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 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}

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.