Multiple Regression
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CHAPTER OUTLINE
- 11.1 Data Analysis for Multiple Regression
- 11.2 Inference for Multiple Regression
- 11.3 Multiple Regression Model Building
Introduction
In Chapters 2 and 10, we studied methods for inference in the setting of a linear relationship between a response variable and a single explanatory variable . In this chapter, we look at situations in which several explanatory variables work together to explain or predict a single response variable.
- Nike investigates an athlete’s body temperature in relation to outside temperature, humidity, type of footwear, and type of apparel.
- Disney Media and Advertising Lab wants to describe the relationship between a viewer’s skin conductivity and different themed advertisements, the volume of the background music, whether it was viewed on an HD or standard-definition television, and the gender and age of the viewer.
We do this by building on the descriptive tools we learned in Chapter 2 and the basics of regression inference from Chapter 10. Many of these tools and ideas carry directly over to the multiple regression setting. For example, we continue to use scatterplots and correlation for pairs of variables. We also continue to use least-squares regression to obtain model parameter estimates.
The presence of several explanatory variables, however, which may assist or substitute for each other in predicting the response, leads to many new ideas. We start this chapter by exploring the use of a linear model with five variables to determine space allocation in a company. We then turn our attention to data analysis and inference in a multiple regression setting.
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EXAMPLE 11.1 A Space Model
Allocation of space or other resource within a business organization is often done using quantitative methods. Characteristics for a subunit of the organization are determined, and then a mathematical formula is used to decide the required needs.
A university has used this approach to determine office space needs in square feet (ft2) for each department.1 The formula allocates 210 ft2 for the department head (HEAD), 160 ft2 for each faculty member (FAC), 160 ft2 for each manager (MGR), 150 ft2 for each administrator and lecturer (LECT), 65 ft2 for each postdoctorate and graduate assistant (GRAD), and 120 ft2 for each clerical and service worker (CLSV). These allocations were not obtained through multiple linear regression but rather determined by a university committee using information on the numbers of each employee type and space availability in the buildings on campus.
The Chemistry Department in this university has 1 department head, 45.25 faculty, 15.50 managers, 41.52 lecturers, 411.88 graduate assistants, and 25.24 clerical and service workers. Note that fractions of people are possible in these calculations because individuals may have appointments in more than one department. For example, a person with an even split between two departments would be counted as 0.50 in each.
EXAMPLE 11.2 Office Space Needs for the Chemistry Department
Let’s calculate the office space needs for the Chemistry Department based on these personnel numbers. We start with 210 ft2 for the department head. We have 45.25 faculty, each needing 160 ft2. Therefore, the total office space needed for faculty is , which is 7240 ft2. We do the same type of calculation for each personnel category and then sum the results.
Here are the calculations in a table:
| Category | Number of employees |
Square footage per employee |
Employees × square footage |
|---|---|---|---|
| HEAD | 1.00 | 210 | 210.0 |
| FAC | 45.25 | 160 | 7,240.0 |
| MGR | 15.50 | 160 | 2,480.0 |
| LECT | 41.52 | 150 | 6,228.0 |
| GRAD | 411.88 | 65 | 26,772.2 |
| CLSV | 25.24 | 120 | 3,028.8 |
| Total | 45,959.0 |
The calculations that we just performed use a set of explanatory variables— HEAD, FAC, MGR, LECT, GRAD, and CLSV—to find the office space needs for the Chemistry Department. Given values of these variables for any other department in the university, we can perform the same calculations to find the office space needs. We organized our calculations for the Chemistry Department in the preceding table. Another way to organize calculations of this type is to give a formula.
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EXAMPLE 11.3 The Office Space Needs Formula
Let’s assume that each department has exactly one head. So the first term in our equation will be the space need for this position, 210 ft2. To this, we add the space needs for the faculty, 160 ft2 for each, or 160 FAC. Similarly, we add the number of square feet for each category of personnel times the number of employees in the category. The result is the office space needs predicted by the space model. Here is the formula:
The formula combines information from the explanatory variables and computes the office space needs for any department. This prediction generally will not match the actual space being used by a department. The difference between the value predicted by the model and the actual space being used is of interest to the people who assign space to departments.
EXAMPLE 11.4 Compare Predicted Space with Actual Space
The Chemistry Department currently uses 50,075 ft2 of space. On the other hand, the model predicts a space need of 45,959 ft2. The difference between these two quantities is a residual:
According to the university space needs model, the Chemistry Department has about 4116 ft2 more office space than it needs.
Because of this, the university director of space management is considering giving some of this excess space to a department that has actual space less than what the model predicts. Of course, the Chemistry Department does not think that it has excess space. In negotiations with the space management office, the department will explain that it needs all the current space and that its needs are not fully captured by the model.
Apply Your Knowledge
Question 11.1
11.1 Check the formula.
The table that appears before Example 11.3 shows that the predicted office space needed by the Chemistry Department is 45,959.0 ft2. Verify that the formula given in Example 11.3 gives the same predicted value.
11.1
45,959.
Question 11.2
11.2 Needs of the Department of Mathematics.
The Department of Mathematics has 1 department head, 57.5 faculty, 2 managers, 49.75 administrators and lecturers, 198.74 graduate assistants, and 10.64 clerical and service workers.
- Find the office needs for the Mathematics Department that are predicted by the model.
- The actual office space for this department is 27,326 ft2. Find the residual and explain what it means in a few sentences.
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These space allocation examples illustrate two key ideas that we need for multiple regression. First, we have several explanatory variables that are combined in a prediction equation. Second, residuals are the differences between the actual values and the predicted values. We now illustrate the techniques of multiple regression, including some new ideas, through a series of case studies. In all examples, we use software to do the calculations.