A Test of the Equivalence of Two Correlation
Coefficients
{From the Institute of Phonetic Sciences (IFA):
http://www.fon.hum.uva.nl/}
Characteristics:
This is a quite insensitive test to decide whether two correlations have
different strengths. In the
standard tests for correlation, a correlation coefficient is tested against
the hypothesis of no correlation, i.e., R = 0. It is possible to
test whether the correlation coefficient is equal to or different from
another fixed value, but this has few uses (when can you make a reasonable guess
about a correlation coefficient?). However, there are situations where you would
like to know whether a certain correlation strength realy is different from
another one.
H0:
Both samples of pairs show the same correlation strength, i.e., R1 = R2.
Assumptions:
The values of both members of both samples of pairs are Normal (bivariate)
distributed.
Scale:
Interval (for the raw data).
Procedure:
The two correlation coefficients are transformed with the Fisher Z-transform (
Papoulis):
Zf = 1/2 * ln( (1+R) / (1-R) )
The difference
z = (Zf1 - Zf2) / SQRT( 1/(N1-3) + 1/(N2-3) )
is approximately Standard Normal distributed.
If both the correlation coefficient and the sample size of one of the
samples are equal to zero, the standard procedure for
correlation coefficients is used on the other values.
Level of Significance:
Use the z value to determine the level of significance.
Approximation:
This is already an approximation which should be used only when both samples (N1
and N2) are larger than 10.
Remarks:
Check whether you realy want to know whether the correlation coefficients
are different. Only rarely is this a usefull question.
A warning is printed next to the significance level if the number of samples is
too small (i.e., less than 11).
You can compute this test by clicking HERE.