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Limited Dependent Variables:
Binary Models
Erik Nesson
Ball State University
MBSW 2013
1
Outline
1. Overview of LDVs
2. Binary Outcome Models
a. Linear Probability Model
b. Logit and Probit
3. Interpretation of Coefficients
a. Odds ratios vs. marginal effects
b. Implementation in Stata
2
Binary Outcome Models
• Dependent variable takes values of 0 or 1
• Model of interest is probability that y=1
conditional on independent variables
 𝑃 𝑦 = 1|𝑋 = G 𝑋, 𝛽
 Here 𝑋represents matrix of covariates and 𝛽
represents vector of coefficients
• Examples
 Is a person obese? Does a person smoke? Does a
person contract a disease?
3
Example: Secondhand Smoke
• Half of adult non-smokers are exposed to
environmental tobacco smoke (ETS)
• Main point of public policies is to reduce ETS
exposure in specific areas
 Ex: smokefree air laws are meant to reduce ETS
exposure at work
• But most research about tobacco control policies
focuses on reducing smoking
• Main question:
 Do tobacco control policies reduce ETS exposure in
the workplace?
4
How to measure ETS exposure?
• NHANES Dataset
 Repeated cross-section dataset covering 1988-1994
and 1999-2004
 Roughly 10k individuals interviewed every year
 All individuals complete extensive survey AND receive
physical
 Contains extensive demographic and health history
information
• Sample
 Non-smoking, employed individuals age 18 to 65
 8,554 individuals
5
Dependent and Independent Variables
1. Indicators of ETS Exposure
a. Q: “… how many hours per day can you smell the
smoke from other people’s cigarettes, cigars, and/or
pipes?”
b. Serum cotinine levels
•
•
Cotinine is major metabolite of nicotine with 8-16 hr half
life
Very low levels can be detected
2. Tobacco Control Policies
a. Cigarette Taxes measured in real $2009
b. The percent of each individual’s state living under a
workplace SFA law (from 0 to 100)
6
Summary Statistics
Smell Smoke at Work
Observable Cotinine Level
Cigarette Excise Tax
Female
Age
Black
Hispanic
Married
Income to Poverty Ratio
B.A. Degree
Some College
H.S. Degree
Less than H.S.
Family Size
Rooms in Home
All Workers
(N=8554)
Mean
Std. Dev.
0.818
2.126
0.629
0.483
1.265
0.644
0.502
0.500
41.260
12.955
0.102
0.303
0.117
0.321
0.658
0.474
3.236
1.639
0.328
0.469
0.307
0.461
0.242
0.429
0.122
0.328
3.145
1.514
6.334
2.116
White Collar Workers
(N=4825)
Mean
Std. Dev.
0.536
1.717
0.572
0.495
1.267
0.633
0.587
0.492
41.625
12.624
0.092
0.290
0.074
0.262
0.664
0.472
3.572
1.556
0.446
0.497
0.316
0.465
0.192
0.394
0.045
0.208
3.003
1.407
6.566
2.189
Blue Collar Workers
(N=3729)
Mean
Std. Dev.
1.388
2.686
0.746
0.435
1.262
0.666
0.330
0.470
40.522
13.571
0.121
0.327
0.203
0.402
0.646
0.478
2.559
1.591
0.089
0.285
0.290
0.454
0.343
0.475
0.278
0.448
3.431
1.674
5.866
1.877
T-Test
0.000
0.000
0.834
0.000
0.007
0.000
0.000
0.248
0.000
0.000
0.105
0.000
0.000
0.000
0.000
7
Self-Reported ETS Exposure at Work
by Job Category
90
White Collar Jobs
80
70
Blue Collar Jobs
Percent
60
Total
50
40
30
20
10
0
No
Yes
Any Secondhand Smoke Exposure
Notes:
Data from NHANES III and NHANES 1999/2000 - NHANES 2003/2004.
8
Tabulation of Observable Cotinine
Levels by Job Category
80%
70%
Percent
60%
White Collar Jobs
Blue Collar Jobs
Total
50%
40%
30%
20%
10%
0%
No
Yes
Any Secondhand Smoke Exposure
Notes:
Data from NHANES III and NHANES 1999/2000 - NHANES 2003/2004.
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Basic Model
• Basic model estimates exposure to ETS as a
function of tobacco control policies, individual
characteristics, other geographic characteristics:
 𝑃 𝐸𝑇𝑆𝑖𝑗𝑡 = 1 𝑇𝐶, 𝑋, 𝑍, 𝜎, 𝜏 = G 𝑇𝐶𝑗𝑡 𝛽 + 𝑋𝑖𝑡 𝛼 +
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Linear Probability Model
• Assume conditional probability is linear in 𝑋
 𝑃 𝑦 = 1|𝑋 = 𝑋𝛽
• Upsides to LPM
 Easy to run: run a linear regression:
• 𝑦 = 𝛽0 + 𝛽1 𝑥1 + ⋯ + 𝛽𝑘 𝑥𝑘
 Interpretation is easy!
