# 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.

• Using a table suggests you have made many tests of such equality. This is an unusual thing to do, perhaps because it may be difficult to interpret the results and perhaps because differences in correlation do not appear to reveal much. Suppose two variables do have different degrees of correlation for two subsets of the data. What does that mean?
– whuber
Commented Oct 27, 2011 at 19:27
• In my mind, if variable A correlates with B differently for males and for females (as shown by the test of equality), I can then say that the relationship between A and B is influenced by gender (or age or education etc.). Note I am not comparing males and females; I am comparing the relationship between A and B. Hence, all I am saying is that if you are a male, the relationship between A and B would be different for you, compared to if you are a female. (Something tells me that this borders on being meaningless!) Commented Oct 28, 2011 at 20:10
• I wouldn't go so far as to say "meaningless," but it is indeed a subtle relationship. One usually characterizes the overt relationships first, such as differences of means and regressions, before getting to this kind of comparison. Its interpretation is rendered difficult by the need to characterize possible confounders and to investigate linearity and homoscedasticity of the relationships.
– whuber
Commented Oct 28, 2011 at 21:41

"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"?

• My table has the following row headings: Demography (i.e. Gender), Pearson's 'r', r squared, p value, sample (n) and Equality of 'r'. The last label 'Equality of r' is what this question is about. Thanks a lot. (Am I missing any other vital information in this summary table?) Commented Oct 27, 2011 at 16:44

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).

Listen to @whuber and to me.

Don't do this.

It seems almost completely meaningless.

• Thanks and I am most grateful for your advice. Please help! How else do I find out if there is a difference, if any, in the relationship between A and B for males and females. I would add that the correlation for male and female on their own are statistically significant but this test of equality on the gender difference is not significant. I am totally confused now. Commented Oct 27, 2011 at 23:45
• Why do you want to know the answer to this question? Suppose the correlation is stronger for men than for women. Then what? Unless it's in a regression framework, I don't see how the answer can help you do anything Commented Oct 28, 2011 at 9:57
• I confess to lacking the imagination to conceive of a realistic situation where these analyses are useful. But I would like to suggest that you focus on your study objective, Adhesh, rather than on trying to analyze your data solely in terms of correlation coefficients. I do have the imagination to envision datasets where these comparisons are not just meaningless, but are also misleading, and I suspect that may be Peter's concern too. You've been dancing around this issue for 43 questions now. What is it you really want to know about your data? Can you phrase this in non-statistical terms?
– whuber
Commented Oct 28, 2011 at 16:20
• @whuber I like the way you have put it! (because it has helped me to rethink what I want to achieve with my data set). I am looking at the relationship between "product research" (A) and "product purchase" (B). Product means any consumer goods. A and B are composite scores (i.e. made up of responses to several questions). I want to see (1) if a relationship exists betwen A and B (which is bivarite in my case) and (2) if differences exist for certain demography (e.g. gender, age). i.e. do males do it differently than females. The second part is where "I am dancing..." because I don't know! Commented Oct 28, 2011 at 18:40
• I think you have formulated a useful question with that last comment, Adhesh. Why don't you consider starting (yet another :-) new thread where you pose it just like that, in a statistically neutral way (that is, without suggesting what analysis you think is needed) and perhaps offer an illustration of the kind of data you have, such as a dozen or so records randomly chosen from the database. I am also still wondering what your clients would be doing with your discoveries: if you do find relationships, just what kinds of recommendations would that lead you to make?
– whuber
Commented Oct 28, 2011 at 19:54