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
Arguments
- x
Kruskal-Wallis results from
kruskal_wallis()- ...
Additional arguments passed to methods. The method for
kruskal_wallisobjects acceptsp_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
Non-parametric equivalent of
tukey_test()andscheffe_test()Follow-up to
kruskal_wallis(), just like Tukey follows ANOVAUses ranks instead of raw values, making it robust to outliers
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) Results
#> ---------------------------------------
#>
#> - Dependent variable: life_satisfaction
#> - Grouping variable: education
#> - P-value adjustment: Bonferroni
#>
#> ----------------------------------------------------------------------------
#> Group 1 Group 2 Z p (unadj) p (adj) Sig
#> Basic Secondary Intermediate Secondary -7.402 <.001 <.001 ***
#> Basic Secondary Academic Secondary -9.465 <.001 <.001 ***
#> Basic Secondary University -11.159 <.001 <.001 ***
#> Intermediate Secondary Academic Secondary -1.973 0.048 0.291
#> Intermediate Secondary University -4.539 <.001 <.001 ***
#> Academic Secondary University -2.790 0.005 0.032 *
#> ----------------------------------------------------------------------------
#>
#>
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05
#>
#> Interpretation:
#> - Positive Z: First group has higher mean rank
#> - Negative Z: First group has lower mean rank
#> - p-values are adjusted for multiple comparisons
# With Holm correction (less conservative)
kw_result %>% dunn_test(p_adjust = "holm")
#> Dunn Post-Hoc Test (Holm) Results
#> ---------------------------------
#>
#> - Dependent variable: life_satisfaction
#> - Grouping variable: education
#> - P-value adjustment: Holm
#>
#> ----------------------------------------------------------------------------
#> Group 1 Group 2 Z p (unadj) p (adj) Sig
#> Basic Secondary Intermediate Secondary -7.402 <.001 <.001 ***
#> Basic Secondary Academic Secondary -9.465 <.001 <.001 ***
#> Basic Secondary University -11.159 <.001 <.001 ***
#> Intermediate Secondary Academic Secondary -1.973 0.048 0.048 *
#> Intermediate Secondary University -4.539 <.001 <.001 ***
#> Academic Secondary University -2.790 0.005 0.011 *
#> ----------------------------------------------------------------------------
#>
#>
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05
#>
#> Interpretation:
#> - Positive Z: First group has higher mean rank
#> - Negative Z: First group has lower mean rank
#> - p-values are adjusted for multiple comparisons
# With Benjamini-Hochberg (controls false discovery rate)
kw_result %>% dunn_test(p_adjust = "BH")
#> Dunn Post-Hoc Test (Benjamini-Hochberg) Results
#> -----------------------------------------------
#>
#> - Dependent variable: life_satisfaction
#> - Grouping variable: education
#> - P-value adjustment: Benjamini-Hochberg
#>
#> ----------------------------------------------------------------------------
#> Group 1 Group 2 Z p (unadj) p (adj) Sig
#> Basic Secondary Intermediate Secondary -7.402 <.001 <.001 ***
#> Basic Secondary Academic Secondary -9.465 <.001 <.001 ***
#> Basic Secondary University -11.159 <.001 <.001 ***
#> Intermediate Secondary Academic Secondary -1.973 0.048 0.048 *
#> Intermediate Secondary University -4.539 <.001 <.001 ***
#> Academic Secondary University -2.790 0.005 0.006 **
#> ----------------------------------------------------------------------------
#>
#>
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05
#>
#> Interpretation:
#> - Positive Z: First group has higher mean rank
#> - Negative Z: First group has lower mean rank
#> - p-values are adjusted for multiple comparisons
# With weights
kw_weighted <- survey_data %>%
kruskal_wallis(life_satisfaction, group = education,
weights = sampling_weight)
kw_weighted %>% dunn_test()
#> Weighted Dunn Post-Hoc Test (Bonferroni) Results
#> ------------------------------------------------
#>
#> - Dependent variable: life_satisfaction
#> - Grouping variable: education
#> - Weights variable: sampling_weight
#> - P-value adjustment: Bonferroni
#>
#> ----------------------------------------------------------------------------
#> Group 1 Group 2 Z p (unadj) p (adj) Sig
