7
Advanced
Regression Analysis
Business Analytics, 1e
By Sanjiv Jaggia, Alison Kelly, Kevin Lertwachara, and Leida Chen
8/17/2020
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7-1
Chapter 7 Learning Objectives (LOs)
LO 7.1 Estimate and interpret regression models
with interaction variables.
LO 7.2 Estimate and interpret nonlinear
regression models.
LO 7.3 Estimate and interpret linear probability
and logistic regression models.
LO 7.4 Use cross-validation techniques to
evaluate regression models.
7-2
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7-2
Use Gender_Gap
7-3
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7-3
Introductory Case: Gender Gap in
Manager Salaries
•
Is there a gender gap in salaries for project managers? Data on 200 managers
in small- to middle-sized firms in Boston:
–
–
–
–
–
Salary (in $1,000’s)
The number of employees at the firm (Size, number of employees)
The manager’s years of experience (Experience, in years)
Whether or not the manager is a female (Female equals 1 if the manager is female, 0 otherwise)
Whether or not the manager has a graduate degree (Grad equals 1 if the manager has a
graduate degree, 0 otherwise)
1. Analyze the determinants of a project manager’s salary.
2. Estimate and interpret a regression model with relevant interaction variables.
3. Determine whether there is evidence of a gender gap in salaries.
7-4
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7-4
7.1: Regression Models with Interaction
Variables (1/15)
•
Recall the sample regression equation 𝑦ො = 𝑏0 + 𝑏1 𝑥1 + 𝑏2 𝑥2 + ⋯ + 𝑏𝑘 𝑥𝑘 .
–
•
•
•
•
–
𝑏𝑗 measures the change in the predicted value of the response given in a unit increase 𝑥𝑗 , holding
all other predictor variables constant.
𝑏𝑗 is the partial (or marginal) effect of 𝑥𝑗 on 𝑦ො (the partial derivative).
–
This partial effect does not depend on other predictor variables.
The interaction effect in a regression model occurs when the partial effect of a
predictor variable on the response depends on the value of another predictor
variable.
Example: An additional bedroom results in a higher increase in house prices for
larger houses than smaller houses. Here, the effect of an additional bedroom
depends on house size.
An interaction variable is the product of predictor variables.
Three types of interaction variables:
–
–
–
The interaction between two dummy variables
The interaction between a dummy variable and a numerical variable
The interaction between two numerical variables
7-5
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7-5
7.1: Regression Models with Interaction
Variables (2/15)
• Consider a regression model with two dummy variables 𝑑1 and
𝑑2 , and an interaction variable 𝑑1 𝑑2 .
𝑦ො = 𝑏0 + 𝑏1 𝑑1 + 𝑏2 𝑑2 + 𝑏3 𝑑1 𝑑2
• The partial effect of 𝑑1 on 𝑦ො is 𝑏1 + 𝑏3 𝑑2, this depends on 𝑑2 .
– 𝑑2 = 0: the partial effect of 𝑑1 on 𝑦ො is 𝑏1
– 𝑑2 = 1: the partial effect of 𝑑1 on 𝑦ො is 𝑏1 + 𝑏3
• The partial effect of 𝑑2 on 𝑦ො is 𝑏2 + 𝑏3 𝑑1, this depends on 𝑑1 .
– 𝑑1 = 0: the partial effect of 𝑑2 on 𝑦ො is 𝑏2
– 𝑑1 = 1: the partial effect of 𝑑2 on 𝑦ො is 𝑏2 + 𝑏3
• Tests of significance can be conducted on the interaction
variables.
7-6
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7-6
Use Salary_MIS
7-7
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7-7
7.1: Regression Models with Interaction
Variables (3/15)
•
Example: Data were collected on the starting salary of business graduates (Salary in
$1,000s) along with their cumulative GPA, whether they have an MIS concentration (MIS
= 1 if yes, 0 otherwise), and whether they have a statistics minor (Statistics = 1 if yes, 0
otherwise).
a.
Estimate and interpret the effect of GPA, MIS, and Statistics on Salary. Predict the
salary of a business graduate with and without a MIS concentration and a statistics
minor. Use a GPA of 3.5 for making predictions.
Extend the model from part a to include the interaction between MIS and Statistics.
Predict the salary of a business graduate with and without a MIS concentration and a
statistics minor. Use a GPA of 3.5 for making predictions.
b.
