306

19.3

REPEATED-MEASUREMENT AND OTHER EXPERIMENTAL DESIGNS

ferent treatment conditions will in many experiments be highly correlated
since they are made on the same subjects. The presence of these correla-
tions will reduce the error term. 'Another advantage resides in the number
of subjecis. |t may be more economical in terms of time and effort to test
the same subjects under each treatment. A further point here is that the
nature of certain experimental problems demands the use of repeated-
measurement designs. One disadvantage of experiments with repeated
measurements is that performance under prior treatments may affect per-
formance under subseguent treatments due to either fatigue, practice,
boredom, or some other circumstance. Effects resulting from such circum-
stances are sometimes called carry-over effects. An investigator may not
be able to clearly decide whether the results observed under the different
treatments are due to those treatments or are due to the carry-over effects.
A further problem associated with repeated measurement designs in the as-
sumption made in the analysis of data. Not only is the usual assumption of
homogeneity of variances made but also an assumption is made regarding
the homogeneity of covariances. This matter is discussed in some detail in
Section 19.9.

ONE-FACTOR EXPERIMENTS WITH REPEATED MEASUREMENTS:
COMPUTATION AND EXPECTATION OF MEAN SOUARES

As indicated above, the data resulting from a one-factor experiment with
repeated measurements may be represented as a table of numbers in which
rows represent experimental subjects and columns represent treatments;
that is, the representation of the data is the same as that for the two-way
classification with one observation per cell. The analysis of such data in-
volves nothing new. The data are analyzed as in the two-way classifica-
tion case with one observation per cell. The reguired computation for-
mulas are given in Section 16.9. Three sums of sguares result: sums of
sguares for subjects (rows), treatments (columns), and interaction.

For a one-factor experiment with repeated measurements, subjects
constitute a random variable and treatments are usually viewed as fixed.
The model is the mixed model for n <— 1. The expectations of the mean
sguares are as follows:

Mean sguares Expectation of mean sguares
Subjects, s,? o,: 4 Co,"
Treatments, s. o,? 4d ot Ro?
interaction, s,,? o" ož

The proper error term for testing differences between treatments is
Sre", that is, F, — s.2/s,,. | No unbiased test of individual differences