Testing Correlation Coefficient
Let $(X_i, Y_i), i=1,2,\cdots, n$ be a random sample of $n$ pairs of observations drawn from a bivariate population with correlation coeffficient $\rho$.
Let $r$ be the observed correlation coefficient between $X$ and $Y$.
We wish to test the hypothesis $H_0 : \rho =\rho_0$ against $H_a : \rho \neq \rho_0$.
Assumptions
- The population from which, the samples drawn, is a bivariate normal.
- The relationship between $X$ and $Y$ is linear.
Step by step procedure
The step by step procedure for testing $H_0: \rho = \rho_0$ 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: | $H_0: \rho=\rho_0$ against $H_a : \rho < \rho_0$ (Left-tailed) |
| Situation B: | $H_0: \rho=\rho_0$ against $H_a : \rho > \rho_0$ (Right-tailed) |
| Situation C: | $H_0: \rho=\rho_0$ against $H_a : \rho \neq \rho_0$ (Two-tailed) |
Step 2 Define the test statistic
The test statistic for testing above hypothesis is
$$ \begin{aligned} Z&=\dfrac{U-\xi}{\sqrt{\frac{1}{n-3}}} \end{aligned} $$
where
$$ \begin{aligned} U&=\frac{1}{2}\log_e \bigg(\frac{1+r}{1-r}\bigg) \end{aligned} $$
and
$$ \begin{aligned} \xi & =\frac{1}{2}\log_e \bigg(\frac{1+\rho_0}{1-\rho_0}\bigg) \end{aligned} $$
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 depending upon the alternative hypothesis.
- For left-tailed alternative hypothesis: Find the $Z$-critical value using
$$ \begin{aligned} P(Z<-Z_\alpha) &= \alpha. \end{aligned} $$
- For two-tailed alternative hypothesis: $Z_{\alpha/2}$.
$$ P(Z<-Z_{\alpha/2} \text{ or } Z> Z_{\alpha/2}) = \alpha. $$
- For right-tailed alternative hypothesis: $Z_\alpha$.
$$ \begin{aligned} P(Z>Z_\alpha) & = \alpha. \end{aligned} $$
Step 5 Computation
Compute the test statistic under the null hypothesis $H_0$ using equation
$$ \begin{aligned} Z_{obs} &= \dfrac{U-\xi}{\sqrt{\frac{1}{n-3}}} \end{aligned} $$
Step 6 Decision (Traditional Approach)
Based on the critical values.
- For left-tailed alternative hypothesis: Reject $H_0$ if
$Z_{obs}\leq -Z_\alpha$. - For right-tailed alternative hypothesis: Reject $H_0$ if
$Z_{obs}\geq Z_\alpha$. - For 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 |
|---|---|---|
| $H_a: \rho<\rho_0$ | Left-tailed | $p$-value $= P(Z\leq Z_{obs})$ |
| $H_a: \rho>\rho_0$ | Right-tailed | $p$-value $= P(Z\geq Z_{obs})$ |
| $H_a: \rho\neq \rho_0$ | Two-tailed | $p$-value $= 2P(Z\geq abs(Z_{obs}))$ |
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.
