discuss is labeled “Model Summary.” From the summary you can see that when Expend is
the sole predictor the correlation between Expend and SAT is 2 .381, just as we saw be-
fore. But when we add LogPctSAT the correlation jumps to .941, which is a very long way
from the correlation of 2 .381 that we obtained with Expend alone.
A couple of things need to be said here. In multiple regression the correlations are
always going to be positive, whereas in simple Pearson correlation they can be either posi-
tive or negative. There is a good reason for this, but I don’t want to elaborate on that now. (If
the correlations are always positive, how do we know when the relationship is negative? We
look at the sign of the regression coefficient, and I’ll come to that in a minute.) You might
recall that in Table 15.2 we saw that the simple correlation between SAT and LogPctSAT
was 2 .93 whereas the correlation between SAT and Expend was 2 .38. While LogPctSAT
adds a great deal to the regression that just used Expend, adding Expend to the correlation,
versus just using LogPctSAT adds much less. We will look at this more closely in a minute.
In the subtable named Model Summary you will also see the squared correlations. The
squared correlation in multiple regression has the same meaning that it had in simple re-
gression. Using Expend alone we were able to explain ( 2 .381)^25 .145 5 14.5% of the
variation in SAT scores (not shown in table). Using both Expend and LogPctSAT we can
explain .941^25 .886 5 88.6% of the variability in the SAT score. To the right of these val-
ues you will see a column labeled Adj.R square. You can ignore that column. The adjusted
R squared is actually a less biased estimate of the true squared correlation in the popula-
tion, but we never report it. Simply useR and not adjusted R.
The third subtable in Exhibit 15.1 is labeled ANOVA, which is an analysis of variance
testing the significance of the regression. The Fis a test on whether the multiple correlation
coefficient in question is significantly different from 0. This is the same kind of test that we
saw in Chapters 9 and 10, though it uses an Fstatistic instead of t. When we have only one
predictor (Expend) the correlation is 2 .38, as we saw in Table 15.2, and the probability of
getting a correlation that high if the null hypothesis is true was.006. This is well less than
.05 and we can declare that correlation to be significantly different from 0. When we move
to multiple regression and include the predictor LogPctSAT along with Expend, we have
two questions to ask. The first is whether the multiple correlation using both predictors to-
gether is significantly different from 0.00, and the second is whether each of the predictor
variables is contributing at greater than chance levels to that relationship. From the ANOVA
table we see an F 5 182.856, with an associated probability of 0.000 to three decimal
places. This tells us that using both predictors our correlation is significantly greater than 0.
I will ask about the significance of the individual predictors in the next section.
Now we come to the most interesting part of the output. In the subtable labeled
“Coefficients” we see the full set of regression coefficientswhen using both predictors at
the same time. Just as a simple regression equation was of the form
,
a multiple regression equation is written aswhere X 1 and X 2 are the predictors and b 0 is the intercept. From the table we can see that,
with both predictors, the coefficient for Expend (call it b 1 ) is 11.130, and for LogPctSAT
the coefficient is 2 78.205. From the sign of these coefficients we can tell whether the re-
lationship is positive or negative. These values, plus the intercept, give us our regression
equation.The value of 1147.133 is the intercept, often denoted b 0 and here denoted simply as
“(constant)“. This is the predicted value of SAT if both Expend and LogPctSAT were 0.00,YN =1147.113 1 11.130( Expend ) 2 78.205(LogPctSAT )YN =b 1 X 11 b 2 X 21 b 0YN =bX 1 a15.1 Multiple Linear Regression 523regression
coefficients