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Two Correlation coefficients

Example:
p <= 0.00216

The correlation coefficients (R1, R2)
R1 R2

The sample sizes (N1, N2)
N1 N2


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).


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