7-8
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7-8
7.1: Regression Models with Interaction
Variables (4/15)
7-9
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7-9
7.1: Regression Models with Interaction
Variables (5/15)
෣ = 44.0073 + 6.6227𝐺𝑃𝐴 + 6.6071𝑀𝐼𝑆 + 6.7309𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠
• 𝑆𝑎𝑙𝑎𝑟𝑦
• All predictor variables exert a positive and significant influence on the
starting salary.
• The MIS concentration and a statistics minor are predicted to fetch an
additional starting salary of $6,607 and $6,731.
• No MIS concentration and no statistics minor
– 𝐺𝑃𝐴 = 3.5, 𝑀𝐼𝑆 = 0, 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠 = 0
෣ = 44.0073 + 6.6227 ∗ 3.5 + 6.6071 ∗ 0 + 6.7309 ∗ 0
– 𝑆𝑎𝑙𝑎𝑟𝑦
– The predicted salary is $67,187.
• MIS concentration and statistics minor
– 𝐺𝑃𝐴 = 3.5, 𝑀𝐼𝑆 = 1, 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠 = 1
෣ = 44.0073 + 6.6227 ∗ 3.5 + 6.6071 ∗ 1 + 6.7309 ∗ 1
– 𝑆𝑎𝑙𝑎𝑟𝑦
– The predicted salary is $80,525.
7-10
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7-10
7.1: Regression Models with Interaction
Variables (6/15)
•
•
•
•
෣ = 44.0993 + 6.7109𝐺𝑃𝐴 + 5.3250𝑀𝐼𝑆 + 5.5350𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠 +
𝑆𝑎𝑙𝑎𝑟𝑦
𝟑. 𝟒𝟗𝟏𝟓(𝑴𝑰𝑺 ∗ 𝑺𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄𝒔)
The interaction is positive and significant. The coefficient indicates that
graduates with an MIS concentration and Statistics minor earn $3,492 more
than graduates with an MIS concentration or Statistics minor.
This model has a higher adjusted 𝑅2 and provides a better fit.
No MIS concentration and no statistics minor
– 𝐺𝑃𝐴 = 3.5, 𝑀𝐼𝑆 = 0, 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠 = 0, 𝑀𝐼𝑆 ∗ 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠 = 0
–
෣ = 44.0993 + 6.7109 ∗ 3.5 + 5.3250 ∗ 0 + 5.5350 ∗ 0 + 3.4915(0 ∗ 0)
𝑆𝑎𝑙𝑎𝑟𝑦
– The predicted salary is $67,588.
• MIS concentration and statistics minor
– 𝐺𝑃𝐴 = 3.5, 𝑀𝐼𝑆 = 1, 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠 = 1, 𝑀𝐼𝑆 ∗ 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠 = 1
෣ = 44.0993 + 6.7109 ∗ 3.5 + 5.3250 ∗ 0 + 5.5350 ∗ 0 + 3.4915(1 ∗ 1)
– 𝑆𝑎𝑙𝑎𝑟𝑦
– The predicted salary is $81,939.
7-11
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7-11
7.1: Regression Models with Interaction
Variables (7/15)
• Consider a regression model with a numerical variable 𝑥, a
dummy variable 𝑑, and an interaction variable 𝑥𝑑.
𝑦ො = 𝑏0 + 𝑏1 𝑥 + 𝑏2 𝑑 + 𝑏3 𝑥𝑑
• The partial effect of 𝑥 on 𝑦ො is 𝑏1 + 𝑏3 𝑑, this depends on 𝑑.
– 𝑑 = 0: the partial effect of 𝑥 on 𝑦ො is 𝑏1
– 𝑑 = 1: the partial effect of 𝑥 on 𝑦ො is 𝑏1 + 𝑏3
• The partial effect of 𝑑 on 𝑦ො is 𝑏2 + 𝑏3 𝑥, this depends on 𝑥.
– Difficult to interpret because 𝑥 is numerical
– Common to interpret this partial effect at the sample mean 𝑥ҧ
– At 𝑥,ҧ the partial effect of 𝑑 on 𝑦ො is 𝑏2 + 𝑏3 𝑥ҧ
– 𝑏3 > 0: the partial effect on 𝑦ො will be greater (smaller) at values higher
(lower) than 𝑥ҧ
7-12
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7-12
Use BP_Race
7-13
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7-13
7.1: Regression Models with Interaction
Variables (8/15)
•
Example: A public policy researcher in Atlanta surveyed 150 adult men in
the 55–60 age group. Data were collected on their systolic pressure,
weight (in pounds), and race (Black = 1 for African American, 0 otherwise).
a.
