Post-hoc Tests for ANOVA
Explaining significant differences
in 1-way ANOVA
Multiple comparisons
• Post hoc procedures vs. a priori
comparisons (we look at only post hoc)
• Comparisonwise error rate - alpha for
each comparison
• Experimentwise error rate - the probability
of making a Type I error for the set of all
possible comparisons
alphae = 1 - (1-alpha)c
Multiple comparisons
• If you are comparing 3 groups (A, B, C), than you can do 3
total comparisons
A – B
A – C
B – C
• The experimentwise error rate without any adjustments would
be:
alphae = 1 - (1-alpha)c
= 1 – (1-.05)3
= 1 - .953
= 1 - .86
= .14
• Using SPSS, LSD post hoc comparison does
not control experimentwise error rate (e.g.
comparisonwise error rate = 0.05 per
comparison).
– This is considered a liberal post hoc analysis
• Tukey method for making comparisons
maintains an experimentwise error rate
– based on the distribution called the Q distribution
– makes comparisons without inflating the
experimentwise error rate
Post-hoc analysis on SPSS
– Compare Means
– Click “post-hoc” button
–Select method of analysis
Dependent Variable: distance
Tukey HSD
Results from
tire data
Tukey test
(I) brand
firestone
goodyear
michelin
uniroyal
bridg estone
(J) brand
goodyear
michelin
uniroyal
bridg estone
firestone
michelin
uniroyal
bridg estone
firestone
goodyear
uniroyal
bridg estone
firestone
goodyear
michelin
bridg estone
firestone
goodyear
michelin
uniroyal
Mean
Difference
(I-J)
-7.00000
.80000
-3.00000
-12.30000*
7.00000
7.80000*
4.00000
-5.30000
-.80000
-7.80000*
-3.80000
-13.10000*
3.00000
-4.00000
3.80000
-9.30000*
12.30000*
5.30000
13.10000*
9.30000*
*. The mean difference is significant at the .05 level.
Results from Tire data
brand
bridgestone
goodyear
uniroyal
firestone
michelin
Total
Mean
200.50
195.20
191.20
188.20
187.40
192.50
N
10
10
10
10
10
50
SD
5.44
9.02
5.55
3.88
5.27
7.61
Bridgestone Goodyear Uniroyal Firestone Michelin
In words
• Bridgestone is different than firestone,
Michelin, and Uniroyal
• Goodyear is different than Michelin
• Uniroyal is different than Bridgestone
• Firestone is different than Bridgestone
• Michelin is different than Goodyear and
Bridgestone