#> Basic Secondary Intermediate Secondary -7.351 <.001 <.001 ***
#> Basic Secondary Academic Secondary -9.444 <.001 <.001 ***
#> Basic Secondary University -10.927 <.001 <.001 ***
#> Intermediate Secondary Academic Secondary -2.007 0.045 0.269
#> Intermediate Secondary University -4.484 <.001 <.001 ***
#> Academic Secondary University -2.735 0.006 0.037 *
#> ----------------------------------------------------------------------------
#>
#>
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05
#>
#> Interpretation:
#> - Positive Z: First group has higher mean rank
#> - Negative Z: First group has lower mean rank
#> - p-values are adjusted for multiple comparisons
# Multiple variables
kw_multi <- survey_data %>%
kruskal_wallis(life_satisfaction, trust_government,
group = education)
kw_multi %>% dunn_test()
#> Dunn Post-Hoc Test (Bonferroni) Results
#> ---------------------------------------
#>
#> - Grouping variable: education
#> - P-value adjustment: Bonferroni
#>
#>
#> --- life_satisfaction ---
#>
#> ----------------------------------------------------------------------------
#> Group 1 Group 2 Z p (unadj) p (adj) Sig
#> Basic Secondary Intermediate Secondary -7.402 <.001 <.001 ***
#> Basic Secondary Academic Secondary -9.465 <.001 <.001 ***
#> Basic Secondary University -11.159 <.001 <.001 ***
#> Intermediate Secondary Academic Secondary -1.973 0.048 0.291
#> Intermediate Secondary University -4.539 <.001 <.001 ***
#> Academic Secondary University -2.790 0.005 0.032 *
#> ----------------------------------------------------------------------------
#>
#>
#> --- trust_government ---
#>
#> ---------------------------------------------------------------------------
#> Group 1 Group 2 Z p (unadj) p (adj) Sig
#> Basic Secondary Intermediate Secondary 0.955 0.340 1.000
#> Basic Secondary Academic Secondary 0.552 0.581 1.000
#> Basic Secondary University -0.037 0.971 1.000
#> Intermediate Secondary Academic Secondary -0.378 0.706 1.000
#> Intermediate Secondary University -0.821 0.411 1.000
#> Academic Secondary University -0.490 0.624 1.000
#> ---------------------------------------------------------------------------
#>
#>
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05
#>
#> Interpretation:
#> - Positive Z: First group has higher mean rank
#> - Negative Z: First group has lower mean rank
#> - p-values are adjusted for multiple comparisons
# Grouped analysis
kw_grouped <- survey_data %>%
group_by(region) %>%
kruskal_wallis(life_satisfaction, group = education)
kw_grouped %>% dunn_test()
#> Dunn Post-Hoc Test (Bonferroni) Results
#> ---------------------------------------
#>
#> - Grouping variable: education
#> - P-value adjustment: Bonferroni
#>
#>
#> Group: region = East
#> --------------------
#>
#> --- life_satisfaction ---
#>
#> ---------------------------------------------------------------------------
#> Group 1 Group 2 Z p (unadj) p (adj) Sig
#> Basic Secondary Intermediate Secondary -1.886 0.059 0.356
#> Basic Secondary Academic Secondary -3.158 0.002 0.010 **
#> Basic Secondary University -3.429 <.001 0.004 **
#> Intermediate Secondary Academic Secondary -1.230 0.219 1.000
#> Intermediate Secondary University -1.705 0.088 0.529
#> Academic Secondary University -0.592 0.554 1.000
#> ---------------------------------------------------------------------------
#>
#>
#> Group: region = West
#> --------------------
#>
#> --- life_satisfaction ---
#>
#> ----------------------------------------------------------------------------
#> Group 1 Group 2 Z p (unadj) p (adj) Sig
#> Basic Secondary Intermediate Secondary -7.345 <.001 <.001 ***
#> Basic Secondary Academic Secondary -9.023 <.001 <.001 ***
#> Basic Secondary University -10.774 <.001 <.001 ***
#> Intermediate Secondary Academic Secondary -1.587 0.113 0.675
#> Intermediate Secondary University -4.226 <.001 <.001 ***
#> Academic Secondary University -2.830 0.005 0.028 *
#> ----------------------------------------------------------------------------
#>
#>
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05
#>
#> Interpretation:
#> - Positive Z: First group has higher mean rank
#> - Negative Z: First group has lower mean rank
#> - p-values are adjusted for multiple comparisons