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Details of the Regression Equation

In this example, y is TEENBIRTH, Xj is INCOME, and x2 is VIOLRATE. R2i.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:

which we can read as follows: The total variance in TEENBIRTH equals the variance accounted for by INCOME, plus the variance accounted for by VIOLRATE once the effect of INCOME has been accounted for. Calculating R2i.23, then:

Taking the partial contribution of this relation to the dependent variable:

The contribution of this correlation to the variance of TEENBIRTH is .4817762, or .23211. INCOME accounts for 49% of the variance in TEENBIRTH. Adding contributions, we get .49 + .23211, or 72.2%, which is the squared multiple-R in table 22.17.

There are three coefficients in the regression equation: a, Ъъ and b2. These are the coefficients in formula 22.3. Each of the b coefficients is the product of the standardized regression coefficient for each independent variable with the ratio of the standard deviation of the independent variable to the standard deviation of the dependent variable.

The standardized coefficient for the relation between x (TEENBIRTH) and x2 (INCOME) is:

These figures are given in table 22.17 as the ‘‘Std. Coef,’’ or standardized coefficients.

Thus:

and:

These figures, bl and b2, are given in table 22.17 as the “Coefficients,” and are the unstandardized regression coefficients. The method for calculating the value for a in the multiple regression equation is beyond the scope of this book. (For more about deriving multiple regression equations, consult Pedhazur [1997] or Gujarati [2003].)

But there’s more. If we add up the variances accounted for by the zero-order correlations of INCOME and VIOLRATE on TEENBIRTH, we get 49% + 11.56% = 60.56%. According to the results in table 22.17, however, the income in a state and the rate of violent crimes together account for 72.2% of the variance in teenage births. In other words, the two variables acting together account for more than they do separately, and this is the case despite the fact that the independent variables are moderately correlated (r = .340) with each other.

In fact, INCOME explains 43.82% of the variance in motor vehicle deaths (r = — .662 and r2 is .4382) and VIOLRATE explains 6% of the variance in motor vehicle deaths (r = .245 and r2 is .0600). Together, though, INCOME and VIOLRATE have a multiple-R of .762 and an R2 of .581. Here again, the two variables explain more variance working together than they explain working separately.

In other words, it’s the complex association of per capita income and the level of violence that explains so much variance in both the rate of teenage births and the rate of motor vehicle deaths. It turns out that lots of things are best explained by a series of variables acting together (box 22.2).

 
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