Statistical Methods for Psychology

coefficient (slope) that we obtain is the same coefficient we find in the multiple regres-

sion solution.

• We can think of the multiple correlation as the simple Pearson correlation between the
  criterion (call it Y) and another variable (call it ) that is the best linear combination of
  the predictor variables.
  The Educational Testing Service, which produces the SAT, tries to have everyone put a
  disclaimer on results broken down by states that says that the SAT is not a fair way to com-
  pare the performance of different states. Having gone through this example you can see
  that one reason that they say this is that different states have different cohorts of students
  taking the exam, and this makes the test inappropriate as a way of judging a state’s
  performance, even if it is a good way of judging the performance of individuals. We could
  create a new variable that is the SAT score adjusted for LogPctSAT, but I would be very
  wary of using that measure to compare states. It is possible that it would be fair, but it is
  also possible that there are a number of other variables that I have not taken into account.

15.2 Using Additional Predictors

Before we look at other characteristics of multiple regression we should ask what would

happen if we used additional variables to predict SAT. We have two potential variables in

our data that we have not used—the pupil/teacher ratio and teacher’s salaries. We could add

both of them to what we already have, but I am only going to add PTratio. Folklore would

have it that a lower ratio would be associated with better performance. At the same time,

lower pupil/teacher ratios cost money, so PTratio should overlap with Expend and might

not contribute significant new information.

Table 15.3 shows the results of using Expend, LogPctSAT, and PTratio to predict SAT.

There are several things to say about this table.

The regression equation that results from this analysis is now

Notice that Expend and LogPctSAT are still significant (t 5 3.302 and 2 17.293, respec-

tively, but PTratio is far from significant (t 5 .418). This shows us that adding PTratio to our

model did not improve our ability to predict. (Even the simple correlation between PTratio

and SAT was not significant (r 5 .081).) You will see two new columns in Table 15.3, label

Toleranceand VIF (Variance Inflation Factor).When predictor variables are correlated

among themselves we have what is called collinearityor multicollinearity.Collinearity has

the effect of increasing the standard error of a regression coefficient, which increases the

YN =1132.033 1 11.665 Expend 2 78.393 PctSAT 2 0.742 PTratio

YN

15.2 Using Additional Predictors 527

Coefficientsa

Model

1 Constant

Expend

LogPctSAT

PTratio

B

1132.033

11.665

–78.393

.742

Std. Error

39.787

3.533

4.533

1.774

Beta

.212

–1.042

.022

t

28.452

3.302

–17.293

.418

Sig.

.000

.002

.000

.678

Tolerance

.596

.679

.854

VIF

1.679

1.473

1.171

Unstandardized

Coefficients Collinearity Statistics

Standardized

Coefficients

aDependent Variable: SAT

Table 15.3 Adding PTratio to the prediction equation

Tolerance

VIF (Variance
Inflation Factor)

collinearity

multicollinearity