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If you want to test the null hypothesis that the correlation between X and Y in one population is the same as the correlation between X and Y in another population, you can use the procedure developed by R. A. Fisher in 1921 (On the probable error of a coefficient of correlation deduced from a small sample, Metron, 1, 3-32).
First, transform each of the two correlation coefficients in this fashion:
Second, compute the test statistic this way:
Third, obtain p for the computed z.
Consider the research reported in by Wuensch, K. L., Jenkins, K. W., & Poteat, G. M. (2002). Misanthropy, idealism, and attitudes towards animals. Anthrozoös, 15, 139-149.
The relationship between misanthropy and support for animal rights was compared between two different groups of persons – persons who scored high on Forsyth’s measure of ethical idealism, and persons who did not score high on that instrument. For 91 nonidealists, the correlation between misanthropy and support for animal rights was .3639. For 63 idealists the correlation was .0205.
The test statistic,
, p = .031, leading to the conclusion that the correlation in nonidealists is significantly higher than it is in idealists.
See the pdf document Bivariate Correlation Analyses and Comparisons authored by Calvin P. Garbin of the Department of Psychology at the University of Nebraska. Dr. Garbin has also made available a program (FZT.exe) for conducting this Fisher’s z test.
If you are going to compare correlation coefficients, you should also compare slopes. It is quite possible for the slope for predicting Y from X to be different in one population than in another while the correlation between X and Y is identical in the two populations, and it is also quite possible for the correlation between X and Y to be different in one population than in the other while the slopes are identical, as illustrated below:
On the left, we can see that the slope is the same for the relationship plotted with blue o’s and that plotted with red x’s, but there is more error in prediction (a smaller Pearson r ) with the blue o’s. For the blue data, the effect of extraneous variables on the predicted variable is greater than it is with the red data.
On the right, we can see that the slope is clearly higher with the red x’s than with the blue o’s, but the Pearson r is about the same for both sets of data. We can predict equally well in both groups, but the Y variable increases much more rapidly with the X variable in the red group than in the blue.
H_{Æ}: b_{1} = b_{2}
Let us test the null hypothesis that the slope for predicting support for animal rights from misanthropy is the same in nonidealists as it is in idealists.
First we conduct the two regression analyses, one using the data from nonidealists, the other using the data from the idealists. Here are the basic statistics:
Group |
Intercept |
Slope |
SE_{slope} |
SSE |
SD_{X} |
n |
Nonidealists |
1.626 |
.3001 |
.08140 |
24.0554 |
.6732 |
91 |
Idealists |
2.404 |
.0153 |
.09594 |
15.6841 |
.6712 |
63 |
The test statistic is Student’s t, computed as the difference between the two slopes divided by the standard error of the difference between the slopes, that is, on (N – 4) degrees of freedom