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Chapter 14: Problem 13

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 dummy 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
The dummy variables for size class (subcompact, compact, midsize, large) would be \(D1, D2, D3\) respectively. The complete model equation would be: \(y = 尾0 + 尾1x_{1} + 尾2x_{2} + 尾3D1 + 尾4D2 + 尾5D3 + 蔚\). To incorporate interactions between age and size class, the additional predictors will be \(x_{1}*D1, x_{1}*D2, x_{1}*D3\). With these variables, the fully expanded regression model becomes: \(y = 尾0 + 尾1x_{1} + 尾2x_{2} + 尾3D1 + 尾4D2 + 尾5D3 + 尾6(x_{1}*D1) + 尾7(x_{1}*D2) + 尾8(x_{1}*D3) + 蔚\).

Step by step solution

01

Define Dummy Variables

Define dummy variables for the categorical variable 'size class of the car'. Since there are four categories: subcompact, compact, midsize, large. Let's assign the following dummy variables: Let \(D1\) represent subcompact, \(D2\) represent compact, \(D3\) represent midsize. If \(D1, D2, D3\) are all 0, then it's large size.
02

Write Out the Complete Model Equation Using Dummy Variables

The regression model that includes the variable age of car, engine size and size class of car is written as:\(y = 尾0 + 尾1x_{1} + 尾2x_{2} + 尾3D1 + 尾4D2 + 尾5D3 + 蔚\) where \(尾0\) is the intercept, \(尾1\) is the coefficient for age of car, \(尾2\) is the coefficient for engine size, \(尾3\), \(尾4\), \(尾5\) are coefficients for the respective size class.
03

Incorporate Interaction Between Age and Size Class

To incorporate interaction between age and size class, additional predictors are needed. These are the product of age ( \(x_{1}\) ) and each of the dummy variables:The interaction variables would be \(x_{1}*D1\), \(x_{1}*D2\), \(x_{1}*D3\).The resulting model is:\(y = 尾0 + 尾1x_{1} + 尾2x_{2} + 尾3D1 + 尾4D2 + 尾5D3 + 尾6(x_{1}*D1) + 尾7(x_{1}*D2) + 尾8(x_{1}*D3) + 蔚\)where \(尾6\), \(尾7\), \(尾8\) are coefficients for the respective interaction terms.

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Key Concepts

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

Fuel Efficiency
Fuel efficiency is a critical measure for any vehicle, indicating the distance a car can travel per unit of fuel, commonly miles per gallon (mpg) in the U.S. When analyzing factors that influence fuel efficiency, regression models are highly useful. They allow us to quantify the relationship between several variables, like a car's age and engine size, and its fuel efficiency.

Understanding how different characteristics affect fuel efficiency helps manufacturers and consumers make informed decisions. For instance, a newer, smaller engine might be associated with higher fuel efficiency, which could influence both design choices and consumer preferences. By using regression analysis, we can predict fuel efficiency based on a set of vehicle features, providing valuable insights into performance and environmental impact.
Categorical Variables in Regression
Categorical variables represent types or categories of data, such as the size class of a car. These variables can take on a limited, fixed number of possible values, which don't have a natural numeric representation. To include such variables in a regression model, we transform them into a series of dummy variables.

Dummy variables are binary (\(0\text{ or }1\)) indicators, representing the presence or absence of a category. In our example, we create dummy variables for each size class of car (except for the reference category). The introduction of dummy variables allows the regression model to distinguish between different car sizes and assess their individual impact on fuel efficiency. Each dummy coefficient tells us the difference in fuel efficiency compared to the reference category, while controlling for other factors in the model.
Interaction Terms in Regression
Interaction terms in regression are crucial when the effect of one variable on the dependent variable depends on the level of another variable. By including interaction terms, we can capture the combined effect of two variables working together. In our car example, this means examining how the size class's impact on fuel efficiency changes with the age of the car.

