OTHER EXPERIMENTAL DESIGNS

on within subjects
m are Rin—l)<
interactions summed
R(In—iXC-hs

is the S/R variance
in effects and R x C
itios are as follows:

nificant, whereas the
-column interaction

the homogeneity of
lysis of variance in-
e stated in the form
reasonable violations
st. The analysis-of-
t to violations of the
of variance involving
' only regarding the
eneity of covariance.
ted measurements N
e covariances, rjjsisj,
The homogeneity of
f the same population
were made for N sub-
e, or matrix, might be

49.10 ASSUMPTIONS UNDERLYING REPEATED-MEASUREMENT DESIGNS 321

In this table variances appear along the main diagonal, and covariances ap-
pear on either side of the main diagonal. The homogeneity of variance-
covariance assumption means that the variances and covariances in the
population sampled are as follows:

1 2 "a 4

BR atn—

Here p is the value of the population correlation coefficient, and po" is the
population covariance. If the homogeneity of variance-covariance as-
sumption is not satisfied in repeated measurement designs, the F test is
positively biased. "This means that more significant differences will be
found, and more null hypotheses rejected, than would have been the case
had the F test not been biased.

A method exists for testing the homogeneity of variance-covariance as-
sumption. This method is due to Box (1953), and a description of it is
found in more advanced texts such as Winer (1971). Application of the
procedure involves considerable arithmetical labor, and the computation
reguired is best done on a computer.

If the homogeneity of variance-covariance assumption is not satisfied,
Box (1954) has suggested a procedure which for a one-factor experiment
with repeated measurements uses the same F ratio, F, < s,'/s,.', that
would be appropriate if the assumption had been satisfied. Different
degrees of freedom are, however, used on entering the F table. Instead of
(C — I) and (R—I)(C— Il), the Box procedure uses | and (R—1I).
Since in this design R is the number of subjects, (R — 1) <(N — 1). This
is a conservative procedure and is based on a maximal departure of the ob-
served covariance matrix from the homogeneity assumption. This proce-
dure in most situations will be negatively biased and will lead to too few
significant differences. Since the investigator will usually wish to proceed
without applying a proper test of the homogeneity of the variance-
covariance assumption, the following has been suggested. Test for column
effects by using the F test with | degree of freedom associated with the
numerator and (R — 1) with the denominator. If the result is significant at
the desired level, no further test is reguired, because this is a conservative
test that works against obtaining a significant difference. If the result is not
significant, test the F ratio by using (C — 1) and (R — 1X(C — 1) degrees
of freedom. If this is not significant, no further test is reguired, because
this is a libera! procedure that works in the direction of too many signifi-
cant differences. If the conservative procedure indicates that the F ratio is
not significant, and the liberal procedure indicates that it is significant, then
a test of the homogeneity of variance-covariance assumption is reguired.

When the Box procedure as described above is applied to tests used in