Introduction
This sheet deals specifically with formulas for linear models and related bivariate analyses.
Material related to categorical data will be published elsewhere.
Bivariate Summations
\[\begin{eqnarray} S_{XY} &=& \sum x_iy_i - \frac{\sum x_i\sum y_i}{n}\\ S_{XX} &=& \sum x_i^2 - \frac{(\sum x_i)^2}{n}\\ S_{YY} &=& \sum y_i^2 - \frac{(\sum y_i)^2}{n}\\ \end{eqnarray}\]
Correlation
Pearson’s correlation coefficient
\[\begin{eqnarray} r = \frac{S_{XY}}{\sqrt{S_{XX} \times S_{YY}}} \end{eqnarray}\]
Linear Regression Estimates
Slope Estimate \[\begin{eqnarray} b_1 = \frac{S_{XY}}{S_{XX}} \end{eqnarray}\]
Intercept Estimate \[\begin{eqnarray} b_0 = \bar{y} -b_1\bar{x} \end{eqnarray}\]
Standard error of the Slope \[\begin{eqnarray*} S.E.(b1) = \sqrt{\frac{s^2}{S_{XX}}} \end{eqnarray*}\]
where \(s^2 = \frac{SSE}{n-2}\)
and SSE \(= S_{YY} - b_1S_{XY}\)