Estimate and interpret the effect of Weight and Black on systolic pressure.
Predict the systolic pressure of black and nonblack adult men with a
weight of 180 pounds.
Extend the model in part a to include the interaction between Weight and
Black. Predict the systolic pressure of black and nonblack adult men with
a weight of 180 pounds.
b.
7-14
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7-14
7.1: Regression Models with Interaction
Variables (9/15)
7-15
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7-15
7.1: Regression Models with Interaction
Variables (10/15)
෣ = 80.2085 + 0.3901𝑊𝑒𝑖𝑔ℎ𝑡 + 6.9082𝐵𝑙𝑎𝑐𝑘
a. 𝑆𝑦𝑠𝑡𝑜𝑙𝑖𝑐
– 𝑊𝑒𝑖𝑔ℎ𝑡 = 180, 𝐵𝑙𝑎𝑐𝑘 = 1
෣ = 80.2085 + 0.3901 ∗ 180 + 6.9082 ∗ 1
• 𝑆𝑦𝑠𝑡𝑜𝑙𝑖𝑐
• The predicted systolic pressure is 157.
– 𝑊𝑒𝑖𝑔ℎ𝑡 = 180, 𝐵𝑙𝑎𝑐𝑘 = 0
෣ = 80.2085 + 0.3901 ∗ 180 + 6.9082 ∗ 0
• 𝑆𝑦𝑠𝑡𝑜𝑙𝑖𝑐
• The predicted systolic pressure is 150.
෣ = 70.8312 + 0.4362𝑊𝑒𝑖𝑔ℎ𝑡 + 30.2482𝐵𝑙𝑎𝑐𝑘 − 0.1118𝑊𝑒𝑖𝑔ℎ𝑡 ∗ 𝐵𝑙𝑎𝑐𝑘
b. 𝑆𝑦𝑠𝑡𝑜𝑙𝑖𝑐
– 𝑊𝑒𝑖𝑔ℎ𝑡 = 180, 𝐵𝑙𝑎𝑐𝑘 = 1, 𝑊𝑒𝑖𝑔ℎ𝑡 ∗ 𝐵𝑙𝑎𝑐𝑘 = 180
෣ = 70.8312 + 0.4362 ∗ 180 + 30.2482 ∗ 1 − 0.1118 ∗ 180 ∗ 1
• 𝑆𝑦𝑠𝑡𝑜𝑙𝑖𝑐
• The predicted systolic pressure is 159.
– 𝑊𝑒𝑖𝑔ℎ𝑡 = 180, 𝐵𝑙𝑎𝑐𝑘 = 0, 𝑊𝑒𝑖𝑔ℎ𝑡 ∗ 𝐵𝑙𝑎𝑐𝑘 = 0
෣ = 70.8312 + 0.4362 ∗ 180 + 30.2482 ∗ 0 − 0.1118 ∗ 180 ∗0
• 𝑆𝑦𝑠𝑡𝑜𝑙𝑖𝑐
• The predicted systolic pressure is 149.
7-16
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7-16
7.1: Regression Models with Interaction
Variables (11/15)
• Consider a regression model with two numerical variables, 𝑥1
and 𝑥2 , and an interaction variable 𝑥1 𝑥2 .
𝑦ො = 𝑏0 + 𝑏1 𝑥1 + 𝑏2 𝑥2 + 𝑏3 𝑥1 𝑥2
• The partial effect of 𝑥1 on 𝑦ො is 𝑏1 + 𝑏3 𝑥2 , this depends on 𝑥2 .
• The partial effect of 𝑥2 on 𝑦ො is 𝑏2 + 𝑏3 𝑥1 , this depends on 𝑥1 .
• The partial effects of both variables are difficult to interpret.
• Consider the partial effects at the sample means 𝑥ҧ1 and 𝑥ҧ2 .
– At 𝑥ҧ1 , the partial effect of 𝑥2 on 𝑦ො is 𝑏2 + 𝑏3 𝑥ҧ1 .
– 𝑏3 > 0: the partial effect on 𝑦ො will be greater (smaller) at values higher
(lower) than 𝑥ҧ1
7-17
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7-17
Use Marketing_MSA
7-18
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7-18
7.1: Regression Models with Interaction
Variables (12/15)
•
Example: Consider the data collected on the number of applications received
(Applicants), marketing expense (Marketing, in $1,000s), and the percentage
employed within three months (Employed).
a. Estimate and interpret the effect of Marketing and Employed on the number of
applications received. For a given marketing expense of $80,000, predict the number of
applications received if 50% of the graduates were employed within three months.
