BIOS14: Multiple Regression（多元回归） using R

NOTES

1 Multiple reegression model

1.1 multiple regression equation

$$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_4+...+{\beta }_k{x}_k+\epsilon$$
Assumption:
A1: $E(\epsilon )=0$
A2: The variance of $\epsilon$ is constant, $var(\epsilon )={\sigma }^2$
A3: The $\epsilon$'s are independent.
A4: The values of xiare not random and are not exact linear functions of the other explanatory variables
A5: The $\epsilon$'s are normally distributed

1.2 estimated multiple regression equation

$$\hat{y}={\hat{\beta }}_0+{\hat{\beta }}_1{x}_1+{\hat{\beta }}_2{x}_2+{\hat{\beta }}_3{x}_3+{\hat{\beta }}_4{x}_4+...+{\hat{\beta }}_k{x}_k$$

1.3 method of least squares

$$Q=\sum (y_i-{\hat{y}}_i{)}^2=\sum (y_i-{\hat{\beta }}_0-{\hat{\beta }}_1{x}_1-...-{\hat{\beta }}_k{x}_k{)}^2$$
minimize:
$$\{\begin{array}{l}\frac{\partial Q}{\partial {\beta }_0}{|}_{{\beta }_0={\hat{\beta }}_0}=0\\ \frac{\partial Q}{\partial {\beta }_i}{|}_{{\beta }_i={\hat{\beta }}_i}=0,i=1,2,...,k\end{array}$$