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 normal population with correlation coeffficient $\rho$.
Let $r_{xy}$ or $r$ be the observed correlation coefficient between $x$ and $y$. A test of significance for a linear relationship between the variables $x$ and $y$ can be performed using the sample correlation coefficient $r_{xy}$.
We wish to test the hypothesis $H_0 : \rho =0$ (there is no significant linear relationship between $x$ and $y$) against $H_a : \rho \neq 0$ (there is a significant linear relationship between $x$ and $y$).
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 $H_0: \rho = 0$ 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 | $H_0: \rho=0$ against $H_a : \rho<0$ (Left-tailed) |
| Situation B | $H_0: \rho=0$ against $H_a : \rho>0$ (Right-tailed) |
| Situation C | $H_0: \rho=0$ against $H_a : \rho\neq 0$ (Two-tailed) |
Step 2 Define the test statistic
The test statistic for testing above hypothesis is
$$ \begin{aligned} t=\dfrac{r}{\sqrt{1-r^2}}\sqrt{n-2} \end{aligned} $$
The test statistic $t$ follows Students’ $t$ distribution with $n-2$ degrees of freedom.
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.
- left-tailed alternative hypothesis: Find the $t$-critical value using
$$ \begin{aligned} P(t<-t_{\alpha,n-1}) & = \alpha. \end{aligned} $$ - right-tailed alternative hypothesis: $t_\alpha$.
$$ \begin{aligned} P(t>t_{\alpha, n-1}) & = \alpha. \end{aligned} $$ - two-tailed alternative hypothesis: $t_{\alpha/2}$.
$$ \begin{aligned} P(|t| > t_{\alpha/2,n-1}) &= \alpha. \end{aligned} $$
Step 5 Computation
Compute the test statistic under the null hypothesis $H_0$ using
$$ \begin{aligned} t_{obs} =\dfrac{r}{\sqrt{1-r^2}}\sqrt{n-2} \end{aligned} $$
Step 6 Decision (Traditional Approach)
Traditional approach is based on the critical value.
- For left-tailed alternative hypothesis: Reject $H_0$ if
$t_{obs}\leq -t_{\alpha,n-1}$. - right-tailed alternative hypothesis: Reject $H_0$ if
$t_{obs}\geq t_{\alpha,n-1}$. - two-tailed alternative hypothesis: Reject $H_0$ if
$|t_{obs}|\geq t_{\alpha/2, n-1}$.
OR
Step 6 Decision ($p$-value Approach)
It is based on the $p$-value.
| Alternative Hypothesis | Type of Hypothesis | $p$-value |
|---|---|---|
| $H_a: \rho<0$ | Left-tailed | $p$-value $= P(t\leq t_{obs})$ |
| $H_a: \rho>0$ | Right-tailed | $p$-value $= P(t\geq t_{obs})$ |
| $H_a: \rho\neq 0$ | Two-tailed | $p$-value $= 2P(t\geq t_{obs})$ |
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.
