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Tukey's HSD Test

Created:December 1, 2024
Last Updated:April 2, 2025

This calculator helps you perform multiple comparisons between group means after getting a significant ANOVA result. It identifies which specific groups differ significantly from each other while controlling for multiple comparisons. The calculator provides basic test information, group statistics, an ANOVA table, pairwise comparisons, visualizations, and an interpretation in APA format. Click here to populate the sample data for a quick example.

Important considerations before using Tukey's HSD Test:

  • Tukey's HSD test should only be performed after a significant one-way ANOVA result.
  • Each group must have at least 2 samples for the test to work.
  • Groups with zero variance (all identical values) will cause the test to fail.

Calculator

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2. Select Columns & Options

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Tukey's HSD Test

Definition

Tukey's HSD (Honestly Significant Difference) Test is a post-hoc test used after ANOVA to determine which specific groups differ from each other. It controls the family-wise error rate while conducting multiple pairwise comparisons.

Formula

Test Statistic:

HSD=qα,k,N−kMSEnHSD = q_{\alpha,k,N-k}\sqrt{\frac{MSE}{n}}

Where:

  • qα,k,N−kq_{\alpha,k,N-k} = studentized range statistic
  • MSEMSE = Mean Square Error from ANOVA, which is equivalent to Mean Square Within (MSWithinMS_{Within})
  • nn = sample size per group
  • kk = number of groups
  • NN = total sample size

Key Assumptions

Independence: Observations must be independent
Normality: Data within each group should be normally distributed
Homogeneity of Variance: Groups should have equal variances
Equal Sample Sizes: Preferably, groups should have equal sample sizes

Practical Example

Step 1: State the Data

Comparing three fertilizer treatments on plant growth (cm):

Treatment ATreatment BTreatment C
254235
303833
284031
324534
294132
Step 2: Calculate Group Statistics
  • Treatment A: xˉ=28.8\bar{x} = 28.8, n=5n = 5
  • Treatment B: xˉ=41.2\bar{x} = 41.2, n=5n = 5
  • Treatment C: xˉ=33.0\bar{x} = 33.0, n=5n = 5
  • MSE (from ANOVA) = 5.35.3
Step 3: Calculate HSD

For α=0.05\alpha = 0.05, k=3k = 3 groups, df=12df = 12:

q0.05,3,12=3.77q_{0.05,3,12} = 3.77HSD=3.775.35=3.884HSD = 3.77\sqrt{\frac{5.3}{5}} = 3.884
Step 4: Compare Group Differences

∣B−A∣=∣41.2−28.8∣=12.4>3.884|B - A| = |41.2 - 28.8| = 12.4 > 3.884 (significant)

∣C−A∣=∣33.0−28.8∣=4.2>3.884|C - A| = |33.0 - 28.8| = 4.2 > 3.884 (significant)

∣B−C∣=∣41.2−33.0∣=8.2>3.884|B - C| = |41.2 - 33.0| = 8.2 > 3.884 (significant)

Step 5: Draw Conclusions

All pairwise comparisons show significant differences at α=0.05\alpha = 0.05. Treatment B produced significantly higher growth than both A and C, and Treatment C produced significantly higher growth than Treatment A.

Code Examples

R
library(emmeans)
library(tidyverse)

# Example data
data <- data.frame(
  group = rep(c("A", "B", "C"), each = 5),
  values = c(25, 30, 28, 32, 29,  # Group A
             42, 38, 40, 45, 41,  # Group B
             35, 33, 31, 34, 32)  # Group C
)

data |>
  group_by(group) |>
  summarize(mean = mean(values), sd = sd(values), n = n())

# Fit ANOVA model
model <- aov(values ~ group, data = data)
summary(model)

q <- qtukey(0.95, nmeans = 3, df = 12)
mse <- summary(model)[[1]]["Residuals", "Mean Sq"]
hsd <- q * sqrt(mse / 5)
print(hsd)

# Get emmeans and perform Tukey's HSD
emmeans_result <- emmeans(model, "group")
pairs(emmeans_result, adjust = "tukey")
Python
import pandas as pd
from statsmodels.stats.multicomp import pairwise_tukeyhsd

# Example data
group1 = [25, 30, 28, 32, 29]
group2 = [42, 38, 40, 45, 41]
group3 = [35, 33, 31, 34, 32]

# Create DataFrame
data = pd.DataFrame({
    'values': group1 + group2 + group3,
    'group': ['A']*5 + ['B']*5 + ['C']*5
})

# Perform Tukey's HSD
tukey = pairwise_tukeyhsd(
    endog=data['values'],
    groups=data['group'],
    alpha=0.05
)

print(tukey)

Alternative Tests

Consider these alternatives:

  • Bonferroni Test: More conservative, controls family-wise error rate
  • Scheffé's Test: More flexible for complex comparisons
  • Games-Howell Test: When variances are unequal

Verification

Related Calculators

One-Way ANOVA Calculator

Two-Way ANOVA Calculator

Dunnett's Test Calculator

Dunn's Test Calculator