Statistical Methods for Psychology

One laborious way to do this would be to put all the subjects’ contributions on a com-

mon footing by equating subject means without altering the relationships among the scores

obtained by that particular subject. Thus, we could set , where is the

mean of the ith subject. Now subjects would all have the same means ( ), and any

remaining differences among the scores could be attributable only to error or to treatments.

Although this approach would work, it is not practical. An alternative, and easier, approach

is to calculate a sum of squares between subjects (denoted as either or ) and

remove this from before we begin. This can be shown to be algebraically equivalent

to the first procedure and is essentially the approach we will adopt.

The solution is represented diagrammatically in Figure 14.1. Here we partition the

overall variation into variation between subjects and variation within subjects. We do the

same with the degrees of freedom. Some of the variation within a subject is attributable to

the fact that his scores come from different treatments, and some is attributable to error;

this further partitioning of variation is shown in the third line of the figure. We will always

think of a repeated-measures analysis as firstpartitioning the into and

. Depending on the complexity of the design, one or both of these partitions may
then be further partitioned.
The following discussion of repeated-measures designs can only begin to explore the
area. For historical reasons, the statistical literature has underemphasized the importance
of these designs. As a result, they have been developed mostly by social scientists, particu-
larly psychologists. By far the most complete coverage of these designs is found in Winer,
Brown, and Michels (1991). Their treatment of repeated-measures designs is excellent and
extensive, and much of this chapter reflects the influence of Winer’s work.

SSwithin subj

SStotal SSbetween subj

SStotal

SSbetween adj SSs

X¿i.= 0

X¿ij=Xij 2 Xi Xi

Introduction 463

Table 14.1 Hypothetical data for simple repeated-measures designs

Treatment

Subject 1 2 3 Mean

1 2 4 7 4.33

2 10 12 13 11.67

3 22 29 30 27.00

4 30 31 34 31.67

Mean 16 19 21 18.67

Figure 14.1 Partition of sums of squares and degrees of freedom

Partition of Sums of Squares Partition of Degrees of Freedom

Total variation

Between subjects

Between treatments Error

Within subjects

kn 21

n(k 2 1)

(n 2 1)(k 2 1)

n 21

k 21