Let’s try to solve the puzzle of the relation between teenage births and the rate of motor vehicle deaths. We can take a stab at this using multiple regression by trying to predict the rate of teenage births without the data from motor vehicle deaths.

From our previous analyses with elaboration tables and with partial correlation, we already had an idea that income might have something to do with the rate of teen births. We know from the literature (Handwerker 1998) that poverty is associated with violence and with teenage pregnancy, and that this is true across ethnic groups, so I’ve added a variable on violent crimes that I think might be a proxy for the amount of violence against persons in each of the states.

Table 22.16 Correlation Matrix for Variables Associated with the Percentage of Teenage Births in the United States

TEENBIRTH

INCOME

VIOLRATE

MVD

TEENBIRTH

1.00

INCOME

— .700

1.00

VIOLRATE

.340

.190

1.00

MVD

.778

— .662

.245

1.00

Table 22.16 shows the correlation matrix for the four variables: TEENBIRTH (the percentage of births to teenagers in the 50 U.S. states during 1996); INCOME (the mean per capita income for each of the 50 states during 1996); VIOLRATE (the rate of violent crime—rape, murder, assault, and robbery—per 100,000 population in the 50 states in 1995); and MVD (the number of motor vehicle deaths per 100 million vehicle miles in each of the 50 states in 1995).

We see right away that the mean per capita income predicts the rate of births to teenagers (r = —.700) almost as well as does the rate of motor vehicle deaths (r = .778), and that mean income also predicts the rate of motor vehicle deaths rather well (r = — .662).

This is a clue about what might be going on: An antecedent variable, the level of income, might be responsible for the rate of motor vehicle deaths and the rate of teenage births. (By the way, did you notice that the strong correlations above were negative? The greater the mean income in the state, the lower the rate of teenage births and the lower the rate of motor vehicle deaths. Remember, correlations can vary from — 1.0 to + 1.0 and the strength of the correlation has nothing to do with its direction.)

And one more thing: The rate of violent crimes against people is moderately correlated with the rate of teenage births (r = .340), but is only weakly correlated with the mean income (r = .190) and with the rate of motor vehicle deaths (r = .245).

The task for multiple regression is to see how the independent variables predict the dependent variable together. If the correlation between mean per capita income and the rate of births to teenagers is — .700, that means that the independent variable accounts for 49% (— .700^{2}) of the variance in the dependent variable. And if the correlation between the rate of violent crimes and the rate of births to teenagers is .340, then the independent variable accounts for 11.56% (.340^{2}) of the variance in the dependent variable.

We can’t just add these variances-accounted-for together, though, because the two independent variables are related to each other—each of the independent variables accounts for some variance in the other.

Table 22.17 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

Table 22.17 shows what the output looks like from SYSTAT® when I asked the program to calculate the multiple correlation, R, for INCOME and VIOLRATE on TEENBIRTH.

Table 22.17 tells us that the regression equation is:

For example, the violence rate for Wisconsin was 281 crimes per 100,000 residents in 1995 and the average income in Wisconsin was $21,184 in 1996. The regression equation predicts that the teenage birthrate 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 versus accounting for 72.2% of the variance in the dependent variable.