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The Pearson correlation coefficient significance

The test of significance for Pearson product-moment correlation coefficient is used to verify the hypothesis determining the lack of linear correlation between an analysed features of a population and it is based on the Pearson's linear correlation coefficient calculated for the sample. The closer to 0 the value of coefficient $r_p$ is, the weaker dependence joins the analysed features.

Basic assumptions:

measurement on the interval scale,
normality of distribution of residuals or an analysed features in a population.

Hypotheses:

$\begin{array}{cl} \mathcal{H}_0: & R_p = 0, \\ \mathcal{H}_1: & R_p \ne 0. \end{array}$

The test statistic is defined by: $\begin{displaymath} t=\frac{r_p}{SE}, \end{displaymath}$

where $\displaystyle SE=\sqrt{\frac{1-r_p^2}{n-2}}$ .

The value of the test statistic can not be calculated when $r_p=1$ or $r_p=-1$ or when $n<3$ .

The test statistic has the t-Student distribution with $n-2$ degrees of freedom.

The p-value, designated on the basis of the test statistic, is compared with the significance level $\alpha$ :

$\begin{array}{ccl} $ if $ p \le \alpha & \Longrightarrow & $ reject $ \mathcal{H}_0 $ and accept $ \mathcal{H}_1, \\ $ if $ p > \alpha & \Longrightarrow & $ there is no reason to reject $ \mathcal{H}_0. \\ \end{array}$

EXAMPLE cont. (age-height.pqs file)

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The Pearson correlation coefficient significance

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