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Chapter 14: Problem 15

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

Expert verified
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

01

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.
02

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.
03

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).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 鈥� with categories such as 'subcompact', 'compact', 'midsize', and 'large' 鈥� 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.

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Most popular questions from this chapter

The relationship between yield of maize, date of planting, and planting density was investigated in the article 鈥淒evelopment of a Model for Use in Maize Replant Decisions鈥� (Agronomy Journal [1980]: \(459-464)\). Let $$ \begin{aligned} y &=\text { percent maize yield } \\ x_{1} &=\text { planting date }(\text { days after } \text { April } 20) \\ x_{2} &=\text { planting density }(10,000 \text { plants } / \mathrm{ha}) \end{aligned} $$ The regression model with both quadratic terms \((y=\alpha\) $$ +\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\beta_{4} x_{4}+e \text { where } x_{3}=x_{1}^{2} $$ and \(x_{4}=x_{2}^{2}\) ) provides a good description of the relationship between \(y\) and the independent variables. a. If \(\alpha=21.09, \beta_{1}=.653, \beta_{2}=.0022, \beta_{3}=2.0206\), and \(\beta_{4}=0.4\), what is the population regression function? b. Use the regression function in Part (a) to determine the mean yield for a plot planted on May 6 with a density of 41,180 plants/ha. c. Would the mean yield be higher for a planting date of May 6 or May 22 (for the same density)? d. Is it legitimate to interpret \(\beta_{1}=.653\) as the average change in yield when planting date increases by one day and the values of the other three predictors are held fixed? Why or why not?

The following statement appeared in the article 鈥淒imensions of Adjustment Among College Women鈥� (Journal of College Student Development [1998]: 364): Regression analyses indicated that academic adjustment and race made independent contributions to academic achievement, as measured by current GPA. Suppose \(\begin{aligned} y &=\text { current GPA } \\ x_{1} &=\text { academic adjustment score } \\ x_{2} &=\text { race }(\text { with white }=0, \text { other }=1) \end{aligned}\) What multiple regression model is suggested by the statement? Did you include an interaction term in the model? Why or why not?

A manufacturer of wood stoves collected data on \(y=\) particulate matter concentration and \(x_{1}=\) flue temperature for three different air intake settings (low, medium, and high). a. Write a model equation that includes indicator variables to incorporate intake setting, and interpret each of the \(\beta\) coefficients. b. What additional predictors would be needed to incorporate interaction between temperature and intake setting?

Consider a regression analysis with three independent variables \(x_{1}, x_{2}\), and \(x_{3}\). Give the equation for the following regression models: a. The model that includes as predictors all independent variables but no quadratic or interaction terms; b. The model that includes as predictors all independent variables and all quadratic terms; c. All models that include as predictors all independent variables, no quadratic terms, and exactly one interaction term; d. The model that includes as predictors all independent variables, all quadratic terms, and all interaction terms (the full quadratic model).

When coastal power stations take in large quantities of cooling water, it is inevitable that a number of fish are drawn in with the water. Various methods have been designed to screen out the fish. The article 鈥淢ultiple Regression Analysis for Forecasting Critical Fish Influxes at Power Station Intakes" (Journal of Applied Ecology [1983]: 33-42) examined intake fish catch at an English power plant and several other variables thought to affect fish intake: \(\begin{aligned} y &=\text { fish intake (number of fish) } \\ x_{1} &=\text { water temperature }\left({ }^{\circ} \mathrm{C}\right) \\ x_{2} &=\text { number of pumps running } \\ x_{3} &=\text { sea state }(\text { values } 0,1,2, \text { or } 3) \\ x_{4} &=\text { speed }(\mathrm{knots}) \end{aligned}\) Part of the data given in the article were used to obtain the estimated regression equation $$ \hat{y}=92-2.18 x_{1}-19.20 x_{2}-9.38 x_{3}+2.32 x_{4} $$ (based on \(n=26\) ). SSRegr \(=1486.9\) and SSResid = 2230.2 were also calculated. a. Interpret the values of \(b_{1}\) and \(b_{4}\) b. What proportion of observed variation in fish intake can be explained by the model relationship? c. Estimate the value of \(\sigma\). d. Calculate adjusted \(R^{2} .\) How does it compare to \(R^{2}\) itself?
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