On Thu, 26 Jun 2003, jackson_high wrote (edited):
> I have a problem. I set up two correlation-coefficients both
> correlated to a common feature. How can I now test those two
> coefficients for significance in the sense of differing
> significantly?? And how can I do this in SPSS?
> Thanks for any quick help!!
If I understand you correctly, you have a correlation r_xy between
two variables X and Y, and a correlation r_xz between variables X and Z,
for the same sample of instances (persons, cases, whatever); and you
wish to test whether r_xy differs from r_xz.
Equivalently, you wish to test whether (r_xy - r_xz) differs from 0.
Glass & Stanley (1970) report the following procedure, where n is the
sample size and r_yz is the correlation between Y and Z:
Compute Z = Numerator/Denominator, where
Numerator = (r_xy - r_xz)*<square root of n>, and
Denominator = square root of (A +B - C - D*E), where
A = (1 - (r_xy)^2)^2
B = (1 - (r_xz)^2)^2
C = 2*(r_yz)^3
D = (2*r_yz - (r_xy)*(r_xz))
E = (1 - (r_xy)^2 - (r_xz)^2 - (r_yz)^2)
in all of which * = multiply and ^ = exponentiation: k^2 = k*k, e.g.
The sampling distribution of Z (when the null hypothesis that the true
value of Numerator is zero is true) is approximately normal (Gaussian)
with mean 0 and variance 1. When the alternative hypothesis is true,
the mean of thesampling distribution is not 0, but its variance remains
approximately equal to 1.
(Null hyp.: rho(xy) = rho(xz); alternative hyp.: rho(xy) <> rho(xz))
Glass & Stanley cite
Olkin, "Correlations revisited" in 'Improving Experimental Design and
Statistical Analysis', ed. J.C. Stanley: Rand McNally, 1967; and
Olin & Siotani, "Asymptotic distribution functions of a correlation
matrix": Stanford, California, Stanford University Laboratory for
Quantitative Research in Education, Report No. 6, 1964.
I do not know whether this can be performed automatically in SPSS, but
SPSS will certainly calculate the three r's for you and report the
corresponding n (notice that it is assumed that data are complete: that
is, there are no cases with missing values on one or two of the three
variables involved), and you can carry out the computation above with a
calculator. If you have lots of these to do, instead of just one or
two, you'll want to automate the calculation, perhaps by devising a
macro in SPSS syntax, or perhaps by carrying out the whole thing in R.
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