Question

Consider the dependent variable \(y=\) fuel efficiency of a car (mpg). a. Suppose that you want to incorporate size class of car, with four categories (subcompact, compact, midsize, and large), into a regression model that also includes \(x_{1}=\) age of car and \(x_{2}=\) engine size. Define the necessary indicator variables, and write out the complete model equation. b. Suppose that you want to incorporate interaction between age and size class. What additional predictors would be needed to accomplish this?

Short answer

a. The complete model equation incorporating size class via indicator variables is \(y = c + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{4}x_{4} + \beta_{5}x_{5} + \epsilon\). b. Additional predictors would be \(x_{6} = x_{1} * x_{3}\), \(x_{7} = x_{1} * x_{4}\), and \(x_{8} = x_{1} * x_{5}\) to incorporate interaction between age and size class.

Step-by-step solution

Title: Create Indicator Variables

Incorporating the variable 'size class of car' with four categories into the regression model will require defining indicator variables. Each category within the variable will have its own indicator variable where a particular car is assigned a '1' if it falls within that category and a '0' if it does not. Given that there are four categories ('subcompact', 'compact', 'midsize', and 'large'), we need three indicator variables by treating one category as a base case. Let's use 'large' as the base case. Define \(x_{3}\) as '1' if the car is 'subcompact', and '0' otherwise. Define \(x_{4}\) as '1' if the car is 'compact', and '0' otherwise. Define \(x_{5}\) as '1' if the car is 'midsize', and '0' otherwise.

Title: Write Out the Complete Model Equation

With the parameters \(x_{1}\) (age of car), \(x_{2}\) (engine size), and \(x_{3}, x_{4}, x_{5}\) (indicator variables for size class of car), a complete model equation can be written as \(y = c + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \beta_{4}x_{4} + \beta_{5}x_{5} + \epsilon\), where 'y' is the dependent variable(fuel efficiency of a car in mpg), c is a constant, 'x' represent the independent variables and '\(\beta\)' are coefficients for each independent variable, and \(\epsilon\) is the error term.

Title: Create Interaction Variables

To incorporate interaction between age and size class, we need to create interaction predictors for each non-base category. These new predictors will be the product of age (\(x_{1}\)) and the size class indicator variables (\(x_{3}, x_{4}, x_{5}\)). Therefore, interaction predictors would be \(x_{6} = x_{1} * x_{3}\) (age*subcompact), \(x_{7} = x_{1} * x_{4}\) (age*compact), and \(x_{8} = x_{1} * x_{5}\) (age*midsize).

Key concepts

Fuel Efficiency

Understanding the concept of fuel efficiency is critical when analyzing the performance of vehicles. Fuel efficiency, typically measured in miles per gallon (mpg), represents how far a car can travel on a specific amount of fuel. It's an important metric for consumers concerned with the cost of driving and the environmental impact of their car.

When it comes to analyzing factors affecting fuel efficiency using a regression model, several variables come into play. Age of the car (\(x_{1}\)) and engine size (\(x_{2}\)) are examples of quantitative variables that could predict changes in fuel efficiency; older cars or those with larger engines might be less efficient. By incorporating these factors into a regression analysis, we gain insight into how significant and in what way these aspects influence the efficiency of a vehicle.

Indicator Variables

In regression analysis, indicator variables are used to represent categorical data, allowing the inclusion of non-numeric data types into a model. These are binary variables, typically taking the value 1 if a condition is met, and 0 otherwise.

In the context of our exercise, the size class of a car — with categories such as 'subcompact', 'compact', 'midsize', and 'large' — can be incorporated into a regression model using indicator variables. For instance, if 'large' is chosen as the base category, then a car falling into this category would have all indicator variables (\(x_{3}\), \(x_{4}\), \(x_{5}\)) set to 0. The 'absence' of these variables indicates the base category.

By creating these binary variables, the model can now reflect the influence of a car's size class on its fuel efficiency, which is particularly important given that car size is often a strategic choice by manufacturers affecting both the vehicle's design and its performance.

Interaction Terms

When analyzing data, interaction terms are crucial for uncovering relationships between variables that are not simply additive. These terms arise when the effect of one predictor variable on the outcome depends on the level of another predictor.

In our fuel efficiency example, we might suspect that the relationship between a car's age and its fuel efficiency might differ depending on the car's size class. To explore this, we incorporate interaction terms in the model. These are created by multiplying the age of the car (\(x_{1}\)) with each of our size class indicators (\(x_{3}\), \(x_{4}\), \(x_{5}\)). The inclusion of these terms allows us to observe whether the impact of age on fuel efficiency is different for a subcompact car versus a midsize or compact car.

By including interaction terms, the model becomes more sophisticated and likely more accurate in predicting real-world behaviors, since it can now account for complex, interdependent effects among variables.