Chapter 14 Correlation and Regression

Usage: To check for a relationship between variables.

Requirements: Two or more variables that are interval/ratio.

Steps to calculating a correlation coefficient:

Write out null and alternative hypotheses.
Calculate r-value (correlation coefficient).
Determine critical t-value (yep!) for rejection of null hypothesis.
A. Use degrees of freedom (df) and alpha level.
Calculate t-value.
Compare your calculated t-value to the critical t-value.
A. If your calculated value is greater than the critical value, you reject the null hypothesis.
B. If your calculated value is less than the critical value, you fail to reject the null hypothesis.
Calculate effect size (\(r^2\)).
Estimate regression equation.
Test significance of regression equation.
A. Run an ANOVA.

14.1 Example Setup

\(\Large H_0: \rho = 0\)

\(\Large H_1: \rho \neq 0\)

\(\Large n = number\ of\ participants\)

\(\Large SS_{X} = \sum X^2 - \frac{(\sum X)^2}{n}\)

\(\Large SS_{Y} = \sum Y^2 - \frac{(\sum Y)^2}{n}\)

\(\Large SP = \sum XY - \frac{\sum X * \sum Y}{n}\)

\(\Large r = \frac{SP}{\sqrt{SS_X * SS_Y}}\)

\(\Large df = n - 2\)

\(\Large \alpha = .05\)

\(\Large CV = [check\ table]\)

\(\Large t = \frac{r}{\sqrt{\frac{(1 - r^2)}{(n - 2)}}}\)

If t is greater than the CV, reject the null hypothesis.
If t is less than the CV, fail to reject the null hypothesis.

\(\Large r^2 = r * r\)

Regression line equation: \(\hat Y = bX + a\)

\(\Large b = \frac{SP}{SS_X}\)

\(\Large a = \bar Y - (b*\bar X)\)

\(\Large SEoE = \sqrt{\frac{(1 - r^2)*SS_Y}{n - 2}}\)

\(\Large H_0: b = 0\)

\(\Large H_1: b \neq 0\)

\(\Large df_{regression} = 1\)

\(\Large df_{residual} = n - 2\)

\(\Large SS_{regression} = r^2*SS_Y\)

\(\Large SS_{residual} = (1 - r^2)*SS_Y\)

\(\Large MS_{regression} = \frac{SS_{regression}}{df_{regression}}\)

\(\Large MS_{residual} = \frac{SS_{residual}}{df_{residual}}\)

\(\Large F = \frac{MS_{regression}}{MS_{residual}}\)

\(\Large \alpha = .05\)

\(\Large CV = [check\ table]\)

Remember that a t-test is used when hypothesis testing correlation coefficients!

Remember that \(df = n - 2\) for tests of correlation coefficients!