Analysis of Covariance - Analysis of Variance Designs - page 460

Table 16.2. Observed and adjusted group means for the
data shown in Table 16.1
Group Mean of observed Mean of adjusted

membership code dependent variable dependent variable

1 2.00 2.827

2 6.33 5.461

predict a value on the dependent variable of 6.18794. The observed
value on the dependent variable is 6 and so the model overpredicted
the score of Case 5 by a small margin. Not all of the scores were over-
predicted; the model underpredicted the observed values twice (Cases 1
and 6).
The last row of Table 16.1 presents the means for the dependent vari-
able, the covariate, and the adjusted dependent variable. Note that the
adjusted overall mean is the same as the observed overall mean because
the center of the distribution of dependent values has not shifted.
The observed and adjusted means of the groups are presented in
Table 16.2. The observed means for Groups 1 and 2 are 2.00 and 6.33,
respectively. Note that the adjusted group means are different from the
observed means. The adjusted mean of Group 1 is 2.827, and the adjusted
mean of Group 2 is 5.461. We should note that these adjusted group
means are not the simple average of the adjusted scores for the cases in the
respective groups; rather, they are the adjusted means for the two groups
at the mean value of the covariate (which is 6.33). In an ANCOVA, it is
these adjusted means, not the observed means, that are being evaluated
by the F ratio associated with the group effect.

16.5.2 HOMOGENEITY OF REGRESSION
We have just seen that the assumption of linearity of regression is tested on
the sample as a whole typically by visually inspecting a scatterplot of the
data. In contrast, the assumption of homogeneity of regression focuses on
the individual groups and is typically evaluated by performing a statistical
analysis.
Every linear regression model has a value for the slope of the line; the
slope indexes how steeply the function rises or falls with respect to the
x axis. The slope of interest in testing the assumption of homogeneity
of regression is associated with the covariate as it predicts the dependent
variable, as we described in Section 16.5.1. What is of interest here is
not the results based on the sample as a whole; instead, we examine each
group separately. Homogeneity of regression assumes that the slope of the
regression line is the same for each group. When the slopes of the regression
models for the individual groups are significantly different, that is, when
the slope for one group differs significantly from the slope of at least one