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Kruskal-Wallis Test

Includes: H statistic (uncorrected), H statistic (tie-corrected), Dunn’s post-hoc comparisons (Bonferroni), optional boxplot/descriptives, optional homogeneity-of-variance tests.
Purpose: Nonparametric one-way ANOVA alternative for comparing two or more independent groups.

Overview

The Kruskal–Wallis test compares k independent groups by ranking all observations together and asking whether the groups have systematically different mean ranks.

  • Works for continuous or ordinal outcomes.
  • Does not assume normality.
  • With similarly-shaped group distributions, it is often interpreted as a difference in location (typical values); more generally, it is a test that at least one group tends to yield larger/smaller values than another.

BESHStatNG reports:

  • H (standard Kruskal–Wallis statistic),
  • Hcor (tie-corrected H),
  • corresponding two-sided p-values (chi-square approximation, df = k − 1),
  • optional Dunn’s multiple comparisons (pairwise, Bonferroni-adjusted),
  • optional descriptive statistics, boxplot, and variance-homogeneity tests.

Example dataset

Download the CSV used in the screenshots:

In the screenshots, the dataset is arranged in wide format (each column is a group):

  • Relaxation response and biofeedback
  • Relaxation response alone
  • Untreated

Screenshots (BESHStatNG)

Input tab

Kruskal-Wallis – Input

Options tab

Kruskal-Wallis – Options

Results (test + Dunn post-hoc)

Kruskal-Wallis – Results

Results (descriptives + boxplot)

Kruskal-Wallis – Descriptives + plot

Results (homogeneity of variances)

Kruskal-Wallis – Variance homogeneity

When to use it

Use the Kruskal–Wallis test when:

  • you have k ≥ 2 independent groups (different subjects in each group),
  • the outcome is numeric or at least ordinal,
  • you want a robust comparison when normality is questionable and/or outliers are present.

Key requirements / considerations:

  • Independence: observations are independent within and between groups.
  • Similar distribution shapes: if shapes/spreads differ strongly, interpret the test as “at least one group tends to produce larger values” rather than strictly a median shift.

Inputs in Excel

BESHStatNG uses the shared “Group by Column / Group by ID” dialog.

Option A: Group by Column (wide format)

  • Select a rectangular range where each column is one group.
  • If the first row contains text, it is treated as the group name.

This is the most common layout for Kruskal–Wallis in Excel.

Option B: Group by ID (long format)

Use this when you have two columns:

  • Group ID (labels: A/B/C… or 1/2/3…)
  • Data (numeric values)

BESHStatNG splits the values by unique group IDs and runs the test.

Output destination

  • Output range (current sheet)
  • New worksheet
  • New workbook

Options

On the Options tab you can add:

Homogeneity of variances (optional)

You can also compute up to four variance-homogeneity tests:

  • Fligner–Killeen
  • Levene (median-centered / Brown–Forsythe style)
  • Squared Ranks
  • Bartlett

These are described in detail here:

Output and how to read it

Main Kruskal–Wallis table

BESHStatNG writes a “Kruskal-Wallis Test” table containing:

  • n: total sample size \(N\) across all groups
  • Number of groups: \(k\)
  • Test statistics H + Two sided P-value (ties un-corrected)
  • Test statistics Hcor + Two sided p-value (ties corrected)

Example interpretation (from the screenshots):

  • \(H=5.6959\), p = 0.0580 (uncorrected)
  • \(H_{cor}=5.7254\), p = 0.0571 (tie-corrected)
  • p is close to 0.05 but slightly above, so evidence for a group difference is suggestive but not strong at the 5% level.

Dunn’s multiple comparison table (optional / shown in screenshots)

If post-hoc comparisons are produced, BESHStatNG adds:

  • Mean rank diff.: difference in mean ranks between each pair of groups
  • Z: normal approximation statistic for the pairwise comparison
  • Two sided P-value: Bonferroni-adjusted two-sided p-value

Interpretation: - A small adjusted p-value (e.g., < 0.05) suggests that that pair of groups differs in central tendency (in terms of ranks), controlling the family-wise error rate across all pairwise comparisons.

What it does (math and implementation details)

Let there be \(k\) groups. Group \(j\) has \(n_j\) observations \(x_{j1},\dots,x_{jn_j}\). Let

\[ N=\sum_{j=1}^k n_j. \]

A) Ranking (midranks for ties)

Pool all observations and assign ranks \(1,\dots,N\).
Ties are given the average rank (midranks).

Let \(R_j\) be the sum of ranks in group \(j\), and \(\bar R_j = R_j/n_j\) the mean rank.

B) Kruskal–Wallis H statistic

\[ H = \frac{12}{N(N+1)} \sum_{j=1}^k \frac{R_j^2}{n_j} - 3(N+1). \]

Under \(H_0\) (all groups have the same distribution), \(H\) is approximated by a chi-square distribution with \(k-1\) degrees of freedom:

\[ p = 1 - F_{\chi^2(k-1)}(H). \]

C) Tie correction

Let the pooled data contain tied groups of sizes \(t_1,t_2,\dots\). Define

\[ T=\sum_{m}(t_m^3-t_m). \]

BESHStatNG reports the tie-corrected statistic:

\[ H_{cor} = \frac{H}{1-\frac{T}{N^3-N}}. \]

The p-value is computed as:

\[ p_{cor} = 1 - F_{\chi^2(k-1)}(H_{cor}). \]

D) Dunn’s post-hoc comparisons (Bonferroni)