•
𝜕𝑃 𝑦=1|𝑋
𝜕𝑥𝑖
= 𝛽𝑖
• In words, “Every unit increase in 𝑥𝑖 is associated with a 𝛽𝑖
percentage point change in the probability that y=1.”
 Prediction is easy!
• 𝑃 𝑦 = 1|𝑋 = 𝑥 = 𝛽0 + 𝛽1 𝑥1 + ⋯ + 𝛽𝑘 𝑥𝑘
11
Binary Outcome Models
• Downsides to LPM
 G 𝑋𝛽 may not be linear
 Predicted probability does not need to be
between 0 and 1
 Constant marginal effect may not be reasonable!
12
Linear Probability Results
• Table shows marginal effects with standard
errors in parentheses
Self-Reported Exposure
White
Collar
Blue Collar
All Workers Workers
Workers
Cigarette Excise Tax 0.035
0.055*
0.021
(0.03)
(0.03)
(0.07)
Work SFA Law
-0.001*
0.000
-0.002***
(0.00)
(0.00)
(0.00)
Observable Cotinine Levels
White
Collar
Blue Collar
All Workers Workers
Workers
0.019
0.015
0.026
(0.03)
(0.04)
(0.05)
-0.002*** -0.003*** -0.001
(0.00)
(0.00)
(0.00)
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Logit or Probit
• Assume that 𝐺(𝑧) is a function such that
𝐺 𝑧 ∈ 0,1 for any value of 𝑧
 As 𝑧 → ∞ 𝐺(𝑧) → 1 and as 𝑧 → −∞ 𝐺(𝑧) → 0
• For Logit 𝐺 𝑋𝛽 =
exp 𝑋𝛽
1+exp 𝑋𝛽
• For Probit 𝐺 𝑋 ′ 𝛽 = Φ 𝑋𝛽
 Note: Φ 𝑍 is the standard normal CDF
14
Interpretation of Coefficients
• Continuous variable:
 If 𝑃 𝑦 = 1|𝑋 = G 𝑋𝛽 then what is marginal effect,
𝜕𝑃 𝑦=1|𝑋
i.e.
?
𝜕𝑥𝑖
 Using some
𝜕𝑃 𝑦=1|𝑋
calculus:
𝜕𝑥𝑖
= g 𝑋𝛽 𝛽𝑖
• Where g 𝑋𝛽 is the density function associated with G 𝑋𝛽
 Some notes:
1.
2.
3.
𝜕𝑃 𝑦=1|𝑋
𝜕𝑥𝑖
doesn’t only depend on 𝛽𝑖
What values of 𝑋 should we use?
Marginal effect isn’t constant like in linear probability
model
15
Interpretation of Coefficients
• Discrete variable:
 If 𝑃 𝑦 = 1|𝑋 = G 𝑋𝛽 then marginal effect of
𝑥1 increasing from 0 to 1 =
G 𝛽0 + 𝜷𝟏 + 𝛽2 𝑥2 + ⋯ + 𝛽𝑘 𝑥𝑘
− G 𝛽0 + 𝛽2 𝑥2 + ⋯ + 𝛽𝑘 𝑥𝑘
 Some notes:
1. Again, marginal effect doesn’t only depend on 𝛽𝑖
2. What values of 𝑋 should we use?
3. Marginal effect isn’t constant like in linear probability
model
16
Calculating Marginal Effects
• For both continuous and discrete variables,
other coefficients and variable values enter
into marginal effects calculation
• Three common approaches:
1. Marginal effect at the mean: Use mean values of
other variables
2. Average marginal effect: Calculate marginal
effect for each observation
3. Marginal effect at some other value of
coefficients
17
Calculating Marginal Effects
• Calculating marginal effect at the mean for 𝑥𝑖
 Continuous coefficient
• Find g 𝑋𝛽 by plugging in mean values for 𝑋
• Multiply g 𝑋𝛽 by 𝛽𝑖
 Discrete coefficient
• Find G 𝑋 ′ 𝛽 when 𝑥𝑖 = 1 by plugging in mean values
for 𝑋 and 1 for 𝑥𝑖
• Find G 𝑋 ′ 𝛽 when 𝑥𝑖 = 0 by plugging in mean values
for 𝑋 and 0 for 𝑥𝑖
• Subtract two values
18
Calculating Marginal Effects
• Calculating average marginal effect for 𝑥𝑖
 Continuous coefficient
• Find g 𝑋𝛽 for each observation and multiply by 𝛽𝑖
• Find mean value for all observations
 Discrete coefficient
•
•
•
•
Find G 𝑋𝛽 when 𝑥𝑖 = 1 for each observation
Find G 𝑋𝛽 when 𝑥𝑖 = 0 for each observation
Subtract two values for each observation
Find mean value for all observations
19
Implementation in Stata
• Code for estimating marginal effect at the mean
in Stata:
 logit y x
 margins, dydx(varlist) atmeans
• Code for estimating average marginal effect in
Stata:
 logit y x
 margins, dydx(varlist)
• Usually Stata is smart enough to determine which
independent variables are binary
20
Odds Ratios
• Common in other fields to run a logit model and report
an odds ratio
• What are odds?
