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

KEY TERMS 553
a predicted portion and an unpredicted, or residual,
portion. Overall, the predicted portion of the Y score
variability is measured by r 2 , and the residual portion
is measured by 1 – r 2 .
Predicted variability = SS regression
= r 2 SS Y
Unpredicted variability = SS residual
= (1 – r 2 )SS Y
3. The residual variability can be used to compute the
standard error of estimate, which provides a measure
of the standard distance (or error) between the predicted
Y values on the line and the actual data points.
The standard error of estimate is computed by
standard error of estimate = Î SS residual
n 2 2 5 ÏMS residual
4. It is also possible to compute an F-ratio to evaluate the
significance of the regression equation. The process is
called analysis of regression and determines whether
the equation predicts a significant portion of the variance
for the Y scores. First a variance, or MS, value is
computed for the predicted variability and the residual
variability,
MS regression
5
SS regression
df regression
MS residual
5 SS residual
df residual
where df regression
= 1 and df residual
= n – 2. Next, an
F-ratio is computed to evaluate the significance of the
regression equation.
F =
MS regression
MS residual
with df = 1, n − 2
5. Multiple regression involves finding a regression
equation with more than one predictor variable. With
two predictors (X 1
and X 2
), the equation becomes
Ŷ = b 1
X 1
+ b 2
X 2
+ a
with the values for b 1
, b 2
, and a computed using
Equations 16.16, 16.17, and 16.18.
6. For multiple regression, the value of R 2 describes the
proportion of the total variability of the Y scores that
is accounted for by the regression equation. With two
predictor variables,
R 2 = b 1 SP X1Y 1 b 2 SP X2Y
SS Y
The predicted variability = SS regression
= R 2 SS Y
.
Unpredicted variability = SS residual
= (1 – R 2 )SS Y
7. The residual variability for the multiple-regression
equation can be used to compute a standard error of
estimate, which provides a measure of the standard
distance (or error) between the predicted Y values
from the equation and the actual data points. For
multiple regression with two predictors, the standard
error of estimate is computed by
standard error of estimate = Î SS residual
n 2 3 = ÏMS residual
8. Evaluating the significance of the two-predictor
multiple-regression equation involves computing
an F-ratio that divides the MS regression
(with df = 2)
by the MS residual
(with df = n – 3). A significant
F-ratio indicates that the regression equation
accounts for a significant portion of the variance
for the Y scores.
9. An F-ratio can also be used to determine whether a
second predictor variable (X 2
) significantly improves
the prediction beyond what was already predicted by
X 1
. The numerator of the F-ratio measures the additional
SS that is predicted by adding X 2
as a second
predictor.
SS additional
= SS regression with X1 and X2
– SS regression with X1 alone
This SS value has df = 1. The denominator of the
F-ratio is the MS residual
from the two-predictor regression
equation.
10. A partial correlation measures the relationship
that exists between two variables after a third
variable has been controlled or held constant.
KEY TERMS
linear relationship (531)
linear equation (531)
slope (532)
Y-intercept (532)
Regression (533)
regression line (533)
least-squared-error solution (533)
regression equation for Y (534)
standard error of estimate (538)
predicted variability (SS regression
) (540)
unpredicted variability (SS residual
) (540)
multiple regression (544)
partial correlation (551)