Introduction to Multiple and Logistic Regression
The principles of simple linear regression lay the foundation for more sophisticated regression methods used in a wide range of challenging settings. In this section, we explore multiple regression, which introduces the possibility of more than one predictor, and logistic regression, a technique for predicting categorical outcomes with two possible categories.
Multiple regression extends simple two-variable regression to the case that still has one response but many predictors (denoted x1, x2, x3, …). The method is motivated by scenarios where many variables may be simultaneously connected to an output.
We will consider eBay auctions of a video game called Mario Kart for the Nintendo Wii. The outcome variable of interest is the total price of an auction, which is the highest bid plus the shipping cost. We will try to determine how total price is related to each characteristic in an auction while simultaneously controlling for other variables. For instance, all other characteristics held constant, are longer auctions associated with higher or lower prices? And, on average, how much more do buyers tend to pay for additional Wii wheels (plastic steering wheels that attach to the Wii controller) in auctions? Multiple regression will help us answer these and other questions.
The data set mario_kart includes results from 141 auctions.[1] Four observations from this data set are shown in Table 1, and descriptions for each variable are shown in Table 2. Notice that the condition and stock photo variables are indicator variables. For instance, the cond_new variable takes value 1 if the game up for auction is new and 0 if it is used. Using indicator variables in place of category names allows for these variables to be directly used in regression. Multiple regression also allows for categorical variables with many levels, though we do not have any such variables in this analysis, and we save these details for a second or third course.
| Table 1. Four Observations from the mario-kart data set. | |||||
|---|---|---|---|---|---|
| price | cond_new | stock_photo | duration | wheels | |
| 1 | 51.55 | 1 | 1 | 3 | 1 |
| 2 | 37.04 | 0 | 1 | 7 | 1 |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| 140 | 38.76 | 0 | 0 | 7 | 0 |
| 141 | 54.51 | 1 | 1 | 1 | 2 |
| Table 2. Variables and their descriptions for the mario-kart data set. | |
|---|---|
| Variable | Description |
| price | final auction price plus shipping costs, in US dollars a coded two-level categorical variable, which takes value 1 when the game is new and 0 if the game is used |
| stock_photo | a coded two-level categorical variable, which takes value 1 if the primary photo used in the auction was a stock photo and 0 if the photo was unique to that auction |
| duration | the length of the auction, in days, taking values from 1 to 10 |
| wheels | the number of Wii wheels included with the auction (a Wii wheel is a plastic raci