95,
                          ci.type = "perc", nboot = 100)
#> # A tibble: 1 × 7
#>   .y.        n effsize conf.low conf.high method    magnitude
#> * <chr>  <int>   <dbl>    <dbl>     <dbl> <chr>     <ord>    
#> 1 uptake    12   0.829     0.75      0.94 Kendall W large    
effectsize::kendalls_w(form, data = df_co2)
#> Warning: 1 block(s) contain ties.
#> Kendall's W |       95% CI
#> --------------------------
#> 0.83        | [0.76, 1.00]
#> 
#> - One-sided CIs: upper bound fixed at [1.00].
compute_friedman_test(formula = form, df = df_co2)
#>                 formula                 method kendall_w        Q df  n            p
#> 1 uptake ~ conc | Plant Friedman rank sum test 0.8288423 59.67665  6 12 5.235869e-11
set.seed(1)
compute_friedman_test(formula = form, df = df_co2, ci = TRUE)
#>                 formula                 method kendall_w kendall_w_lower kendall_w_upper        Q df  n            p
#> 1 uptake ~ conc | Plant Friedman rank sum test 0.8288423       0.7565647       0.9340429 59.67665  6 12 5.235869e-11