Section 13.3 Summary

1. Multiple regression describes the linear relationship between one response variable $y$ and more than one predictor variable, $x_{1 , }{x}_{2, }{x}_{3,}\dots$. The multiple regression equation is an extension of the regression equation: $\hat{y} = b_{0 } + b_1{x}_{1 }+ b_2{x}_{2 }+ \dots + b_{k }{x}_{k }$ where $k$ represents the number of $x$ variables in the equation, and $b_{1 , }{b}_{2, }{b}_{3, }\cdots b_k$ represent the multiple regression coefficients.
2. The multiple coefficient of determination $R^2$ represents the proportion of the variability in the response $y$ that is explained by the multiple regression equation. The adjusted coefficient of determination $R_{\text{adj}}^2$ adjusts the value of $R^2$ as a penalty for having too many unhelpful $x$ variables in the equation.
3. The multiple regression model is an extension of the regression model from Section 13.1. The population multiple regression equation is $y = {\beta }_{0 } + {\beta }_{1 }{x}_{1 }+ {\beta }_{2 }{x}_{2 }+ \cdots + {\beta }_{k }{x}_{k }+ \epsilon$. The $F$ test is performed to assess the significance of the overall model.
4. To determine whether a particular $x$ variable has a significant linear relationship with the response variable $y$, we perform the $t$ test for the significance of that $x$ variable. One may perform as many such $t$ tests as there are $x$ variables in the model, which is $k$ assuming the overall $F$ test is significant.
5. Dummy variables are 0/1 variables that allow, via recoding, categorical variables to be included in the multiple regression model.
6. The Strategy for Building a Multiple Regression Model brings together all we have learned about multiple regression modeling.