Statistical Methods for Psychology

with from the overall analysis of variance and evaluate the t on degrees of free-

dom. When the variances are heterogeneous but the sample sizes are equal, we do not use

, but instead use the individual sample variances and evaluate t on 2(n 2 1) degrees of

freedom. Finally, when we have heterogeneity of variance and unequal sample sizes, we use

the individual variances and correct the degrees of freedom using the Welch–Satterthwaite ap-

proach (see Chapter 7). (For an evaluation of this approach, albeit for a slightly different test

statistic, see Games and Howell, 1976.)

The indiscriminate use of multiple t tests is typically brought up as an example of a

terrible approach to multiple comparisons. In some ways, this is an unfair criticism. It isa

terrible thing to jump into a set of data and lay waste all around you with t tests on each

and every pair of means that looks as if it might be interesting. The familywise error rate

will be outrageously high. However, if you have only one or two comparisons to make and

if those comparisons were truly planned in advance (you cannot cheat and say, “Oh well,

I would have planned to make them if I had thought about it”), the t-test approach has much

to recommend it. With only two comparisons, for example, the maximum FWwould be ap-

proximately .10 if each comparison were run at a5.05, and would be approximately .02

if each comparison were run at a5.01. For a discussion of the important role that individ-

ual contrasts can play in an analysis, see Howell (2008).

In the study on morphine tolerance described previously, we would probably not use

multiple t tests simply because too many important comparisons should be considered. (In

fact, we would probably use one of the post hoc procedures for making all pairwise com-

parisons unless we can restrict ourselves to relatively few comparisons.) For the sake of an

example, however, consider two fundamental comparisons that were clearly predicted by

the theory and that can be tested easily with a t test. The theory predicted that a rat that had

received three previous morphine trials and was then tested in the same environment using a

saline injection would show greater pain sensitivity than would an animal that had always been

tested using saline. This involves a comparison of group M-S with group S-S. Furthermore,

the theory predicted that group Mc-M would show less sensitivity to pain than would group

M-M, because the former would be tested in an environment different from the one in which it

had previously received morphine. Because the sample variances are similar and the sample

sizes are equal, we will use as the pooled variance estimate and will evaluate the result

on degrees of freedom.

Our general formula for t, replacing individual variances with , will then be

Substituting the data from our example, the contrast of group M-S with group S-S yields

And group Mc-M versus group M-M yields

t=

XMc-M 2 XM-M

B

2 MSerror

n

=

29.00 2 10.00

B

2(32.00)

8

=

19

18

=6.72

XMc-M=29.00 XM-M=10.00 MSerror=32.00

t=

XM-S 2 XS-S

B

2 MSerror

n

=

4.00 2 11.00

B

2(32.00)

8

=

- 7

18

=-2.47

XM-S=4.00 XS-S=11.00 MSerror=32.00

t=

Xi 2 Xj

B

MSerror

n

1

MSerror

n

=

Xi 2 Xj

B

2 MSerror

n

MSerror

dferror

MSerror

MSerror

MSerror dferror

370 Chapter 12 Multiple Comparisons Among Treatment Means