Pairwise Comparisons (Correlated Observations)
Prerequisites
Difference
between Two Means (Independent Groups), All
Pairwise Comparisons Among Means, Difference
Between Two Means, Correlated Pairs, Specific
Comparisons
Learning Objectives
 Compute the Bonferroni correction
 Calculate pairwise comparisons using the Bonferroni correction
In the section on all pairwise
comparisons among independent groups, the Tukey HSD Test was the recommended procedure. When you have one
group with several scores from the same subjects, the Tukey test
makes an assumption that is unlikely to hold: The variance of
difference scores is the same for all pairwise differences between
means.
The standard practice for pairwise comparisons with
correlated observations is to compare
each pair of means using the method outlined in the section "Difference
Between Two Means, Correlated Pairs" with the addition
of the Bonferroni correction described
in the section "Specific
Comparisons." For example, suppose you were going to
do all pairwise comparisons among four means and hold the familywise
error rate at 0.05. Since there are six possible pairwise
comparisons among four means, you would use the 0.05/6 = 0.0083
for the per comparison error rate.
As an example, consider the case study "Stroop."
There were three tasks each performed by 47 subjects. In the "words"
task, subjects read the names of 60 color words written in black
ink; in the "color" task, subjects named the colors
of 60 rectangles; in the "interference" task, subjects
named the ink color of 60 conflicting color words. The times to
read the stimuli were recorded. In order to do compute all pairwise
comparisons, the difference in times for each pair of conditions
for each subject is calculated. Table 1 shows these scores for
5 of the 42 subjects.
Data
for all 47 subjects
The means, standard deviations, and standard error of the mean (Sem), t, and p for all 47 subjects
are shown in Table 2. The t's are computed by dividing the means
by the standard errors of the mean. Since there are 47 subject,
the degrees of freedom is 46. Notice how different the standard
deviations are. For the Tukey test to be valid, all population
values of the standard deviation would have to be the same.
Using the Bonferroni correction for three comparisons, the p
value has to be below 0.05/3 = 0.0167 for an effect to be significant
at the 0.05 level. For these data, all p values are far below
that, and therefore all pairwise differences are significant.
