Correlation and Regression (Further)

This lesson extends correlation and regression to include Spearman's rank correlation coefficient, hypothesis testing for correlation, and deeper analysis of the product-moment correlation coefficient (PMCC).

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Recap: Pearson's Product-Moment Correlation Coefficient

The PMCC $r$r measures the strength and direction of a linear relationship between two variables:

$r = \frac{S_{xy}}{\sqrt{S_{xx} S_{yy}}}$r=Sxx​Syy​​Sxy​​

where $S_{xy} = \sum(x_i - \bar{x})(y_i - \bar{y})$Sxy​=∑(xi​−xˉ)(yi​−yˉ​), $S_{xx} = \sum(x_i - \bar{x})^2$Sxx​=∑(xi​−xˉ)2, $S_{yy} = \sum(y_i - \bar{y})^2$Syy​=∑(yi​−yˉ​)2.

Value of $r$r	Interpretation
$r = 1$r=1	Perfect positive linear correlation
$r = -1$r=−1	Perfect negative linear correlation
$r = 0$r=0	No linear correlation

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Hypothesis Test for the PMCC

To test whether there is a significant linear correlation in the population:

$H_0: \rho = 0$H0​:ρ=0 (no linear correlation in the population) $H_1: \rho \neq 0$H1​:ρ=0 (two-tailed) or $\rho > 0$ρ>0 / $\rho < 0$ρ<0 (one-tailed)

Compare the sample $r$r with the critical value from the PMCC table for $n$n data points at the chosen significance level.

If $|r| > \text{critical value}$∣r∣>critical value, reject $H_0$H0​.

Worked Example

For $n = 10$n=10 pairs, $r = 0.65$r=0.65. Test at 5% (two-tailed).