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In-Depth Information
We often speak about the effect of the independent variable. This form
of expression comes from the experimental setting in which we intend to
draw a causal inference by asserting that the presence of the treatment -
the presumed cause - produced or caused the changes in behavior - the
effect - that we measured. The way we quickly summarize the validity of
this presumed causal relationship is by examining the means of the groups
in the study. If the treatment was “effective,” the mean of the one group
should be significantly different from the mean of the other group.
Whenwecomputeanysumofsquares,ameanmustbesubtracted
from a “score,” and that must be true in computing the between-groups
sum of squares. We therefore deal with the mean differences in what may
seem to be an indirect manner, although it does get the job done: The
group means become the scores from which a mean is subtracted. The
mean that is subtracted is the grand mean. Thus,
between-groups sum of squares
= (group mean
grand mean) 2
.
In our notation system the formula for computing the between-groups
sum of squares is therefore written as
SS A = ( Y j
Y T ) 2 ,
(3.2)
where SS A is the b etween-groups sum of squares, Y j is the mean for a
given group, and Y T is the grand mean.
The between-groups source of variance is one of the portions or parti-
tions of the total variance. Here is one way to understand what this means.
In Figure 3.1 the total variability of the dependent variable is depicted.
When we examine the between-groups source of variance, we are asking
if any variation we see in the total set of scores is related to or associ-
ated with the group to which the participant belongs. This “association,”
should any be observed, is treated as variance that is “explained” by the
independent variable. In our example, the lower values are generally asso-
ciated with participants who took the mood survey in the red room, and
the higher values are generally associated with participants who took the
mood survey in the blue room.
The idea that the independent variable is associated with or that it
explains some of the total variability in the dependent variable can be
understood from the standpoint of prediction. Suppose we are attempt-
ing to predict the scores on the dependent variable. Simply treating all
of the participants as a single large group, and knowing nothing about
them, our best prediction in the long run of any one participant's score
will be the overall or grand mean, which gives us little in the way of
precise prediction (it's better than selecting any number at random). If
the independent variable is shown to explain some of the variance of the
dependent variable, then considering it in the prediction effort should
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