Multiple comparisons
Section of Analysis of variance, part of Statistical inference
Multiple comparisons
| Type | Section |
| Taught in | Analysis of variance |
| Part | Statistical inference |
Comparing every pair of means with its own 95% interval inflates the chance that some interval misses. With five varieties there are ten pairs, and the chance that at least one of ten 95% intervals misses can be far above 5%. Tukey’s intervals control that simultaneous error.
Definition 1 (Tukey intervals) In a one-way design with k groups of equal size r, and n=kr, the probability is 1-\alpha that every one of the \binom k2 differences \mu_i-\mu_j simultaneously satisfies
\overline Y_{\cdot i}-\overline Y_{\cdot j}-D\sqrt{\mathrm{MS}_{\mathrm{error}}}<\mu_i-\mu_j<\overline Y_{\cdot i}-\overline Y_{\cdot j}+D\sqrt{\mathrm{MS}_{\mathrm{error}}}, \qquad D=\frac{Q_{k,\,n-k;\,1-\alpha}}{\sqrt r},
where Q_{k,\,n-k;\,1-\alpha} is the quantile of the studentised range for k groups and n-k error degrees of freedom.
The studentised range is the law of the largest minus the smallest of k independent standard normals, divided by an independent \sqrt{V/m} with V\sim\chi^2_m. Bounding the largest difference among the group means bounds all of them at once. A pair whose interval excludes zero is declared different at familywise level \alpha.
For the corn, k=5, r=3 and n-k=10, so Q_{5,10;0.95}=4.654 and the half-width of every interval is 4.654\sqrt{10.57/3}=8.74. The ten intervals for \mu_i-\mu_j are
| j=2 | j=3 | j=4 | j=5 | |
|---|---|---|---|---|
| i=1 | (-14.87,\ 2.60) | (-19.60,\ -2.13) | (-8.10,\ 9.37) | (-11.74,\ 5.74) |
| i=2 | (-13.47,\ 4.00) | (-1.97,\ 15.50) | (-5.60,\ 11.87) | |
| i=3 | (2.76,\ 20.24) | (-0.87,\ 16.60) | ||
| i=4 | (-12.37,\ 5.10) |
Two exclude zero: variety 3 yields more than variety 1 and more than variety 4. No other pair can be separated at familywise level 0.05.
Used in
See also
One-way analysis of variance · ANOVA sum of squares decomposition · ANOVA F distribution · Contrast · Tukey intervals · Randomised block design · Between and within · One-factor analysis of variance