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

546 CHAPTER 16 | Introduction to Regression
The goal of the multiple-regression equation is to produce the most accurate estimated
values for Y. As with the single predictor regression, this goal is accomplished with a leastsquared
solution. First, we define “error” as the difference between the predicted Y value
from the regression equation and the actual Y value for each individual. Each error is then
squared to produce uniformly positive values, and then we add the squared errors. Finally,
we calculate values for b 1
, b 2
, and a that produce the smallest possible sum of squared
errors. The derivation of the final values is beyond the scope of this text, but the final equations
are as follows:
b 1
5 sSP X1Y dsSS X2 d 2 sSP X1X2 dsSP X 2Y d
sSS X1
dsSS X2
d 2 sSP X1X2
d 2 (16.16)
b 2
5 sSP X2Y dsSS X1 d 2 sSP X1X2 dsSP X1Y d
sSS X1
dsSS X2
d 2 sSP X1X2
d 2 (16.17)
a = M Y
– b 1
M X1
– b 2
M X2
(16.18)
In these equations, you should recognize the following SS and SP values:
SS X1
is the sum of squared deviations for X 1
SS X2
is the sum of squared deviations for X 2
SP X1Y
is the sum of products of deviations for X 1
and Y
SP X2Y
is the sum of products of deviations for X 2
and Y
SP X1X2
is the sum of products of deviations for X 1
and X 2
Note: More detailed information about the calculation of SS is presented in Chapter 4
(pp. 112–113) and information concerning SP is in Chapter 15 (pp. 490–492). The following
example demonstrates multiple regression with two predictor variables.
EXAMPLE 16.6
We use the data in Table 16.2 to demonstrate multiple regression. Note that each individual
has a Y score and two X scores that are used as predictor variables. Also note that we have
already computed the SS values for Y and for both of the X scores, as well as all of the SP
values. These values are used to compute the coefficients, b 1
and b 2
, and the constant, a, for
the regression equation.
Ŷ = b 1
X 1
– b 2
X 2
+ a
b 1
5 sSP dsSS d 2 sSP dsSP d
X1Y X2 X1X2 X2Y 54(64d 2 42(47)
5 5 0.672
sSS X1
dsSS X2
d 2 sSP X1X2
d 2 62(64) 2 s42d 2
b 2
5 sSP X2Y dsSS X1 d 2 sSP X1X2 dsSP X1Y d
sSS X1
dsSS X2
d 2 sSP X1X2
d 2
5
47s62d 2 42s54d
62s64d 2 s42d 2 5 0.293
a = M Y
– b 1
M X1
– b 2
M X2
= 7 – 0.672(4) – 0.293(6) = 7 – 2.688 – 1.758 = 2.554
Thus, the final regression equation is
Ŷ = 0.672X 1
+ 0.293X 2
+ 2.554