## Testing Homogeneity of Two Correlation Coefficient

Let $\rho_1$ be the correlation coefficient between $X$ and $Y$ in one population and $\rho_2$ be the correlation coefficient between $X$ and $Y$ in another population.

Let $n_1$ be the sample pair of observations from the first population with sample correlation coefficient $r_1$ and $n_2$ be the sample pair of observations from the second population with sample correlation coefficient $r_2$.

The hypothesis testing problem can be structured in any one of the three situations as follows:

Situation Hypothesis Testing Problem
Situation A $H_0: \rho_1=\rho_2$ against $H_a : \rho_1<\rho_2$ (Left-tailed)
Situation B $H_0: \rho_1=\rho_2$ against $H_a : \rho_1>\rho_2$ (Right-tailed)
Situation C $H_0: \rho_1=\rho_2$ against $H_a : \rho_1\neq\rho_2$ (Two-tailed)

## Formula

The test statistic for testing above hypothesis is $$\begin{eqnarray*} Z=\dfrac{Z_1-Z_2}{\sqrt{\frac{1}{n_1-3}+\frac{1}{n_2-3}}} \end{eqnarray*}$$ where \begin{aligned} Z_1=\frac{1}{2}\log_e \bigg(\frac{1+r_1}{1-r_1}\bigg) \end{aligned} and \begin{aligned} Z_2=\frac{1}{2}\log_e \bigg(\frac{1+r_2}{1-r_2}\bigg) \end{aligned} The test statistic $Z$ follows standard normal distribution $N(0,1)$.