# How to test whether correlation measures differ when controlling or not for a third variable?

If the correlation between demographic dissimilarity and satisfaction is $r=.-14$ and the partial correlation, with career development partialled out, between demographic dissimilarity and satisfaction is $r=-.06$ across a very large sample of size $n$, what is the appropriate test to determine if these correlations are significantly different?

Steiger (1980) appears to be the authoritative article on this but the article and most other sources assume one is comparing the correlation between x/y and v/y with dependent groups. This one has stumped our school's statistic resource center.

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I don't doubt a particular test statistic aiming to accomplish what your asking for exists, but I will offer some alternatives that you may be interested in that offer different answers (but probably still interesting) given the nature of the question.

Like Rolando already stated, the extent to which the partial correlation is moderated depends on the correlation between the three variables. Below I have inserted a picture of the situation in the form of a path diagram (and you can just pretend the box with X is demographic dissimilarity, Z is career development, and Y is satisfaction). The test for whether the direct correlation between X and Y is different than the partial correlation of X and Y controlling for Z amounts to stating whether the blue line and the red line in the below picture are non-zero. So simply examining both of those correlations will be informative. If the product of those correlations is near zero, then the direct correlation between X and Y will be largely unchanged when partialling out the variance of Z.

Another test you may be interested in would be a likelihood ratio test. In this example you have a case of nested models, and hence you can test the model with only the direct correlation between X and Y and the model that includes both X and Z. This can be more easily extended to multiple Z's as well. But this is not the same as testing whether the effect of X is moderated by Z(s), but whether the model including X and Z is a better fit to the data than the model only including X.

Some might interpret this question as asking about moderating and mediating effects. By the names of the variables though I wouldn't immediately suggest these are what you are after. Kline (2010) has a few references regarding these, so those might be a good forray to search. Although to conduct those tests it appears you need to make much stronger assumptions about causality (Frazier, Barron, & Tix 2004 PDF).

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For this particular case there's not much of a difference to work with in practical terms, but for the general case, I'm going to go out on a limb and guess that there is no way to conduct a strict test of significance. The partial correlation will be a direct function of the correlations among the 3 variables. Depending on 'career's r with each of the others, we can specify the change from zero-order to partial r for the 2 main variables. So in a sense there's no room for variation and thus no place for a significance test. I think.

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