The first step in the statistical analysis of experimental data is usually an Analysis of Variance (ANOVA), provided its assumptions are met. The hypothesis tested by ANOVA is the equality of population means across several groups:
H₀: μ₁ = μ₂ = ... = μₖ
H₁: At least two means differ.
However, ANOVA does not identify which specific groups have significantly different means.
When ANOVA yields a significant result, researchers typically turn to a multiple comparisons test to evaluate pairwise differences among group means. In this post, we will discuss one such test—Tukey’s test, likely the most commonly used in this context.
Key Features of Tukey’s HSD Test
Tukey’s HSD test allows for all possible pairwise comparisons (unplanned comparisons), meaning researchers do not need to pre-specify which comparisons will be made during the experimental design phase. For this reason, Tukey’s test is considered a post hoc test.
Procedure for Tukey’s HSD Test
To apply Tukey’s HSD test, we calculate the Honestly Significant Difference (HSD) between two means—the smallest difference required for them to be considered statistically different at a given significance level. The HSD is given by:
Where:
q(k, df, α) = Studentized range statistic (from tables, based on the number of groups k, residual degrees of freedom df, and significance level α).
MSE = Mean Square Error from ANOVA.
r = Number of replicates per group.
Two means are considered significantly different (at the chosen significance level) if their absolute difference is greater than or equal to the HSD.
How to Use the Studentized Range (q) Table
Below is an excerpt from the q-table. The bolded value corresponds to a comparison involving six treatments (k=6) and 24 residual degrees of freedom (df=24) at a 5% significance level (α=0.05).
Table Values of q for α=5%
💡 Note:
In statistical software and English literature, the term HSD (Honestly Significant Difference) is commonly used.
In Brazil, this is often called the minimum significant difference (represented by the Greek letter Δ).
The term Least Significant Difference (LSD) refers specifically to Fisher’s test, whereas Tukey’s test uses HSD, a name coined by its creator, John W. Tukey.
Example: Blood Pressure Reduction Study
Consider the blood pressure reduction data in Table 1, analyzed using ANOVA (Table 2). The F-test was significant at the 5% level, indicating that at least one mean differs from the others. Group means are shown in Table 3.
Table 1: Blood pressure reduction (mmHg)
Table 2: ANOVA
Table 3: Means of blood pressure reduction by group
Our goal is to identify which means differ significantly using Tukey’s HDS test.
HSD Calculation
q = 4.3727 (from q-table, k=6, df=24, α=5%)
MSE = 36.00 (Mean Square Error from ANOVA)
r = 5 (replicates per group)
Pairwise Comparisons
We compare the means pairwise, marking significant differences (at α=5%) with an asterisk (*).
Conclusion
According to Tukey’s HSD test (α=5%):
The mean of Treatment A was significantly higher than those of B and the Control.
The mean of Treatment D was significantly higher than those of B, C, E, and the Control.
