Statistical Methods for Psychology

and

Given an additional assumption about sphericity, which we will discuss in the next

section, both of these lead to respectable Fratios that can be used to test the relevant null

hypotheses.

Usually, however, we are cautious about assuming that there is no Subject 3 Treatment

interaction. In much of our research it seems more reasonable to assume that different sub-

jects will respond differently to different treatments, especially when those “treatments”

correspond to phases of an ongoing experiment. As a result we usually prefer to work with

the more complete model.

The full model (which includes the interaction term) leads to the following ratios:

and

Although the resulting Ffor treatments is appropriate, the Ffor subjects is biased. If

we did form this latter ratio and obtained a significant F, we would be fairly confident that

subject differences really did exist. However, if the Fwere not significant, the interpreta-

tion would be ambiguous. A nonsignificant Fcould mean either that or that

. Because we usually prefer this second model, and hate ambiguity,
we seldom test the effect due to Subjects. This represents no great loss, however, since we
have little to gain by testing the Subject effect. The main reason for obtaining
in the first place is to absorb the correlations between treatments and thereby remove sub-
ject differences from the error term. A test on the Subject effect, if it were significant,
would merely indicate that people are different—hardly a momentous finding. The impor-
tant thing is that both underlying models show that we can use as the denominator
to test the effect of treatments.

14.3 The Covariance Matrix

A very important assumption that is required for any Fratio in a repeated-measures design

to be distributed as the central (tabled) Fis that of compound symmetry of the covariance

matrix.^1 To understand what this means, consider a matrix ( ) representing the covariances

among the three treatments for the data given in Table 14.1.

a

N

=

A 1 A 2 A 3

A 1 154.67 160.00 160.00

A 2 160.00 176.67 170.67

A 3 160.00 170.67 170.00

gN

MSerror

SSbetween subj

ks^2 p.0 but ...s^2 pt

ks^2 p= 0

E(MStreat)

E(MSerror)

=

s^2 e1s^2 pt 1 nu^2 t

s^2 e1s^2 pt

E(MSbetween subj)

E(MSerror)

=

s^2 e 1 ks^2 p

s^2 e1s^2 pt

E(MStreat)

E(MSerror)

=

s^2 e 1 nu^2 t

s^2 e

Section 14.3 The Covariance Matrix 465

(^1) This assumption is overly stringent and will shortly be relaxed somewhat. It is nonetheless a sufficient assumption,
and it is made often.