How to describe a test of differences between correlation of the same two variables across two groups? If I have two (Pearson) correlation coefficients between Variables A and B (for example .657 for males and .876 for females) and I want to test if there is a difference based on gender (as in this case), which of the following phrases should I use to describe this process?


*

*Test for equality of the correlation coefficients

*Test for the difference in the correlation coefficients


I want to use one of the above as a row heading in a table that describes the above results, so I am looking for the one that is more theoretically grounded in statistical terminology.
 A: "Test for equality" is more theoretically correct because you test the (likelihood of) null hypothesis, and it states that the coefficients are equal in the population. But "Test for difference" will be more common way to put it, because it is the alternative hypothesis - the inequality - which a researcher and the reader usually "is after" or has a morbid interest in. Why don't you simply label the table "Correlation coefficients for females and males"?
A: You should test for the difference in the correlation coefficients because if the test is positive, you reject the null hypothesis (equality of the two coefficients).
And if your test is negative, you conclude that you can not reject the null hypothesis, and therefore can not conclude that there is a difference, which does not mean that the two coefficients are equal (they most likely are not exactly the same) but it means that given the level of information in your data, you can not establish that they are different.
Given enough data points, you could have a significant difference between the correlation of variables A and B for males and females, even if the actual coefficient are very close (e.g. 0.40 and 0.41). In such a situation, you would state the statistically significant difference between the coefficients and you may add (based on your knowledge of the issue at hand) that such a small difference is likely to be of little consequence on the real life event under study (e.g. the difference may not be big enough to require that different drugs be developped for each gender).
A: Listen to @whuber and to me.
Don't do this. 
It seems almost completely meaningless. 
