The q and t ratios reported with the multiple comparison tests follwing one-way ANOVA.
When computing the multiple comparison test, the first step is to compute a q or t ratio. We include these ratios with the results, because some people need to check our results against text books or other programs. The value of q won't help you interpret the results, so you can ignore the q or t ratio in most cases.
If you are curious, read on to see how these values are defined.
The first step is to compute the standard error of the difference between two means, based on the Mean Square Residual (MS) and the two sample sizes Na and Nb. The standard error of the difference (SED) is computed as:
SED=sqrt(MS*(1/Na + 1/Nb))
Note that even though the goal is to compare two particular groups, the SED is computed from the Mean Square Residual of the overall ANOVA, which takes into account the variability within all the groups.
The table below shows the equations used to compute the q or t ratio from the difference between two means (D) and the standard error of that difference (SED) defined above.
| Tukey | q = sqrt(2)*D/SED |
| Newman-Keuls | q = sqrt(2)*D/SED |
| Dunnett | q = D/SED |
| Bonferroni, Sidak and Holm-Sidak | t = D/SED |
For historical reasons (but no logical reason), the q ratio reported by both the Tukey test and the Newmann-Keuls differ from the one reported by Dunnett's test by a factor of the square root of 2, so cannot be directly compared. The t ratio reported by the Bonferroni test is identical to the q ratio reported by the Dunnet test, but they use different names (t vs q).
For the sample one-way ANOVA data built-in to Prism 5, the MS is 105.4. When comparing the first two groups, Na and Nb equal 6 and 5. The SED, therefore, equals 6.217. The difference between the first two means equals 38.33. The q ratio for the Dunnett test and the t ratio for the Bonferroni test equal 38.33/6.217 or 6.165. The q ratio from the Tukey or Newman-Keuls test is increased by a factor of 1.41 (the square root of 2) so equal 8.719.