Student-Newman-Keuls Test for Mean Comparison

Numerous methods exist for multiple comparisons following a significant Analysis of Variance (ANOVA). Most involve pairwise comparisons to identify which specific means differ significantly. Among these, the best-known is Tukey's Honestly Significant Difference (HSD) test, which relies on the studentized range distribution (q).

Tukey's test is often considered the gold standard, especially with unequal sample sizes or when confidence intervals are required. However, for equal sample sizes where confidence intervals are not the focus, the Student-Newman-Keuls (SNK) test offers greater statistical power. This post explores the SNK procedure.

While Tukey's test uses a single, conservative critical value for all comparisons—making it robust against Type I errors (false positives)—it can sometimes be overly cautious. A key practical advantage, though, is its universal availability in statistical software, which is not always the case for the SNK test.

2. Explanation of the SNK Procedure

Like Tukey's test, the SNK is based on the studentized range statistic (q). Its key feature is that it is a sequential (stepwise) procedure. Imagine we have four group means, ranked from smallest to largest:

                                                       x̄₁ < x̄₂ < x̄₃ < x̄₄

The SNK procedure works as follows:

1.           Compare the largest and smallest means (a span of *m* = 4 means).

2.           Next, compare the largest with the second smallest, and the second largest with the smallest (both spanning *m* = 3 means).

3.           Finally, compare the remaining adjacent pairs (spanning *m* = 2 means).

For each comparison, calculate a critical difference (dₘ):

           

where:

·             q(α, m, df) is the critical value from the studentized range distribution for significance level α, with m means in the range, and df degrees of freedom (from the ANOVA residual).

·             MSres is the residual mean square from the ANOVA.

·             r is the number of observations per group (assuming balanced data).

3. Example Section

Consider the (fictitious) data on blood pressure reduction presented in Table 1.

                                   Table 1.Blood pressure reduction (mmHg) 

                         

These data were analyzed by ANOVA (Table 2), where the F value was significant at the 5% level. Thus, at least one mean differs from the others. The sample means are shown in Table 3.

                                       Table 2: Analysis of variance (ANOVA)  

                                                                      

             Table 3: Means of blood pressure reduction (mmHg)

                  

The largest mean is 29 (Group D), and the smallest is 2 (Control). We can calculate the critical difference dₘ for comparing these extremes, where *m* = 6. From the ANOVA, we have residual degrees of freedom (df) = 24 and MSres = 36.00. Each group has *r* = 5 observations. The critical value q(0.05, 6, 24) is 4.3727. Thus:

                           

The observed difference between D and Control is 29 - 2 = 27. Since 27 > 11.733, the difference is statistically significant at the 5% level.

We then proceed to compare pairs spanning *m* = 5 means (e.g., D vs. B and A vs. Control). The new critical value is q(0.05, 5, 24) = 4.1663. Thus:

The differences between groups D and B (29 – 8 = 21) and between A and the control (21 – 2 = 19) are greater than 11.179.  So, both are significant at 5% level.

Attention: The analysis does not stop here; it continues stepwise through the ordered means.

4. "Comparison with Tukey’s Test" Section 

We performed pairwise comparisons using both Tukey’s HSD and the SNK tests. The results are summarized in Table 5, which groups means that are not statistically different from each other.

Table 5:  Comparison: Tukey's test and SNK test 

The SNK test identified more significant differences than Tukey's, demonstrating its greater power. For instance, SNK detected that treatment D is significantly greater than A, a difference that Tukey's test did not find.

This increased power, however, comes with a trade-off: a slightly higher risk of Type I error. Therefore, the choice between Tukey and SNK must be carefully justified based on the study's objectives and the desired balance between rigor (conservatism) and sensitivity (power).

 

Feature	Tukey's HSD	Student-Newman-Keuls (SNK)
Type	Single-step	Sequential (Stepwise)
Error Control	Controls experiment-wise error	Does not control experiment-wise error
Power	Less powerful (conservative)	More powerful (liberal)
Software Availability	Excellent	Limited
Best For	Confirmatory analysis, unequal n	Exploratory analysis, equal n

              Advantages and Disadvantages

         By performing all pairwise comparisons, the Student-Newman-Keuls test has greater                power and can be the best option when confidence intervals are not needed and sample          sizes are equal.

Further reading

- Seaman, M.A., Levin, J.R., Serlin, R.C. (1991). Psychological Bulletin 110: 577–586.
- Day, R.W., Quinn, G.P. (1989). Ecological Monographs 59: 433–463.
- Zar, J.H. Biostatistical Analysis.
- Dean, A., Voss, D. Design and Analysis of Experiments.
- Montgomery, D.C. Design and Analysis of Experiments.

Translation & Editing Note: This post was translated from Portuguese and edited for clarity with the assistance of an AI language model. The statistical methodology, calculations, and conclusions were rigorously verified by the author.