# testing whether two correlation coefficients are different from a non zero value

Hello I have 2 bivariate normal populations with correlation coefficients p1 and p2 respectively. I could easily obtain samples from both. I am interested in testing whether the difference in these correlations is a equal to c. In other words the null hypothesis is H0: p1 - p2 = c . I know you could test H0: p1 - p2 = 0 using fisher transformation but don't this this is possible for this case.

• Perhaps establish a confidence interval for the difference and see if it includes $c$? – mdewey Jan 1 at 16:53
• How would you go about obtaining a confidence interval when you don't know the distribution of the estimator p1_hat - p2_hat? – bgsimeon Jan 1 at 22:43

## 1 Answer

One approach might be to use confidence intervals. Suppose we have two coefficients $$r_1$$ and $$r_2$$ based on independent samples of $$n_1$$ and $$n_2$$ observations respectively. Transform each into the appropriate Fisher $$z$$. The the standard error of the difference $$z_1 - z_2$$ is

$$\sqrt{\frac{1}{n_1 - 3} + \frac{1}{n_2 - 3}}$$

So a confidence interval can be established for any desired confidence coefficient by multiplying the standard error by the corresponding normal deviate and taking that value either side of $$z_1 - z_2$$. The convert back to the scale of $$r$$ if desired and check whether $$c$$ lies within the limits.