Repeat the analysis with 80% employed within three months.
b. Extend the model in part a to include the interaction between Marketing and Employed.
For a given marketing expense of $80,000, predict the number of applications received if
50% of the graduates were employed within three months. Repeat the analysis with 80%
employed within three months.
7-19
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7-19
7.1: Regression Models with Interaction
Variables (13/15)
7-20
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7-20
7.1: Regression Models with Interaction
Variables (14/15)
෣
𝐴𝑝𝑝𝑙𝑖𝑐𝑎𝑛𝑡𝑠
= −49.5490 + 0.3550𝑀𝑎𝑟𝑘𝑒𝑡𝑖𝑛𝑔 + 1.1049𝐸𝑚𝑝𝑙𝑜𝑦𝑒𝑑
a.
–
–
–
Given Employment, a $10,000 increase in the marketing expense is predicted to
bring in 3.55 more applicants.
Given Marketing, a 10% increase in Employed brings in 10.149 more applicants.
𝑀𝑎𝑟𝑘𝑒𝑡𝑖𝑛𝑔 = 90, 𝐸𝑚𝑝𝑙𝑜𝑦𝑒𝑑 = 50
෣
• 𝐴𝑝𝑝𝑙𝑖𝑐𝑎𝑛𝑡𝑠
= −49.5490 + 0.3550 ∗ 90 + 1.1049 ∗ 50
• The predicted number of applicants is 30.
–
𝑀𝑎𝑟𝑘𝑒𝑡𝑖𝑛𝑔 = 90, 𝐸𝑚𝑝𝑙𝑜𝑦𝑒𝑑 =80
෣
• 𝐴𝑝𝑝𝑙𝑖𝑐𝑎𝑛𝑡𝑠
= −49.5490 + 0.3550 ∗ 90 + 1.1049 ∗ 80
• The predicted number of applicants is 60.
b.
෣
𝐴𝑝𝑝𝑙𝑖𝑐𝑎𝑛𝑡𝑠
= −16.6359 + 0.0865𝑀𝑎𝑟𝑘𝑒𝑡𝑖𝑛𝑔 + 0.5405𝐸𝑚𝑝𝑙𝑜𝑦𝑒𝑑 + 0.0039𝑀𝑎𝑟𝑘𝑒𝑡𝑖𝑛𝑔 ∗ 𝐸𝑚𝑝𝑙𝑜𝑦𝑒𝑑
–
𝑀𝑎𝑟𝑘𝑒𝑡𝑖𝑛𝑔 = 90, 𝐸𝑚𝑝𝑙𝑜𝑦𝑒𝑑 = 50, 𝑀𝑎𝑟𝑘𝑒𝑡𝑖𝑛𝑔 ∗ 𝐸𝑚𝑝𝑙𝑜𝑦𝑒𝑑 = 90 ∗ 50
• The predicted number of applicants is 33.
–
𝑀𝑎𝑟𝑘𝑒𝑡𝑖𝑛𝑔 = 90, 𝐸𝑚𝑝𝑙𝑜𝑦𝑒𝑑 = 80, 𝑀𝑎𝑟𝑘𝑒𝑡𝑖𝑛𝑔 ∗ 𝐸𝑚𝑝𝑙𝑜𝑦𝑒𝑑 = 90 ∗80
• The predicted number of applicants is 59.
7-21
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7-21
Use Gender_Gap
7-22
BUSINESS
ANALYTICS,
1e | Jaggia,
Kelly,
Copyright
© 2021 McGraw-Hill
Education.
AllLertwachara,
rights reserved.Chen
No reproduction or distribution without the prior
written consent of McGraw-Hill Education.
Copyright © 2021 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.
7-22
7.1: Regression Models with Interaction
Variables (15/15)
• Example: The objective outlined in the introductory case is to analyze a
possible gender gap in the salaries of project managers.
7-23
BUSINESS
ANALYTICS,
1e | Jaggia,
Kelly,
Copyright
© 2021 McGraw-Hill
Education.
AllLertwachara,
rights reserved.Chen
No reproduction or distribution without the prior
written consent of McGraw-Hill Education.
Copyright © 2021 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.
7-23