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 549
has df = n – 2. For multiple regression with two predictors, SS residual
= (1 – R 2 )SS Y
and has
df = n – 3. In each case, we can use the SS and df values to compute a variance or MS residual.
MS residual
= SS residual
df
The variance or MS value is a measure of the average squared distance between the
actual Y values and the predicted Y values. By simply taking the square root, we obtain
a measure of standard deviation or standard distance. This standard distance for the
residuals is the standard error of estimate. Thus, for both linear regression and multiple
regression
the standard error of estimate = ÏMS residual
In the computer printout
in Figure 16.8, the
standard error of
estimate is reported in
the Model Summary
table at the top.
For either linear or multiple regression, you do not expect the predictions from the
regression equation to be perfect. In general, there will be some discrepancy between
the predicted values of Y and the actual values. The standard error of estimate provides a
measure of how much discrepancy, on average, there is between the Ŷ values and the actual
Y values.
■ Testing the Significance of the Multiple Regression
Equation: Analysis of Regression
Just as we did with the single predictor equation, we can evaluate the significance of a
multiple-regression equation by computing an F-ratio to determine whether the equation
predicts a significant portion of the variance for the Y scores. The total variability of the
Y scores is partitioned into two components, SS regression
and SS residual
. With two predictor
variables, SS regression
has df = 2, and SS residual
has df = n – 3. Therefore, the two MS values
for the F-ratio are
MS regression
=
SS regression
2
(16.20)
and
MS residual
= SS residual
n 2 3
(16.21)
The data for the n = 10 people in Table 16.2 have produced SS regression
= 50.06 and
SS residual
= 39.94.
Therefore, MS regression
= 50.06
2
= 25.03 and MS residual
= 39.94
7
= 5.71
and F =
MS regression
MS residual
= 25.03
5.71 = 4.38
With df = 2, 7, this F-ratio is not significant with α = .05, so we cannot conclude
that the regression equation accounts for a significant portion of the variance for the
Y scores.
The analysis of regression is summarized in the following table, which is a common
component of the output from most computer versions of multiple regression. In the