Fisher, two correlation coefficient: Help!

Tonyandthewhale

Hi, I'm trying to compare the difference between two correlation coefficients to see if there's a significant difference and t'google has suggested I use a fisher r-to-z transformation. Found a handy calculator online for such; http://faculty.vassar.edu/lowry/rdiff.html and I think I'm plugging the right numbers in so that's not a issue but the problem is I don't know how to interpret the resulting z score.

Let's say I compare two correlation co-efficients, 0.149 and 0.210 with n=60 for both. The one-tailed z score result is 0.3669 which is all well and good but what does that mean? And what am I supposed to say about that in my interpretation?

Basically, what I'm asking is, is there a rule of thumb for when you can describe such a Z score as 'significant?' Or is there further analysis which must be carried out before this score is meaningful in terms of discussing the significance of the difference?

Sorry for rambling on and sorry if none of the above made any sense (stats is not my strong suite) and thanks for any help you might send my way.

hivizman

Actually, you have reported the probability of obtaining the Z-statistic (or a Z-statistic with the same sign but larger in terms of magnitude).

The actual test statistic is given by the software you used as -0.34. I calculated it manually to be -0.336615, so the software appears to be correctly calculating the test statistic.

If your null hypothesis is that the two samples come from underlying populations with the same correlation between the two variables, you cannot reject this null hypothesis and accept the alternative hypothesis that the correlations in the underlying populations are different.

The reason for this is that the test statistic Z follows the normal distribution with mean 0 and variance 1. The way I have formulated the null and alternative hypotheses, you are applying a two-tailed test rather than a one-tailed test. The p-value (probability) for the two-tailed test is twice that for the one-tailed test, that is, 0.7339. This means that, if the correlation coefficients in the underlying populations are equal, the probability of getting a Z-statistic (ignoring the sign) of 0.34 or more is 0.7339 (nearly 3 in 4, so very likely). So the difference between the two sample correlation coefficients isn't big enough to reject the null hypothesis that the underlying population correlations are the same.

A common criterion for rejecting a null hypothesis is that the probability of obtaining the Z-statistic (or a statistic larger in magnitude, ignoring sign) by chance, given that the underlying correlations are equal, is 0.05 or less. If the Z-statistic is 1.96 or more, then we can reject the null hypothesis.

Working back from this, if you have two samples of 60, the sample correlation coefficients need to differ by around 0.4 to give you a Z-statistic that would allow you to reject the null hypothesis of equal correlations in the underlying populations.

Simply put, Fisher's Z-transformation allows you to use the normal distribution tables to assess the significance of the Z-statistic.