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 547
TABLE 16.2
Sample data consisting
of three scores for each
person. Two of the scores,
X 1
and X 2
, are used to
predict the Y score for
each individual.
Person Y X 1
X 2
A 11 4 10 SP X1Y
= 54
B 5 5 6 SP X2Y
= 47
C 7 3 7 SP X1X2
= 42
D 3 2 4
E 4 1 3
F 12 7 5
G 10 8 8
H 4 2 4
I 8 7 10
J 6 1 3
M Y
= 7 M X1
= 4 M X2
= 6
SS Y
= 90 SS X1
= 62 SS X2
= 64
■
Example 16.6 also demonstrates that multiple regression can be a tedious process. As
a result, multiple regression is usually conducted on a computer. To demonstrate this process,
we used the SPSS computer program to perform a multiple regression on the data in
Table 16.2 and the output from the program is shown in Figure 16.8. At this time, focus
on the Coefficients Table at the bottom of the printout. The values in the first column of
Unstandardized Coefficients include the constant, b 1
and b 2
for the regression equation. We
will discuss other portions of the SPSS output later in this chapter.
■ R 2 and Residual Variance
In the same way that we computed an r 2 value to measure the percentage of variance
accounted for with the single-predictor regression, it is possible to compute a corresponding
percentage for multiple regression. For a multiple-regression equation, this
percentage is identified by the symbol R 2 . The value of R 2 describes the proportion of
the total variability of the Y scores that is accounted for by the regression equation. In
symbols,
R 2 5
SS regression
SS Y
or SS regression
5 R 2 SS Y
For a regression with two predictor variables, R 2 can be computed directly from the regression
equation as follows:
R 2 5 b 1 SP X1Y 1 b 2 SP X 2Y
SS Y
(16.19)
For the data in Table 16.2, we obtain a value of
In the computer printout
in Figure 16.8, the
value of R 2 is reported
in the Model Summary
table as 0.557. Within
rounding error, the two
values are the same.
R 2 =
0.672s54d 1 0.293s47d
90
= 50.059
90
= 0.5562 (or 55.62%)
Thus, 55.62% of the variance for the Y scores can be predicted by the regression equation.
For the data in Table 16.2, SS Y
= 90, so the predicted portion of the variability is
SS regression
= R 2 SS Y
= 0.5562(90) = 50.06