Statistical Methods for Psychology

Both of these obtained values of t would be evaluated against t.025(35) 5 2.03, and both

would lead to rejection of the corresponding null hypothesis. We can conclude that with two

groups of animals tested with saline, the group that had previously received morphine in the

same situation will show a heightened sensitivity to pain. We can also conclude that changing

the setting in which morphine is given significantly reduces, if it does not eliminate, the con-

ditioned morphine-tolerance effect. Because we have tested two null hypotheses, each with

a5.05 per comparison, the FWwill approach .10 if both null hypotheses are true, which

seems quite unlikely. In fact, given the position of Jones and Tukey (2000) that it is highly

unlikely that either null hypothesis would be true, and that we can only incorrectly find a sig-

nificant difference in the wrong direction, the probability of an error in this situation is at

most .05. That is important to keep in mind when we speak of the advantages and disadvan-

tages of individual contrasts on pairs of means.

The basic ttest that we have just used is the basis for almost everything to follow. I

may tweak the formula here or there, and I will certainly use a number of different tables

and decision rules, but it remains your basic ttest—even when I change the formula and

call it q.

Linear Contrasts

The use of individual t tests is a special case of a much more general technique involving

what are known as linear contrasts.^3 In particular, t tests allow us to compare one group

with another group, whereas linear contrasts allow us to compare one group or set of

groupswith another group or set of groups. Although we can use the calculational proce-

dures of linear contrasts with post hoc tests as well as with a priori tests, they are discussed

here under a priori tests because that is where they are most commonly used.

To define linear contrasts, we must first define a linear combination.A linear combi-

nation of means takes the form

This equation simply states that a linear combination is a weighted sum of treatment means.

If, for example, the were all equal to 1, Lwould just be the sum of the means. If, on the

other hand, the were all equal to 1k, then Lwould be the mean of the means.

When we impose the restriction that , a linear combination becomes what is

called a linear contrast.By convention we designate the fact that it is a linear contrast by

replacing “L” with the Greek psi ( ). With the proper selection of the , a linear contrast

is very useful. It can be used, for example, to compare one mean with another mean, or the

mean of one condition with the combined mean of several conditions. As an example, con-

sider three means ( , , and ). Letting , and ,

In this case, is simply the difference between the means of group 1 and group 2, with the

third group left out. If, on the other hand, we let 5 1 2, 5 1 2, and 2 1, then

in which case represents the difference between the mean of the third treatment and the

average of the means of the first two treatments.

c

c=(1>2)X 11 (1>2)X 21 (-1)X 3 =

X 11 X 2

2

2 X 3

a 1 > a 2 > a 3 =

c

c=(1)X 11 (-1)X 21 (0)X 3 =X 12 X 2

X 1 X 2 X 3 a 1 =1,a 2 =- 1 a 3 = 0 gaj=0,

c aj

gaj= 0

aj >

aj

L=a 1 X 11 a 2 X 2 1 Á 1 akXk= aajXj

6

12.3 A Priori Comparisons 371

(^3) The words “contrast” and “comparison” are used pretty much interchangeably in this context.
linear
combination
linear contrast