[R] Testing difference between two partial correlations

[Tore Wentzel-Larsen]

Dear list members,

Perhaps more a methodological than a strictly R question (but I have 
searched for solutions in R Site Search, R FAQ..., and I intend to 
implement the solutions in R. Answers containing references to 
existing R code are of course highly appreciated).

What test(s) should be used for testing for differences between two 
partial (Pearson) coefficients, from independent samples, where
the two variables correlated and the covariates 'corrected for' are 
the same in both samples? Samples sizes are about 100-200,
and the number of covariates are 2-3.

I have already tried to generalize a not uncommon test statistic for two 
'non-partial' Pearson correlations, (z1-z2)/sqrt( 1/(n1-3) + 1/(n2-3) ), 
based on Fisher's Z transform z=ln((1+r)/(1-r))/2 for each correlation
coefficient (used e. g. in the commercial program Power and Precision). 
Correction for degrees of freedom as proposed in e. g.  Afifi & Clark: 
Computer-aided Multivariate Analysis (3. ed., section 7.7) suggests the test 
statistic (z1-z2)/sqrt( 1/(n1-q-3) + 1/(n2-q-3) ) (assumed standard normal under 
the null hypothesis of no difference in 'mean' partial correlations; here z1 and z2 
are the two transformed partial correlations, n1 and n2 are sample sizes and q is the 
number of covariates involved). Checks (in R) by non- and semiparametric bootstrapping 
(details may be given), indicate that this test statistic is not very far from standard 
normal, but with heavier tails in some of my (brain morphology) data sets, and also with 
a standard deviation that in some cases deviates a bit from 1 (in both directions; again, 
details may be given). Thus, better and perhaps more robust tests might be an advantage.

Since this is not a question of problems running R, I do not give full detail of 
hardware... I am running under Windows 2000, using R 1.5.1.

Sincerely,
Tore Wentzel-Larsen
Centre for Clinical Research
Health Care Bergen, Norway
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