Hiring A firm that operates a large, direct-to-consumer sales force would like to be able to put in place a system to monitor the progress of new agents. A key task for agents is to open new accounts; an account is a new customer to the business. The goal is to identify “superstar agents” as rapidly as possible, offer them incentives, and keep them with the company. To build such a system, the firm has been monitoring sales of new agents over the past two years. The response of interest is the profit to the firm (in dollars) of contracts sold by agents over their first year. Among the possible predictors of this performance is the number of new accounts developed by the agent during the first three months of work. Some of these agents were located in new offices, whereas others joined an existing office (see the column labeled Office).
(a) Plot the log of profit on the log of the number of accounts opened for both groups in one scatterplot. Use color-coding or distinct symbols to distinguish the groups. Does the coloring explain an unusual aspect of the “black and white” scatterplot? Does a simple regression that ignores the groups provide a reasonable summary?
(b) Add a dummy variable (coded as 1 for new offices and 0 for existing offices) and its interaction with Log Number of Accounts to the model. Does the fit of this model meet the conditions for the MRM? Comment on the consequences of any problem that you identify.
(c) Assuming that the model meets the conditions for the MRM, use the incremental
-test to assess the size of the change in
. (See the discussion of this test in Exercise 45.) Does the test agree with your visual impression? (The value of
for the model with dummy and interaction is 3, with 2 slopes added. You will need to fit the simple regression to get its
for comparison to the multiple regression.)
(d) Compare the conclusion of the incremental
-test to those of the tests of the coefficients of the dummy variable and interaction separately. Do these agree? Explain the similarity or difference.
(e) What do you think about locating new hires in new or existing offices? Would you recommend locating them in one or the other (assuming it could be done without disrupting the current placement procedures)?
Exercise 45
R&D Expenses This data file contains a variety of accounting and financial values that describe companies operating in the information and professional services sectors of the economy. One column gives the expenses on research and development (R&D), and another gives the total assets of the companies. Both of these columns are reported in millions of dollars. This data table adds data for professional services. To estimate regression models, we need to transform both expenses and assets to a log scale.
(a) Plot the log of R&D expenses on the log of assets for both sectors together in one scatterplot. Use color-coding or distinct symbols to distinguish the groups. Does it appear that the relationship is different in these two sectors or can you capture the association with a single simple regression?
A common question asked when fitting models to subsets is “Do the equations for the two groups differ from each other?” For example, does the equation for the information sector differ from the equation for professional services? We’ve been answering this question informally, using the
statistics for the slopes of the dummy variable and interaction. There’s just one small problem: We’re using two tests to answer one question. What’s the chance for a false-positive error? If you’ve got one question, better to use one test.
To see if there’s any difference, we can use a variation on the
-test for
. The idea is to test both slopes at once rather than separately. The method uses the change in the size of
. If the
of the model increases by a statistically significant amount when we add both the dummy variable and interaction to the model, then something changed and the model is different. The form of this incremental, or partial,
-test is
In this formula,
denotes the number of variables in the model with the extra features, including dummy variables and interactions.
full is the
for that model. As usual, a big value for this
-statistic is 4.
(b) Add a dummy variable (coded as 0 for information companies and 1 for those in professional services) and its interaction with Log Assets to the model. Does the fit of this model meet the conditions for the MRM? Comment on the consequences of any problem that you identify.
(c) Assuming that the model meets the conditions for the MRM, use the incremental
-test to assess the size of the change in
. Does the test agree with your visual impression? (The value of
for the model with dummy and interaction is 3, with 2 slopes added. You will need to fit the simple regression of Log R&D Expenses on Log Assets to get the
from this model.)
(d) Summarize the fit of the model that best captures what is happening in these two sectors.