Information Technology Reference
In-Depth Information
CHAPTER FIVE
ANOVA Assumptions
5.1 OVERVIEW
Before conducting an analysis of variance (ANOVA), researchers need to
consider three fundamental statistical assumptions that underlie the anal-
ysis: (a) the error components associated with the scores of the dependent
variable are independent of one another, (b) these errors are normally
distributed, and (c) the variances across the levels or groups of the inde-
pendent variable are equal. Although we discuss these assumptions sepa-
rately, in practice they are interconnected; a violation of one assumption
may and often does affect the others.
A fairly comprehensive literature on the assumptions underlying
ANOVA has developed over the past six decades or so. Good general
summaries of this work can be found in Glass, Peckham, and Sanders
(1972) and Wilcox (1987).
5.2 INDEPENDENCE OF ERRORS
5.2.1 THE CONCEPT OF RESIDUAL OR ERROR VARIANCE
Consider a hypothetical medical research study in which an experimental
group in a study received a certain dose of a drug. If each patient in that
group showed precisely the same level of improvement, then there would
be a zero value remaining when we subtracted the mean of the group
from each patient's improvement score. Thus, knowing that patients were
under the drug treatment would result in perfect (errorless) prediction
of their scores on the dependent variable. Such a situation for the typical
patient is given by the following expression:
Y
i
−
Y
j
=
0,
(5.1)
where
Y
i
is a score of a patient in the group and
Y
j
is the mean of that
group.
Now consider the situation where the residuals are not zero. Under
this condition we appear to have individual differences in reacting to
the treatment. In our example, although the drug might have resulted in
general improvement in the patients, the fact that they showed differences
among themselves suggests that factors in addition to the effects of the
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