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model is linear in that it represents a straight line function. Each predictor
is assigned a weight in the model to maximize the accuracy of predicting
the dependent variable in combination with the other predictors. The
generalformofthemodelisasfollows:
dependent variable
=
a
+
b
1
X
1
+
b
2
X
2
+···+
b
n
X
n
+
error,
(6.9)
where
X
1
,
X
2
, and so on are the predictors. In a one-way design, there
is only one such predictor and it is the independent variable in
the study. In more complex designs, as we will see in subsequent
chapters, the predictors could be additional independent variables
and their interaction effects.
b
1
,
b
2
, and so on are the weights or coefficients associated with their
respective predictors in the model.
a
is the constant in the model representing the
Y
intercept, that is, the
valueatwhichthestraightlinefunctionrepresentedbythemodel
crosses the y axis.
error is measurement error, that is, unaccounted for variance. For
example, in a one-way between-subjects design, participants in the
same group (those who have been treated in the same way with
respect to the “treatment”) may still provide different scores on the
dependent variable.
The full general linear model shown above has a
Y
intercept value
as well as values for the weights. When SPSS or SAS displays the results
for the full model, our output will sometimes display sums of squares
associated with the
Y
intercept and the full (uncorrected) model. The
total sum of squares for the full model includes not only the values for
the between-groups and within-groups sources of variance but also for
the intercept (which is an additional source of variance that is partitioned
by the ANOVA procedure).
The corrected or restricted (reduced) model can be conceived of as
a partial model in that the variance attributable to the
Y
intercept is
not included. The procedures we use for hand calculating the ANOVA
in this topic conform to the reduced model. For the purposes of many
behavioral and social science students who are learning the details of the
ANOVA designs we present here, the results contained in the corrected
model are probably sufficient to meet their needs.
6.9.2 DESCRIPTIVE STATISTICS
The descriptive statistics generated by
Linear Models
are shown in
Figure 6.17. We requested only the mean, standard deviation, and the
number of observations in each group, and that is what is shown in the
figure.