For each pair of groups \(i\) and \(j\), BESHStatNG computes the mean rank difference:

\[ \Delta_{ij} = \bar R_i - \bar R_j \]

and the Z statistic:

\[ Z_{ij}=\frac{|\Delta_{ij}|}{\sqrt{a\left(\frac{1}{n_i}+\frac{1}{n_j}\right)}}, \]

where the tie-adjusted constant \(a\) is:

\[ a = \frac{N(N+1)}{12} - \frac{T}{12(N-1)}. \]

Two-sided p-value (normal approximation):

\[ p_{ij} = 2\Phi(-|Z_{ij}|). \]

BESHStatNG applies Bonferroni correction for the number of pairwise comparisons \(m=k(k-1)/2\):

\[ p^{adj}_{ij}=\min(1,\; m\cdot p_{ij}). \]

R code (reference using standard R functions)

R’s base kruskal.test() returns the tie-corrected Kruskal–Wallis statistic by default.
To match BESHStatNG’s output exactly (both uncorrected and corrected H, plus Dunn + Bonferroni), you can compute the components explicitly.

dat <- read.csv("020kruskalwalliscsv.csv")

# Wide format: each column is a group
vals <- c(dat[[1]], dat[[2]], dat[[3]])
grp  <- factor(rep(names(dat), each = nrow(dat)))

# Drop non-finite values (BESHStatNG ignores invalid cells)
ok <- is.finite(vals)
vals <- vals[ok]
grp  <- grp[ok]

k <- nlevels(grp)
N <- length(vals)

# --- Kruskal–Wallis (R default = tie-corrected) ---
kw <- kruskal.test(vals ~ grp)
kw$statistic   # ~= H_cor in BESHStatNG
kw$p.value

# --- Compute both H (uncorrected) and H_cor (tie-corrected) like BESHStatNG ---
r   <- rank(vals, ties.method = "average")
Rj  <- tapply(r,   grp, sum)
nj  <- tapply(vals, grp, length)

H <- (12/(N*(N+1))) * sum((Rj^2)/nj) - 3*(N+1)
p_unc <- pchisq(H, df = k-1, lower.tail = FALSE)

tab <- table(vals)
T <- sum((tab[tab > 1]^3 - tab[tab > 1]))
C <- 1 - T/(N^3 - N)

Hcor <- H / C
p_cor <- pchisq(Hcor, df = k-1, lower.tail = FALSE)

c(H_uncorrected = H, p_uncorrected = p_unc,
  H_corrected   = Hcor, p_corrected = p_cor)

# --- Dunn's multiple comparisons with Bonferroni ---
# Package option 1 (recommended):
# install.packages("FSA")
FSA::dunnTest(vals ~ grp, method = "bonferroni")

# Package option 2:
# install.packages("dunn.test")
dunn.test::dunn.test(vals, grp, method = "bonferroni", kw = FALSE, list = TRUE)

# --- Homogeneity of variances (matches the options panel) ---
fligner.test(vals ~ grp)
bartlett.test(vals ~ grp)

# Levene/Brown-Forsythe (median-centered):
# install.packages("car")
car::leveneTest(vals ~ grp, center = median)

# Squared Ranks test (BESHStatNG-style, implemented directly)
# (See also: BESHStatNG help page "Homogeneity of Variance")
dev <- abs(vals - ave(vals, grp, FUN = mean))
rk  <- rank(dev, ties.method = "average")
u   <- rk^2

Sj <- tapply(u, grp, sum)
ubar <- mean(u)
d <- (sum(u^2) - N * ubar^2) / (N - 1)

X2 <- (sum((Sj^2)/nj) - N * ubar^2) / d
p_sq <- pchisq(X2, df = k-1, lower.tail = FALSE)

c(SquaredRanks_X2 = X2, p_value = p_sq)

Why R results may be slightly different from BESHStatNG

  • Kruskal–Wallis statistic: kruskal.test() reports the tie-corrected statistic; BESHStatNG reports both uncorrected \(H\) and corrected \(H_{cor}\).
  • Dunn’s test defaults: different R packages may use different default p-value adjustments (Holm vs Bonferroni) and may differ slightly in rounding; set method = "bonferroni" to match the add-in.
  • Variance homogeneity tests: implementations match standard definitions; small numeric differences are usually due to rounding and how missing values are removed.

Quartiles in R

BESHStatNG uses the CDF method (SAS Method 5) for quartiles.
To match its \(Q_1\), Median and \(Q_3\) in R, use quantile(x, probs = c(0.25, 0.5, 0.75), type = 2).

Notes

  • Invalid / missing cells: empty cells, text, logical values, and Excel error values are ignored during data import (only numeric values are used).
  • Post-hoc testing: Dunn’s test is typically interpreted after a significant (or near-significant) Kruskal–Wallis result, but you may still explore pairwise differences cautiously.
  • Multiple testing: BESHStatNG uses Bonferroni adjustment for Dunn’s test (controls family-wise error rate; conservative when many groups).

References

  • Altman D.G. Practical Statistics for medical research. Chapman & Hall, 1991.
  • Dunn O.J. Multiple Comparisons Using Rank Sums. Technometrics, Vol. 6, No. 3 (1964), 241-252.
  • Kruskal W.H., Wallis W.A. (1952) Use of Ranks in One-Criterion Variance Analysis. Journal of the American Statistical Association, 47(260), 583-621.
  • Kruskal W.H. (1952). A nonparametric test for the several sample problem. Annals of Mathematical Statistics, 23, 525-540.

See also