Test for two independent samples correlation coffients r1 r2 differ significantly
Answers
Theorem 1: Suppose r1 and r2 are as in the Theorem 1 of Correlation Testing via Fisher Transformation where r1 and r2 are based on independent samples and further suppose that ρ1 = ρ2. If z is defined as follows, then z ∼ N(0,1).
where
Proof: By Theorem 1 of Correlation Testing via Fisher Transformation for i = 1, 2
By Property 1 and 2 of Basic Characteristics of the Normal Distribution it follows that
where s is as defined above. Since ρ1 = ρ2 it follows that ρ´1 = ρ´2, and so
from which the result follows.
We can use Theorem 1 to test whether the correlation coefficients of two populations are equal based on taking a sample from each population and comparing the correlation coefficients of the samples.
Example 1: A sample of 40 couples from London is taken comparing the husband’s IQ with his wife’s. The correlation coefficient for the sample is .77. Is this significantly different from the correlation coefficient of .68 for a sample of 30 couples from Paris?
H0: ρ1 = ρ2
= FISHER(r1) = FISHER(.77) = 1.020
= FISHER(r2) = FISHER(.68) = 0.829
s = SQRT(1/(n1 – 3) + 1/(n2 – 3)) = SQRT(1/37 + 1/27) = 0.253
z = ( – )/s = (1.020 – .829) / .253 = 0.755
p-value = 2(1 – NORMSDIST(z) = 1 – NORMSDIST(.522)) = 0.45
We next perform either one of the following tests:
p-value = .45 > .05 = α
zcrit = NORMSINV(1 – α/2) = NORMSINV(.975) = 1.96 > .755 = z
In either case the null hypothesis is not rejected.