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.
