Statistical Methods for Psychology

278 Chapter 9 Correlation and Regression

to first-grade children, and then administered the Wide Range Achievement Test (WRAT)

to those same children 2 years later. They obtained, among other findings, the following

correlations:

WRAT WISC-R McCarthy

WRAT 1.00 .80 .72

WISC-R 1.00 .89

McCarthy 1.00

Note that the WISC-R and the McCarthy are highly correlated but that the WISC-R corre-

lates somewhat more highly with the WRAT (reading) than does the McCarthy. It is of in-

terest to ask whether this difference between the WISC-R–WRAT correlation (.80) and the

McCarthy–WRAT correlation (.72) is significant, but to answer that question requires a test

on nonindependent correlations because they both have the WRAT in common and they are

based on the same sample.

When we have two correlations that are not independent—as these are not, because the

tests were based on the same 26 children—we must take into account this lack of independ-

ence. Specifically, we must incorporate a term representing the degree to which the two tests

are themselves correlated. Hotelling (1931) proposed the traditional solution, but a better test

was devised by Williams (1959) and endorsed by Steiger (1980). This latter test takes the form

where

This ratio is distributed as t on N-3 df. In this equation, and refer to the correla-

tion coefficients whose difference is to be tested, and refers to the correlation between

the two predictors. |R| is the determinant of the 3 3 3 matrix of intercorrelations, but you

can calculate it as shown without knowing anything about determinants.

For our example, let

then

A value of 5 1.36 on 23 dfis not significant. Although this does not prove the ar-

gument that the tests are equally effective in predicting third-grade children’s performance

on the reading scale of the WRAT, because you cannot prove the null hypothesis, it is con-

sistent with that argument and thus supports it.

tobt

=1.36

t=(.80 2 .72)

Q

(25)(1 1 .89)

2 a

25

23

b(.075) 1

(.80 1 .72)^2

4

(1 2 .89)^3

ƒRƒ =(1 2 .80^22 .72^22 .89^2 ) 1 (2)(.80)(.72)(.89)=.075

N= 26

r 23 = correlation between the WISC-R and the McCarthy=.89

r 13 = correlation between the McCarthy and the WRAT=.72

r 12 = correlation between the WISC-R and the WRAT=.80

r 23

r 12 r 13

ƒRƒ =(1 2 r^2122 r^2132 r^223 ) 1 (2r 12 r 13 r 23 )

t=(r 122 r 13 )

Q

(N 2 1)(1 1 r 23 )

2 a

N 21

N 23

bƒRƒ 1

(r 121 r 13 )^2

4

(1 2 r 23 )^3