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Kruskal-Wallis Test with Effect Sizes

Source: R/GLM_ANOVA.R
compute_kruskal_wallis_test.Rd

Runs a one-way Kruskal-Wallis rank-sum test and returns the test statistic, p-value, and two effect sizes:

  • etasq: eta-squared for Kruskal-Wallis (\(\eta_H^2\))

  • epsilonsq: epsilon-squared (\(\epsilon^2\))

In simple terms, this tests whether groups differ in their distributions, and quantifies how large that group effect is.

Usage

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

Source

The computation of H, including the correction for ties, is adapted from stats::kruskal.test (R Core Team, R package stats, licensed GPL-2 | GPL-3). etasq and epsilonsq use the same formulas as rstatix::kruskal_effsize and effectsize::rank_epsilon_squared.

Arguments

formula

A one-way formula in the form y ~ group.

df

A data frame containing the variables in formula.

ci

If TRUE, adds percentile bootstrap confidence intervals for both etasq and epsilonsq.

conf.level

Confidence level of the intervals.

nboot

Number of bootstrap resamples.

Value

A one-row data frame with:

  • formula: model formula used

  • method: test name

  • etasq: Kruskal-Wallis eta-squared, \((H-k+1)/(n-k)\)

  • etasq_lower, etasq_upper: confidence interval of etasq (only if ci = TRUE)

  • epsilonsq: epsilon-squared, \(H/(n-1)\)

  • epsilonsq_lower, epsilonsq_upper: confidence interval of epsilonsq (only if ci = TRUE)

  • H: Kruskal-Wallis chi-squared statistic

  • df: degrees of freedom (\(k-1\))

  • p: p-value

Details

epsilonsq is in [0, 1]. etasq is at most 1 and is negative when \(H < k-1\), that is when the groups differ less than expected by chance. Multiplying by 100 gives an approximate percentage-style interpretation of explained rank variance.

Rules of thumb for etasq and epsilonsq. Small, medium and large are the \(\eta^2\) benchmarks of Cohen (1988); tiny, very large and huge convert the Cohen's d benchmarks of Sawilowsky (2009) with \(\eta^2=d^2/(d^2+4)\):

  • tiny: < 0.01 (d < 0.2)

  • small: 0.01 to < 0.06 (d = 0.2)

  • medium: 0.06 to < 0.14 (d = 0.5)

  • large: 0.14 to < 0.26 (d = 0.8)

  • very large: 0.26 to < 0.50 (d = 1.2)

  • huge: >= 0.50 (d = 2.0)

These are rough guides; what counts as a meaningful effect depends on the field. epsilonsq is biased upwards by about \((k-1)/(n-1)\) in small samples.

Rows with a missing value in the outcome or the grouping variable are removed before the test, as in stats::kruskal.test.

The confidence intervals resample rows with replacement nboot times, recompute both effect sizes in each resample and take the \((1-conf.level)/2\) and \(1-(1-conf.level)/2\) quantiles. Use set.seed for reproducible intervals.

References

Ben-Shachar, M. S., Lüdecke, D., & Makowski, D. (2020). effectsize: Estimation of effect size indices and standardized parameters. Journal of Open Source Software, 5(56), 2815. doi:10.21105/joss.02815

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

Kassambara, A. (2026). rstatix: Pipe-friendly framework for basic statistical tests (R package version 1.1.0). doi:10.32614/CRAN.package.rstatix

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

Sawilowsky, S. S. (2009). New effect size rules of thumb. Journal of Modern Applied Statistical Methods, 8(2), 597-599. doi:10.22237/jmasm/1257035100

Tomczak, M., & Tomczak, E. (2014). The need to report effect size estimates revisited. An overview of some recommended measures of effect size. Trends in Sport Sciences, 1(21), 19-25.

Examples

form <- formula(bp_before ~ agegrp)
kruskal.test(formula = form, data = df_blood_pressure)
#> 
#> 	Kruskal-Wallis rank sum test
#> 
#> data:  bp_before by agegrp
#> Kruskal-Wallis chi-squared = 19.564, df = 2, p-value = 5.645e-05
#> 
rcompanion::epsilonSquared(
  x = df_blood_pressure$bp_before,
  g = df_blood_pressure$agegrp,
  group = "row",
  ci = TRUE,
  conf = 0.95,
  type = "perc",
  R = 1000,
  digits = 3
)
#>   epsilon.squared lower.ci upper.ci
#> 1           0.164   0.0668    0.304
rstatix::kruskal_effsize(df_blood_pressure, 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 bp_before   120   0.150     0.04      0.33 eta2[H] large    
compute_kruskal_wallis_test(formula = form, df = df_blood_pressure)
#>              formula                       method     etasq epsilonsq        H df            p
#> 1 bp_before ~ agegrp Kruskal-Wallis rank sum test 0.1501232 0.1644069 19.56442  2 5.644699e-05
set.seed(1)
compute_kruskal_wallis_test(formula = form, df = df_blood_pressure, ci = TRUE)
#>              formula                       method     etasq etasq_lower etasq_upper epsilonsq epsilonsq_lower epsilonsq_upper        H df            p
#> 1 bp_before ~ agegrp Kruskal-Wallis rank sum test 0.1501232  0.05161268   0.2963859 0.1644069      0.06755196       0.3082114 19.56442  2 5.644699e-05