Specific Comparisons (Correlated Observations)
Prerequisites
Hypothesis
Testing, Testing a Single Mean,
t Distribution, Specific
Comparisons, Difference Between
Means, Related Pairs
Learning Objectives
 Determine whether to use the formula for correlated comparisons or
independentgroups comparisons
 Compute t for a comparison for repeatedmeasures data
In the Weapons
and Aggression case study, subjects were asked to read words
presented on a computer screen as quickly as they could. Some
of the words were aggressive words such as injure or shatter.
Others were control words such as relocate or consider. These
two types of words were preceded by words that were either the
names of weapons such as shot gun and grenade or nonweapon words
such as rabbit or fish. For each subject, the mean reading time
across words was computed for these four conditions. The four
conditions are labeled as shown in Table 1. Table 2 shows the
data for five subjects.
One question was whether reading times would be shorter when
the preceding word was a weapon word (aw and cw conditions) than
when it was a nonweapon word (an and cn conditions). In other
words, is
L1 = (an + cn)  (aw
+ cw)
greater than 0? This is tested for significance by computing
L1 for each subject and then testing whether the mean value of
L1 is significantly different from 0. Table 3 shows L1 for the
first five subjects. L1 for Subject 1 was computed by
L1 = (440 + 452) 
(447 + 432) = 892  885 = 13
Once L1 is computed for
each subject, the significance test described in the section "Testing
a Single Mean" can be used. First we compute the mean
and the standard error of the mean for L1.
There were 32 subjects in the experiment. Computing L1
for the 32 subjects, we find that the mean and standard error
of the mean are 5.875 and 4.2646 respectively. We then compute
where M is the sample mean, μ is the hypothesized
value of the population mean (0 in this case), and sM
is the estimated standard error of the mean. The calculations
show that t = 1.378. Since there were 32 subjects, the degrees
of freedom is 32  1 = 31. The t
distribution calculator shows that the twotailed probability
is 0.1782.
A more interesting question is whether the priming
effect (the difference between words preceded with a nonweapon
word and words preceded by a weapon word) is different for aggressive
words than it is for nonaggressive words. That is, do weapon
words prime aggressive words more than they prime nonaggressive
words? The priming of aggressive words is (an  aw). The priming
of nonaggressive words is (cn  cw). The comparison is the difference:
L2 = (an  aw)  (cn
 cw)
Table 4 shows L2 for
five of the 32 subjects.
The mean and standard error of the mean for all
32 subjects are 8.4375 and 3.9128 respectively. Therefore, t =
2.156 and p = 0.039.
Multiple Comparisons
Issues associated with doing multiple comparisons
are the same for related observations as they are for multiple
comparisons among independent groups.
Orthogonal Comparisons
The most straightforward way to assess the degree
of dependence between two comparisons is to correlate them directly.
For the weapons and aggression data, the comparisons L1
and L2 are correlated 0.24. Of course,
this is a sample correlation and only estimates what the correlation
would be if L1 and L2
were correlated in the whole population. Although mathematically
possible, orthogonal comparisons with correlated observations
are very rare.
