19.2 EXPERIMENTS WITH REPEATED MEASUREMENTS 305

19.2 EXPERIMENTS WITH REPEATED MEASUREMENTS

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make freguent use of
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Many experiments in psychelogy and education reguire the repeated mea-
surement of the same subjects under a number of different conditions or
treatments. Such experiments may be single-factor experiments in which
each subject is tested or measured under a number of different experi-
mental conditions. The simplest experiment of this kind would be one in
which the same subjects are tested under two experimental conditions.
Sometimes, when the same subjects are tested under a number of different
treatments, the order of the presentation of treatments to subjects is ran-
domized independently for each subject or a systematic plan for the
ordering of the presentation of treatments to subjects is adopted. The pur-
pose of either randomization, or the use of a systematic plan for the or-
dering of treatments, is to eliminate effects which might result from the
order of the treatments. In some situations randomization is not appropri-
ate because the different levels of the treatment variable have a natural
order. This is the case where performance is measured at different time in-
tervals, as, for example, in the study of changes in dark adaptation with
time, or for different numbers of trials in a simple learning experiment.

Experiments of the type described above are called one-factor experi-
ments with repeated measurements. In such experiments N subjects are
measured under k conditions or treatments. The matrix of data thus ob-
tained is a table of numbers containing N rows and k columns.

Repeated measurements may, however, be used in two-way classifica-
tion or higher-order factorial experiments. For example, in a 2 X 2 facto-
rial experiment four treatment combinations exist, four groups of experi-
mental subjects are used, and each combination is applied to a different
group of subjects. Experiments may be designed in which each of the N
subjects receives ali four treatment combinations. The matrix of data is a
block of numbers containing two rows, two columns, and N layers, each
layer corresponding to a subject. In general, for an R x C factorial experi-
ment with repeated measurements the matrix of data is an RxC xN
block of numbers. The idea involved here can be extended to higher-order
factorial experiments.

Two-way classification experiments may also be conducted with re-
peated measurements over one factor but not over the other factor. Con-
sider an experiment involving two factors with three levels of one factor
and two levels of the other. If this were an independent-groups factorial
experiment, six treatment combinations and six groups of subjects would
be used. The investigator may, however, decide to use two groups of sub-
jects, with each of the two groups receiving only one level of one factor but
all three levels of the other. Thus the experiment has repeated measure-
ments over the factor with three levels, but not over the factor with two
levels.

Experiments with repeated measurements have advantages and disad-
vantages. One advantage is that the measurements obtained under the dif-