## Testing Homogeneity of Two Correlation Coefficient

We wish to test the null hypothesis that the correlation $\rho_1$ between $X$ and $Y$ in one population is the same as the correlation $\rho_2$ between $X$ and $Y$ in another population. Let $n_1$ be the sample pairs of observation from first population with sample correlation coefficient $r_1$ and $n_2$ be the sample pairs of observation from second population with sample correlation coefficient $r_2$.

We wish to test the hypothesis $H_0 : \rho_1 =\rho_2$ against $H_a : \rho_1 \neq \rho_2$.

## Assumptions

• The observations are independent within and between the populations.
• The joint distribution of two variables in each population is bivariate normal.

## Step by step procedure

The step by step procedure for testing $H_0 : \rho_1 =\rho_2$ is as follows:

### Step 1 State the hypothesis testing problem

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)

### Step 2 Define the test statistic

The test statistic for testing above hypothesis is $$\label{1} \begin{eqnarray} Z=\dfrac{Z_1-Z_2}{\sqrt{\frac{1}{n_1-3}+\frac{1}{n_2-3}}} \end{eqnarray}$$ 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 $N(0,1)$.

### Step 4 Determine the critical values

For the specified value of $\alpha$ determine the critical region. \begin{aligned} P(Z<Z_{1-\alpha/2} \text{ or } Z> Z_{\alpha/2}) = \alpha. \end{aligned}

### Step 5 Computation

Compute the test statistic under the null hypothesis $H_0$ using equation \eqref{1}. \begin{aligned} Z_{obs}=\dfrac{Z_1-Z_2}{\sqrt{\frac{1}{n_1-3}+\frac{1}{n_2-3}}} \end{aligned}

### Step 6 Decision (Traditional Approach)

Traditional approach is based on the critical value.

• For left-tailed alternative hypothesis: Reject $H_0$ if $Z_{obs}\leq -Z_{\alpha}$.
• right-tailed alternative hypothesis: Reject $H_0$ if $Z_{obs}\geq Z_{\alpha}$.
• two-tailed alternative hypothesis: Reject $H_0$ if $|Z_{obs}|\geq Z_{\alpha/2}$.

OR

### Step 6 Decision ($p$-value Approach)

It is based on the $p$-value.

Alternative Hypothesis Type of Hypothesis $p$-value
$H_a: \rho_1<\rho_2$ Left-tailed $p$-value $= P(Z\leq Z_{obs})$
$H_a: \rho_1>\rho_2$ Right-tailed $p$-value $= P(Z\geq Z_{obs})$
$H_a: \rho_1\neq \rho_2$ Two-tailed $p$-value $= 2P(Z\geq Z_{obs})$

If $p$-value is less than $\alpha$, then reject the null hypothesis $H_0$ at $\alpha$ level of significance, otherwise fail to reject $H_0$ at $\alpha$ level of significance.