Russel-Research-Method-in-Anthropology

Multivariate Analysis 665
TABLE 21.17
Multiple Regression Output from SYSTAT
Dep Var: TEENBIRTH N: 50 Multiple-R: 0.850 Squared multiple-R: 0.722
Adjusted squared multiple-R: 0.710 Standard error of estimate: 1.829
Effect Coefficient Std Error Std Coef Tolerance t P (2 Tail)
CONSTANT 28.096 1.822 0.0 . 15.423 0.000
VIOLRATE 0.006 0.001 0.491 0.964 6.269 0.000
INCOME 0.001 0.000 0.793 0.964 10.130 0.000
The regression equation predicts that the teenage birth rate for Wisconsin will
be 8.6. The actual rate was 10.6 in 1996. The mean rate of teenage births for
the 50 U.S. states was 12.9. The multiple regression, then, makes a better prediction
than the mean: The difference between 10.6 and 12.9 is 2.3, while the
difference between 8.6 and 10.6 is 2.0.
If you work out all the differences between the predictions from the multiple
regression equation and the predictions from the simple regression equation
involving just the effect of income on teenage births, the difference in the
predictions will be the difference between accounting for 49% of the variance
vs. accounting for 72.2% of the variance in the dependent variable.
Details of the Regression Equation
In this example, y is TEENBIRTH, x 1 is INCOME and x 2 is VIOLRATE.
R 2 1•23 is the variance in TEENBIRTH (the 1 in the subscript) accounted for by
both INCOME and VIOLRATE (the 2 and 3 in the subscript). This relation
is:
R 2 1•23
r 2 12
r 2 1(3•2)
which we can read as follows: The total variance in TEENBIRTH equals the
variance accounted for by INCOME, plus the variance accounted for by VIOL-
RATE once the effect of INCOME has been accounted for. Calculating r 2 1•23,
then:
R 2 1•23
r 13
(r 12
)(r 32
) .340 (.700)(.190) .473
Taking the partial contribution of this relation to the dependent variable:
.473
1.190 2 .481776