Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

542 CHAPTER 16 | Introduction to Regression
F I G U R E 16.6
The partitioning of SS and df for
analysis of regression. The variability
for the original Y scores
(both SS and df) is partitioned
into two components: (1) the
variability that is predicted by
the regression equation and
(2) the residual variability.
SS regression
r 2 SS Y
df regression
SS Y df Y 5 n 2 1
SS residual
5 1 df residual 5 n 2 2
(1 2 r 2 )SS Y
corresponding degrees of freedom. The numerator of the F-ratio is MS regression
, which is the
variance in the Y scores that is predicted by the regression equation. This variance measures
the systematic changes in Y that occur when the value of X increases or decreases. The
denominator is MS residual
, which is the unpredicted variance in the Y scores. This variance
measures the changes in Y that are independent of changes in X. The two MS value are
defined as
MS regression
5
The F-ratio is
SS regression
df regression
with df 5 1 and MS residuals
5 SS residual
df residual
with df 5 n 2 2
F 5
MS regression
MS residual
with df 5 1, n 2 2 (16.13)
The complete analysis of SS and degrees of freedom is diagrammed in Figure 16.6. The
analysis of regression procedure is demonstrated in the following example, using the same
data that we used in Examples 16.1, 16.3, and 16.4.
EXAMPLE 16.5
The data consist of n = 8 pairs of scores with a correlation of r = 0.847 and SS Y
= 156. The
null hypothesis either states that there is no relationship between X and Y in the population
or that the regression equation has b = 0 and does not account for a significant portion of
the variance for the Y scores.
The F-ratio for the analysis of regression has df = 1, n – 2. For these data, df = 1, 6.
With α = .05, the critical value is 5.99.
As noted in the previous section, the SS for the Y scores can be separated into two
components: the predicted portion corresponding to r 2 and the unpredicted, or residual,
portion corresponding to (1 – r 2 ). With r = 0.847, we obtain r 2 = 0.718 and
predicted variability = SS regression
= 0.718(156) = 112.01
unpredicted variability = SS residual
= (1 – 0.718)(156) = 0.282(156) = 43.99
Using these SS values and the corresponding df values, we calculate a variance or MS for
each component. For these data the MS values are:
MS regression
5
SS regression
5 112.01 5 112.01
df regression
1
MS residual
5 SS residual
5 43.99 5 7.33
df residual
6