## 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 3 Specify the level of significance $\alpha$

### 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.