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

SECTION 16.3 | Introduction to Multiple Regression with Two Predictor Variables 551
Therefore, the additional contribution made by adding X 2
to the regression equation can be
computed as
(% with both X 1
and X 2
) − (% with X 1
alone)
= 55.62% − 52.26%
= 3.36%
Because SS Y
= 90, the additional variability from adding X 2
as a predictor amounts to
SS additional
= 3.36% of 90 = 0.0336(90) = 3.024
This SS value has df = 1, and can be used to compute an F-ratio evaluating the significance
of the contribution of X 2
. First,
MS additional
= SS additional
1
5 3.024
1
= 3.024
This MS value is evaluated by computing an F-ratio with the MS residual
value from the
multiple regression as the denominator. (Note: This is the same denominator that was used
in the F-ratio to evaluate the significance of the multiple-regression equation.) For these
data, we obtain
F = MS additional
MS residual
5 3.024
5.71 5 0.5296
With df = 1, 7, this F-ratio is not significant. Therefore, we conclude that adding X 2
to the
regression equation does not significantly improve the prediction compared to using X 1
as
a single predictor. The computer printout shown in Figure 16.8, reports a t statistic instead
of an F-ratio to evaluate the contribution for each predictor variable. Each t value is simply
the square root of the F-ratio and is reported in the right-hand side of the Coefficients table.
Variable X 2
, for example, is reported as VAR00003 in the table and has t = 0.732, which is
within rounding error of the F-ratio we obtained; ÏF = Ï0.5296 = 0.728.
■ Multiple Regression and Partial Correlations
In Chapter 15 we introduced partial correlation as a technique for measuring the relationship
between two variables while eliminating the influence of a third variable. At that time,
we noted that partial correlations serve two general purposes:
1. A partial correlation can demonstrate that an apparent relationship between two
variables is actually caused by a third variable. Thus, there is no direct relationship
between the original two variables.
2. Partial correlation can demonstrate that there is a relationship between two variables
even after a third variable is controlled. Thus, there really is a relationship between
the original two variables that is not being caused by a third variable.
Multiple regression provides an alternative procedure for accomplishing both of these
goals. Specifically, the regression analysis evaluates the contribution of each predictor
variable after the influence of the other predictor has been considered. Thus, you can determine
whether each predictor variable contributes to the relationship by itself or simply
duplicates the contribution already made by another variable.
For example, in Chapter 15 (p. 503) we presented data for a sample of 15 cities showing
a logical relationship among the number of churches, the number of crimes, and the
population size. Although the original data showed a strong, positive correlation between