Statistical Methods for Psychology

Then to test

For a test on the regression coefficient of Expend, we have

This is a standard Student’s t on N 2 p 21 5 50 – 2 21 547 df, and the critical value is

found in Appendix t to be 2.01. Thus, we can reject and conclude that the regression co-

efficient in the population is not equal to 0. We don’t actually need tables of t, because our

printout gives not only t, but also its (two-tailed) significance level. Thus a bas large as

11.130 (for Expend) has a two-tailed probability of .001 under. In other words, the pre-

dicted value of Yincreases with increasing scores on Expend, and Expend thus makes a

significant contribution to the prediction of SAT.

A corresponding test on the coefficient for LogPctSAT would produce

This result is also significant (p 5 .000), meaning that LogPctSAT contributes signifi-

cantly to the prediction of SAT over and abovewhat Expend contributes. When we added

the PTratio to the model, the resulting twas .481, which was not significant. We might con-

sider dropping this predictor from our model, but there will be more on this issue later. It is

important to recognize that a test on a variable is done in the context of all other variables

in the equation. A variable might have a high individual correlation with the criterion, as

does Salary, with a significant Pearson rwith SAT 52 .440 (p 5 .001), but have nothing

useful to contribute once several other variables are included. That is the situation here.

(Salary correlates .87 with Expend, so once we take Expend into account there is little left

over for Salary to explain.)

Some computer programs prefer to print standard errors for, and test, standardized re-

gression coefficients ( ). It makes no difference which you do. Similarly, some programs

provide an Ftest (on 1 and N 2 p 21 df) instead of t. This Fis simply the square of our t,

so again it makes no difference which approach you take.

15.4 Residual Variance

We have just considered the standard error of the regression coefficient, recognizing that

sampling error is involved in the estimation of the corresponding population regression co-

efficient. A somewhat different kind of error is involved in the estimation of the predicted

Ys. In terms of the SAT data, we would hope that the SAT score is, at least in part, a func-

tion of such variables as Expend, LogPctSAT, and so on. (If we didn’t think that, we would

not have collected data on those variables in the first place.) At the same time, we probably

do not expect that the two or three variables we have chosen will predict Yperfectly, even

if they could be measured, and the coefficients estimated, without error. Error will still be

involved in the prediction of Yafter we have taken all of our predictors into account. This

error is called residual varianceor residual errorand is defined as

a(Y^2 Y

N) 2

N 2 p 21

bj

t=

2 78.205

4.471

= 2 17.491

H 0

H 0

t=

11.130

3.264

=3.410

t=

bj

sbj

H 0 : bj*=0,

530 Chapter 15 Multiple Regression

residual variance

residual error