To create interaction terms, we multiply the dummy variables by the continuous variable they interact with. Adding these terms to our regression model allows for differential slopes, suggesting that the relationship between age and fuel efficiency can vary across different car sizes. These terms enhance the model's flexibility and give us a more nuanced understanding of how combined factors influence the dependent variable, like fuel efficiency in various car types.

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Most popular questions from this chapter

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)

For the multiple regression model in Exercise \(14.4\), the value of \(R^{2}\) was \(.06\) and the adjusted \(R^{2}\) was \(.06 .\) The model was based on a data set with 1136 observations. Perform a model utility test for this regression.

The following statement appeared in the article "Dimensions of Adjustment Among College Women" (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 $$ \begin{aligned} y &=\text { current GPA } \\ x_{1} &=\text { academic adjustment score } \\ x_{2} &=\text { race (with white }=0 \text { , other }=1) \end{aligned} $$ What multiple regression model is suggested by the statement? Did you include an interaction term in the model? Why or why not?

The article "Impacts of On-Campus and Off-Campus Work on First-Year Cognitive Outcomes" (Journal of College Student Development \([1994]: 364-370\) ) reported on a study in which \(y=\) spring math comprehension score was regressed against \(x_{1}=\) previous fall test score, \(x_{2}=\) previous fall academic motivation, \(x_{3}=\) age, \(x_{4}=\) number of credit hours, \(x_{5}=\) residence \((1\) if on campus, 0 otherwise), \(x_{6}=\) hours worked on campus, and \(x_{7}=\) hours worked off campus. The sample size was \(n=210\), and \(R^{2}=.543\). Test to see whether there is a useful linear relationship between \(y\) and at least one of the predictors.

This exercise requires the use of a computer package. The authors of the article "Absolute Versus per Unit Body Length Speed of Prey as an Estimator of Vulnerability to Predation" (Animal Behaviour [1999]: \(347-\) 352) found that the speed of a prey (twips/s) and the length of a prey (twips \(\times 100\) ) are good predictors of the time (s) required to catch the prey. (A twip is a measure of distance used by programmers.) Data were collected in an experiment where subjects were asked to "catch" an animal of prey moving across his or her computer screen by clicking on it with the mouse. The investigators varied the length of the prey and the speed with which the prey moved across the screen. The following data are consistent with summary values and a graph given in the article. Each value represents the average catch time over all subjects. The order of the various speed-length combinations was randomized for each subject. $$ \begin{array}{ccc} \begin{array}{c} \text { Prey } \\ \text { Length } \end{array} & \begin{array}{l} \text { Prey } \\ \text { Speed } \end{array} & \begin{array}{l} \text { Catch } \\ \text { Time } \end{array} \\ \hline 7 & 20 & 1.10 \\ 6 & 20 & 1.20 \\ 5 & 20 & 1.23 \\ 4 & 20 & 1.40 \\ 3 & 20 & 1.50 \\ 3 & 40 & 1.40 \\ 4 & 40 & 1.36 \\ 6 & 40 & 1.30 \\ 7 & 40 & 1.28 \\ 7 & 80 & 1.40 \\ 6 & 60 & 1.38 \\ 5 & 80 & 1.40 \\ 7 & 100 & 1.43 \\ 6 & 100 & 1.43 \\ 7 & 120 & 1.70 \\ 5 & 80 & 1.50 \\ 3 & 80 & 1.40 \\ 6 & 100 & 1.50 \\ 3 & 120 & 1.90 \\ & & \\ \hline \end{array} $$ a. Fit a multiple regression model for predicting catch time using prey length and speed as predictors. b. Predict the catch time for an animal of prey whose length is 6 and whose speed is 50 . c. Is the multiple regression model useful for predicting catch time? Test the relevant hypotheses using \(\alpha=.05\). d. The authors of the article suggest that a simple linear regression model with the single predictor \(x=\frac{\text { length }}{\text { speed }}\) might be a better model for predicting catch time. Calculate the \(x\) values and use them to fit this linear regression model. e. Which of the two models considered (the multiple regression model from Part (a) or the simple linear regression model from Part (d)) would you recommend for predicting catch time? Justify your choice.
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