 Odds of an event =
• The odds ratio =
𝑃 𝑦=1|𝑥1 =1
1−𝑃 𝑦=1|𝑥1 =1
𝑃 𝑦=1|𝒙𝟏 =𝟏
1−𝑃 𝑦=1|𝒙𝟏 =𝟏
𝑃 𝑦=1|𝒙𝟏 =𝟎
1−𝑃 𝑦=1|𝒙𝟏 =𝟎
• Odds ratio>1: event is more likely to happen
• Odds ratio<1: event is less likely to happen
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Odds Ratios and Logit
• How do Odds Ratios work in Logit?
 Odds 𝑦 = 1 given 𝑥1 = 1:
exp 𝑍𝛽
𝑃 𝑦 = 1|𝑥1 = 1, 𝑋
1 + exp 𝑍𝛽
=
= exp 𝑍𝛽
exp 𝑍𝛽
1 − 𝑃 𝑦 = 1|𝑥1 = 1, 𝑋
1−
1 + exp 𝑍𝛽
• Note: Z ′ 𝛽 = 𝛽0 + 𝛽1 𝟏 + 𝛽2 𝑥2 + ⋯ + 𝛽𝑘 𝑥𝑘
 Similarly, odds 𝑦 = 1 given 𝑥1 = 0:
𝑃 𝑦 = 1|𝑥1 = 0, 𝑋
= exp 𝑊𝛽
1 − 𝑃 𝑦 = 1|𝑥1 = 0, 𝑋
• Note: W ′ 𝛽 = 𝛽0 + 𝛽1 𝟎 + 𝛽2 𝑥2 + ⋯ + 𝛽𝑘 𝑥𝑘
22
Odds Ratios and Logit
• Then Odds Ratio =
 Plugging in:
exp 𝑍𝛽
exp 𝑊𝛽
exp 𝛽0 +𝛽1 +𝛽2 𝑥2 +⋯+𝛽𝑘 𝑥𝑘
exp 𝛽0 +𝛽2 𝑥2 +⋯+𝛽𝑘 𝑥𝑘
= exp 𝛽1
• Quick notes about the odds ratio
1. Odds ratio does not depend on other coefficients
or independent variables
2. May be difficult to translate into policy
3. Odds ratios are very easy to compute! Simply
exponentiate coefficients
23
Odds Ratios and Logit
• What does an odds ratio of 2 mean?
 Odds of y=1 are 2x greater when x=1 than when x=0
 Could be that
• Odds that y=1|x=1 = 4 and odds that y=1|x=0 = 2
• Odds that y=1|x=1 = 3 and odds that y=1|x=0 = 1.5
• Odds that y=1|x=1 = 2 and odds that y=1|x=0 = 1
• So odds ratios are not equal to marginal effects
• Do not tell us about differences in probability
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Odds Ratios and Marginal Effects
• Some notation:
 𝑃0 : 𝑃(𝑦 = 1|𝑥 = 0) 𝑂0 : Odds that 𝑦 = 1|𝑥 = 0
 𝑃1 : 𝑃(𝑦 = 1|𝑥 = 1) 𝑂1 : Odds that 𝑦 = 1|𝑥 = 1
𝑷𝟎
𝑷𝟏
𝑶𝟎
𝑶𝟏
0.05
0.10
0.50
0.80
0.90
0.10
0.15
0.55
0.85
0.95
0.05
0.11
1.00
4.00
9.00
0.11
0.18
1.22
5.67
19.00
Odds
Ratio
2.11
1.59
1.22
1.42
2.11
Marginal
Effect
0.05
0.05
0.05
0.05
0.05
25
Logit Results
• Table shows odds ratios, standard errors in
parentheses, and marginal effects in brackets
Cigarette Excise
Tax
Work SFA Law
Self-Reported Exposure
White Collar Blue Collar
All Workers Workers
Workers
1.226
1.447 *
1.127
(1.17)
(0.79)
(3.49)
[0.030]
[0.039]
[0.027]
0.993 *** 0.995
0.988 ***
(-0.32)
(-0.86)
(-0.38)
[-0.001]
[-0.001]
[-0.003]
Observable Cotinine Levels
White Collar Blue Collar
All Workers Workers
Workers
1.008
1.000
1.043
(0.02)
(0.05)
(0.02)
[0.001]
[0.000]
[0.003]
0.990 *** 0.987 *** 0.997
(0.00)
(0.00)
(0.00)
[-0.002]
[-0.002]
[-0.000]
26
Other Issues
• Standard errors are complicated
 Be wary of canned programs (like Stata!) which
allow calculation of robust variance/covariance
matrices.
• Interaction terms are also complicated
 Odds ratios can be difficult to interpret
 Marginal effects are better!
27