2.5 Multiple Regression

Let us try a multiple regression model with linear and additive effects of social setting and program effort:

. regress change setting effort
 
      Source |       SS       df       MS              Number of obs =      20
-------------+------------------------------           F(  2,    17) =   23.96
       Model |  1956.19433     2  978.097163           Prob > F      =  0.0000
    Residual |  694.005675    17  40.8238632           R-squared     =  0.7381
-------------+------------------------------           Adj R-squared =  0.7073
       Total |      2650.2    19  139.484211           Root MSE      =  6.3894
 
------------------------------------------------------------------------------
      change |      Coef.   Std. Err.      T    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     setting |   .2705885   .1079405     2.51   0.023      .042854     .498323
      effort |   .9677137   .2250074     4.30   0.000     .4929895    1.442438
       _cons |   -14.4511   7.093841    -2.04   0.058    -29.41779    .5155975
------------------------------------------------------------------------------

Compare the coefficients with Table 2.5 and the anova with Table 2.7.

The estimates suggest that fertility decline increases with social setting, about a quarter of a percentage point per point of setting, when we compare countries with the same effort. Similarly, fertility declines about one percentage point more per point of program effort, when we compare countries with the same social setting.

It is important to keep in mind that both conclusions are based on the model, which assumes linearity and additivity. In fact we can't really compare countries which differ exactly by one point in effort and have the same setting, so goodness of fit will be a central concern. You may also note that I try to avoid using causal language when describing observational data.

Partial Correlations.

Make sure you know how to reproduce the hierarchical anova in Table 2.8 in the notes. Let us check the partial correlation listed on page 27. Here's the coefficient or change and effort controlling for setting:

. display "partial correlation = " sqrt((1449.1-694)/1449.1)
partial correlation = .72186013

Squaring this coefficient we see that program effort is associated with about half the variation left in fertility decline in countries with the same social setting. (It is common to say "after controlling for social setting", but of course we are not able to manipulate any of these variables.)

We could have computed the partial correlation using Stata's pcor command, which you might want to try to verify our result, but the calculation above shows more clearly what we are doing. Also, we could have used stored quantities, if only we had stored the residual sum of squares of the earlier model in a scalar.

Continue with 2.6 One-Way Analysis of variance