314 REPEATED-MEASUREMENT AND OTHER EXPERIMENTAL DESIGNS

19.7 TWO-FACTOR EXPERIMENTS WITH REPEATED
MEASUREMENTS ON ONE FACTOR

A not uncommon lype of experiment in psychological and educational
research is a two-factor experiment with repeated measurements over one nI
factor only. To illustrate, consider an experiment involving four different Š
learning trials under two drug treatments. Two groups of xn subjects each š
may be used. The first group may be tested on the four learning trials .
under the first drug treatment. The second group may be tested on the :
four learning trials under the second drug treatment. Designs of this type
are sometimes called mixed designs, but this term should not be confused
with mixed models, where the. mixing is with respect to fixed and random
factors rather than repeated- and nonrepeated-measurement factors.

The notation for such an experiment may be illustrated in the particu-
lar case where two experimental groups of three subjects are used with
each subject measured under four experimental conditions. The data may
be represented as follows:

Subjects C, C, Ca C, Means

i Xu Xn Xi Xa Xra

zaj R, 2 Xy2 X z X sa Xn Xi.

Z 3 Xua X vsa X usa Xn Xi.
o — Z — — —

Means Xu. X a. X a. Xi. X...

4 Xaia X zz Kosa X,sa Ke

va R, k] Xaus X zas X zas Xaas Xe.s

3 6 X is X 26 X ss Xzas X.
O h — - — u

Means Xn. X a. Xn. Xra. X...

Means Xa. Xa Xa. Ka. X.

Here triple subscripts are used. "The first subscript identifies the
row or group to which the subject belongs, the second subscript identifies
the column or the level of the repeated measurement, the third subscript
identifies the subject. For example, X.;, is a measurement for the fourth
subject in the second group at the first level of the repeated measurement.

For this type of experimental design the total sum of sguares may be
partitioned into two parts, a between-subjects and a within-subjects sum of
sguares. The between-subjects sum of sguares can be further partitioned
into two parts, a row sum of sguares and a subjects-within-groups sum of
sguares. Denote this latter term by S/R. The within-subjects sum of
sguares can be further partitioned into three parts, a column sum of
sguares, a row-by-column interaction, and a third part which is a column-
by-subject interaction pooled over groups or rows. Denote this latter term
by (S x C)/R. Thus, in effect, the total sum of sguares is partitioned into
five separate sums of sguares. These sums of sguares with the associated

a redke