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Find Which Specific Groups Differ After Kruskal-Wallis

Source: R/dunn_test.R
dunn_test.Rd

dunn_test() tells you exactly which groups are different from each other after Kruskal-Wallis finds overall differences. It's the non-parametric equivalent of a Tukey or Scheffe post-hoc test.

Think of it as:

  • Kruskal-Wallis says "there are differences somewhere"

  • Dunn test says "specifically, Group A differs from Group C"

  • A way to make all possible pairwise comparisons using rank-based statistics

Usage

dunn_test(x, ...)

# Default S3 method
dunn_test(x, ...)

Arguments

x

Kruskal-Wallis results from kruskal_wallis()

...

Additional arguments passed to methods. The method for kruskal_wallis objects accepts p_adjust (character): method for adjusting p-values for multiple comparisons. Options: "bonferroni" (default, most conservative), "holm", "BH", "hochberg", "hommel", "BY", "fdr", "none".

Value

Pairwise comparison results showing:

  • Which group pairs are significantly different

  • Z-statistics based on rank differences

  • Adjusted p-values (controlling for multiple comparisons)

Details

Understanding the Results

Z-Statistics: Based on differences in mean ranks between groups

  • Large absolute Z values indicate big rank differences

  • Positive Z: First group has higher mean rank than second

  • Negative Z: Second group has higher mean rank than first

Adjusted P-values: Control for multiple comparisons

  • p < 0.05: Groups are significantly different

  • p >= 0.05: No significant difference between these groups

The Dunn Test Formula

For each pair of groups (i, j): $$Z_{ij} = \frac{\bar{R}_i - \bar{R}_j}{\sqrt{\frac{N(N+1)}{12} \left(\frac{1}{n_i} + \frac{1}{n_j}\right)}}$$

where \(\bar{R}_i\) is the mean rank for group i, \(N\) is the total sample size, and \(n_i\) is the size of group i.

P-Value Adjustment Methods

  • Bonferroni (default): Most conservative, multiplies p by number of comparisons

  • Holm: Step-down method, less conservative than Bonferroni

  • BH: Controls false discovery rate, good for many comparisons

When to Use This

Use Dunn test when:

  • Your Kruskal-Wallis test shows significant differences (p < 0.05)

  • You want to know which specific groups differ

  • Your data violates normality assumptions

  • You have ordinal data or skewed distributions

Relationship to Other Tests

Weighted variants

When the parent kruskal_wallis() result is weighted, the pairwise z statistics use the same frequency-weighted midranks. SPSS NPAR TESTS ignores WEIGHT BY, so weighted results have no SPSS reference (R-only, guarded by an internal invariance suite); see vignette("spss-compatibility") for validation status.

  • Uses ranks instead of raw values, making it robust to outliers

References

Dunn, O. J. (1964). Multiple comparisons using rank sums. Technometrics, 6(3), 241-252.

See also

kruskal_wallis for performing Kruskal-Wallis tests.

tukey_test for parametric post-hoc comparisons after ANOVA.

scheffe_test for conservative parametric post-hoc comparisons.

Other posthoc: levene_test(), pairwise_wilcoxon(), scheffe_test(), tukey_test()

Examples

# Load required packages and data
library(dplyr)
data(survey_data)

# Perform Kruskal-Wallis followed by Dunn post-hoc test
kw_result <- survey_data %>%
  kruskal_wallis(life_satisfaction, group = education)

# Dunn post-hoc comparisons (default: Bonferroni)
kw_result %>% dunn_test()
#> Dunn Post-Hoc Test (Bonferroni) by education
#>   life_satisfaction: 6 comparisons, 5 significant (p < .05)
#> Use summary() for the full comparison table.

# With Holm correction (less conservative)
kw_result %>% dunn_test(p_adjust = "holm")
#> Dunn Post-Hoc Test (Holm) by education
#>   life_satisfaction: 6 comparisons, 6 significant (p < .05)
#> Use summary() for the full comparison table.

# With Benjamini-Hochberg (controls false discovery rate)
kw_result %>% dunn_test(p_adjust = "BH")
#> Dunn Post-Hoc Test (Benjamini-Hochberg) by education
#>   life_satisfaction: 6 comparisons, 6 significant (p < .05)
#> Use summary() for the full comparison table.

# With weights
kw_weighted <- survey_data %>%
  kruskal_wallis(life_satisfaction, group = education,
                 weights = sampling_weight)

kw_weighted %>% dunn_test()
#> Dunn Post-Hoc Test (Bonferroni) by education [Weighted]
#>   life_satisfaction: 6 comparisons, 5 significant (p < .05)
#> Use summary() for the full comparison table.

# Multiple variables
kw_multi <- survey_data %>%
  kruskal_wallis(life_satisfaction, trust_government,
                 group = education)

kw_multi %>% dunn_test()
#> Dunn Post-Hoc Test (Bonferroni) by education
#>   life_satisfaction: 6 comparisons, 5 significant (p < .05)
#>   trust_government: 6 comparisons, 0 significant (p < .05)
#> Use summary() for the full comparison table.

# Grouped analysis
kw_grouped <- survey_data %>%
  group_by(region) %>%
  kruskal_wallis(life_satisfaction, group = education)

kw_grouped %>% dunn_test()
#> Dunn Post-Hoc Test (Bonferroni) by education
#> [region = East]
#>   life_satisfaction: 6 comparisons, 2 significant (p < .05)
#> [region = West]
#>   life_satisfaction: 6 comparisons, 5 significant (p < .05)
#> Use summary() for the full comparison table.