Determine whether to use the formula for correlated comparisons or
independent-groups comparisons

Compute t for a comparison for repeated-measures 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 shotgun or grenade, or non-weapon 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 from five subjects.

Table 1. Description of Conditions.

Variable

Description

aw

The time in milliseconds (msec) to name an aggressive word following a weapon word prime.

an

The time in milliseconds (msec) to name an aggressive word following a non-weapon word prime.

cw

The time in milliseconds (msec) to name a control word following a weapon word prime.

cn

The time in milliseconds (msec) to name a control word following a non-weapon word prime.

Table 2. Data from Five Subjects.

Subject

aw

an

cw

cn

1

447

440

432

452

2

427

437

469

451

3

417

418

445

434

4

348

371

353

344

5

471

443

462

463

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 non-weapon word (an and cn conditions). In other
words, is

L_{1} = (an + cn) - (aw
+ cw)

greater than 0? This is tested for significance by computing
L_{1} for each subject and then testing whether the mean value of
L_{1} is significantly different from 0. Table 3 shows L_{1} for the
first five subjects. L_{1} for Subject 1 was computed by

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 L_{1}.
There were 32 subjects in the experiment. Computing L_{1}
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 s_{M}
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 two-tailed probability
is 0.178.

A more interesting question is whether the priming
effect (the difference between words preceded by a non-weapon
word and words preceded by a weapon word) is different for aggressive
words than it is for non-aggressive words. That is, do weapon
words prime aggressive words more than they prime non-aggressive
words? The priming of aggressive words is (an - aw). The priming
of non-aggressive words is (cn - cw). The comparison is the difference:

L_{2} = (an - aw) - (cn
- cw).

Table 4 shows L_{2} for
five of the 32 subjects.

Table 4. L_{2} for Five Subjects.

Subject

aw

an

cw

cn

L_{2}

1

447

440

432

452

-27

2

427

437

469

451

28

3

417

418

445

434

12

4

348

371

353

344

32

5

471

443

462

463

-29

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.

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 L_{1}
and L_{2} are correlated 0.24. Of course,
this is a sample correlation and only estimates what the correlation
would be if L_{1} and L_{2}
were correlated in the population. Although mathematically
possible, orthogonal comparisons with correlated observations
are very rare.