THER EXPERIMENTAL DESIGNS

al and educational
urements over one
Iving four different
of n subjects each
four learning trials
y be tested on the
esigns of this type
id not be confused
, fixed and random
nent factors.

ted in the particu- 4
sets are used with Ni:
ns. The data may e
C, Means j
Xn X, LI H:
X a X, 2 di
X aa Xi Š:
Xa X, d
Xass X, 4 4
Ka X. 5 4
X sa Devi 4
PE? j

ript identifies the
subscript identifies
the third subsceript

nent for the fourth d

ated measurement.
of sguares may be
lin-subjects sum of
further partitioned
'hin-groups sum of
in-subjects sum of
a column sum of
vhich is a column-
ote this latter term
is partitioned into
vith the associated

19.7 TWO-FACTOR EXPERIMENTS WITH REPEATED MEASUREMENTS ON ONE FACTOR 315

number of degrees of freedom and variance estimates are shown in Table
19.4.

Some comment on the sums of sguares in Table 19.4 is appropriate.
The meaning of the row, column, and interaction sums of sguares is obvi-
ous. These are concerned with variability due to the main effects and the
interaction between the main effects. The subjects-within-groups sum of
sguares, S/R, is simply the variability among subjects for the first group,
added to the variability among subjects for the second group, and so on for
all levels of R. It may be viewed as the variability among subjects with
the variability due to row treatment effects, as it were, removed. The
(C x S)/R term is a column-by-subject interaction for the first group, added
to the column-by-subject interaction for the second group, and so on for ali
levels of R.

The numbers of degrees of freedom associated with row, column, and
R X C interaction sums of sguares are R—lI,C—l,and(R—I1)(C— |),
respectively. "The S/R term has associated with it n— | degrees of
freedom for each group, and for R groups the number of degrees of
freedom is R(n— 1). The (C x S)/R term involves the summing of
the C x S interaction over R groups or levels. The number of degrees
of freedom associated with each level is (n — 1)(C — 1): conseguently
the total number of degrees of freedom associated with this term is
R(n—-iMXCc—1).

Table 19.4

Analysis of variance for two-factor experiments with repeated measurements
on one factor

Source of Variance
variation Sum of sguares df estimate
Kon - 0.
Between subjects C ba MA (X,,—X..) Rn —1 KA
RO o
Rows nC x (X, —XOE R—I s,
kon - v
SIR C MA bi (X,,—- X, R(n— hb Str
CO KR n
Within subjects N 5 N (X.a- XI RuiC—-h sy
€
Columns nR bA (X.—- x? C-1 se
KO € - - 0 —
RxC nN N(X,—-X, —X.EX." (R—- iIMIC- h Sreč
CA nm — - us
(C x SIR MA x PI (Xu — XKas—Xe.E A) R(n—1iXC- h Nr
KO L
Total 9 S (Xu- x." RCn-—1