Statistical Methods for Psychology

272 Chapter 9 Correlation and Regression

A legitimate t test can be formed from the ratio

which is distributed as t on N 2 2 df.^13 Returning to the example in Exhibit 9.1, r 5 .532

and N 5 28. Thus,

This value of t is significant at a5.05 (two-tailed), and we can thus conclude that

there is a significant relationship between SAT scores and scores on Katz’s test. In other

words, we can conclude that differences in SAT are associated with differences in test

scores, although this does not necessarily imply a causal association.

In Chapter 7 we saw a brief mention of the Fstatistic, about which we will have much more

to say in Chapters 11–16. You should know that any t statistic on d degrees of freedom can be

squared to produce an Fstatistic on 1 and d degrees of freedom. Many statistical packages use

the Fstatistic instead of t to test hypotheses. In this case you simply take the square root of that

Fto obtain the t statistics we are discussing here. (From Exhibit 9.1 we find an Fof 10.251. The

square root of this is 3.202, which agrees with the t we have just computed for this test.)

As a second example, if we go back to our data on stress and psychological symptoms

in Table 9.2, and the accompanying text, we find r 5 .506, and N 5 107.

Here again we will reject We will conclude that there is a significant relation-

ship between stress and symptoms. Differences in stress are associated with differences in

reported psychological symptoms.

The fact that we have an hypothesis test for the correlation coefficient does not mean

that the test is always wise. There are many situations where statistical significance, while

perhaps comforting, is not particularly meaningful. If I have established a scale that pur-

ports to predict academic success, but it correlates only r 5 .25 with success, that test is

not going to be very useful to me. It matters not whether r 5 .25 is statistically significantly

different from .00, it explains so little of the variation that it is unlikely to be of any use.

And anyone who is excited because a test-retest reliability coefficient is statistically signif-

icant hasn’t really thought about what they are doing.

Testing the Significance of b

If you think about the problem for a moment, you will realize that a test on bis equivalent to a

test on rin the one-predictor case we are discussing in this chapter. If it is true that Xand Yare

related, then it must also be true that Yvaries with X—that is, that the slope is nonzero. This

suggests that a test on bwill produce the same answer as a test on r, and we could dispense with

a test for baltogether. However, since regression coefficients play an important role in multiple

regression, and since in multiple regression a significant correlation does not necessarily imply

a significant slope for each predictor variable, the exact form of the test will be given here.

We will represent the parametric equivalent of b(the slope we would compute if we

had Xand Ymeasures on the whole population) as .b*^14

H 0 : r=0.

t=

.529 1105

312 .529^2

=

.529 1105

1 .720

=6.39

r¿=.529

t=

.532 126

312 .532^2

=

.532 126

1 .717

=3.202

t=

r 1 N 22

312 r^2

(^13) This is the same Student’s tthat we saw in Chapter 7.
(^14) Many textbooks use binstead of , but that would lead to confusion with the standardized regression coefficient.b*