91Ó°ÊÓ

Explanatory variables are the independent variables in a model that help explain changes in the dependent variable. For instance, when predicting the gas mileage of a car, the explanatory variables could include:

• Type of driving (city or highway) - represented as \( x_{1} \)
  
• Weight of the car - represented as \( x_{2} \)
  
• Tire pressure - represented as \( x_{3} \)

Each of these variables contributes information to predict the outcome variable (gas mileage, in this case). Explanatory variables can be continuous, like weight and tire pressure, or categorical, like driving type.

Indicator variables, also known as dummy variables, are used to represent categorical data numerically, typically as '0' or '1'. For example, the driving type (city or highway) is a categorical variable, which can be translated into an indicator variable:

• If driving in the city, \( x_{1} = 0 \)
  
• If driving on the highway, \( x_{1} = 1 \)

This numerical representation allows us to include categorical data in our regression models. By defining such indicator variables, we can manage and quantify the impact of different categories on the dependent variable.

When two explanatory variables interact, their combined effect on the dependent variable is not merely additive but multiplicative. An interaction term allows us to capture this combined effect. For instance, suppose we suspect that the weight of the car \( x_{2} \) and tire pressure \( x_{3} \) interact. We include an interaction term in the model:
\[ \beta_{4} x_{2} x_{3} \]Incorporating this term into our model, the equation becomes:
\[ y = \beta_{0} + \beta_{1} x_{1} + \beta_{2} x_{2} + \beta_{3} x_{3} + \beta_{4} x_{2} x_{3} + \text{\textgreek{e}} \]Where \( \beta_{4} \) measures the interaction effect between weight and tire pressure on gas mileage. Interaction terms are crucial for capturing the complexity of relationships between variables.

In an additive model, the effects of explanatory variables on the dependent variable are summed linearly. For our gas mileage prediction model, the additive model is:
\[ y = \beta_{0} + \beta_{1} x_{1} + \beta_{2} x_{2} + \beta_{3} x_{3} + \text{\textgreek{e}} \]
Here, \( y \) represents gas mileage, \( \beta_{0} \) is the intercept, and each \( \beta_{i} \) represents the coefficient for the corresponding explanatory variable \( x_{i} \). The additive model assumes that each variable independently contributes to predicting the result. By adding the coefficients together, it creates a clearer picture of the combined influence of all variables.

Linear terms refer to variables that enter the model in a linear fashion, meaning their impact on the dependent variable is proportional and constant. Our linear regression model:
\[ y = \beta_{0} + \beta_{1} x_{1} + \beta_{2} x_{2} + \beta_{3} x_{3} + \text{\textgreek{e}} \]
includes linear terms for each explanatory variable \( x_{i} \). Each term consists of a coefficient \( \beta_{i} \) and the variable itself. Linear terms are straightforward because any change in an explanatory variable leads to a direct change in the dependent variable, proportional to its coefficient. This makes it easier to interpret the influence of each variable on the outcome, facilitating clearer insights and predictions.