Multiple Regression

Suppose a regional transportation manager wants to understand how changes in the population and changes in prices affect the number of riders using the bus system. To help understand this, the following variables are measured:

• Weekly Riders [WR], which measures the average number of tickets sold each week, in thousands.
• Price Per Ride [PPR], which records the cost of a single, one-way bus ticket in dollars.
• Population [Pop], which records the population of the metropolitan region being served, in thousands of people.
• Income [Inc], which records the average disposable income of individuals in the metro area.
• Parking Rate [PR], which is the average cost for a day of parking in a garage downtown.

Check the boxes next to each explanatory variable to add or remove them to the linear regression model. As you add or remove variables, the four charts will update to reflect the new model. Use the questions below to guide your exploration.

Practice Questions:

1. Start by looking at the single regression equations and their residual plots. For which variable(s) does a linear relationship appear to be good choice for the regression?
   View answer

   Pop is the only variable that does not have a discernible U-shaped pattern, so it is the variable that best fits with a linear regression.
2. Leave Pop as the first explanatory variable, and observe the partial regression when Inc is added to create a multiple regression. What is the general relationship between Pop and Inc? How might you explain this?
   View answer

   When Inc is added to a multiple regression with Pop, we see that as average income goes up, we expect weekly riders to decrease. This could be because for a given population size, when average income is higher, people are more likely to buy cars and less likely to take the bus.
3. Experiment with different combinations of variables, and see if you can find a variable that has a positive relationship with Weekly Riders when combined with some variables, and an inverse relationship with Weekly Riders when combined with others.
   View answer

   One example of this is the parking rate (PR) variable. When PPR is the explanatory variable and PR is the added variable, we see that PR has a positive relationship with weekly riders. However, when Pop is the explanatory variable and PR is added, the directional effect of PR on weekly riders is reversed (the slope of the added variable plot is negative).