DISCRETE CHOICE
MODELING IN
HEALTH ECONOMICS
COPYRIGHT OF:
ABHINAV ANAND
JYOTI ARORA
SHRADDHA RAMSWAMY
INTRODUCTION
 Studies suggest that self rated health score is a reliable
predictor of health status
We investigate impact of a
host of personal and status
characteristics such as age,
gender etc on the health
perception of US Citizens
DATA
 Dataset : NHANES Epidemiological Followup Study :1992
 Health status , represented by Yi coded as follows
POOR
FAIR
GOOD
• Yi = 1
• Yi = 2
• Yi = 3
VERY
GOOD
EXCELLENT
• Yi = 5
• Yi = 4
 Age is measured in years, education is measured in terms of
number of years of schooling completed and dichotomous
variable is created for gender (female = 1) and race (black = 1).
METHODOLOGY
1) Ordered Logit model for
the first part of our enquiry.
2) Sequential Logit Model for
the second part of our enquiry
ORDERED LOGIT MODEL SPECIFICATION
 A multinomial choice model where the values taken by the
dependent variable takes a natural order.
 Yi* is latent variable such that
5
 Yi = j when αj-1 < Yi*< αj where j= 1,2,3,4,5 and
 Yi* = β’Xi + ui where u follows logistic distribution.
α1
α2
α3
α4
ORDERED LOGIT MODEL
Where F( ) is a cdf and j = 1,2,3,4,5 and i is theith individual
We assume that u follows logistic distribution
CONTD….





P(Yi =1/Xi ) = F [ α1 – β’X]
P(Yi = 2/Xi) = F [α2 – β’X] – F[α1-β’X]
P(Yi = 3/Xi) = F [α3 – β’X] – F [α2 -β’X]
P(Yi = 4/Xi) = F [α4– β’X] – F [α3 – β’X]
P(Yi = 5/Xi) = 1– F [α4-β’X]
Where F ( ) is defined as above.
 For estimating the model we specify 5 dummy variables for the
ith individual with the following rule
 Zij = 1 if Yi = j where j = 1,2,3,4,5.
= 0 otherwise
ORDERED LOGIT MODEL ESTIMATION
 Using MLE
5
5
Assuming independent observations, we get
3712
5
F
Using Newton Raphson formula.
F
RESULTS
COMMAND:
proc logistic data = sasuser.nhanes descending;
model health = age gender race edu south;
run;
Analysis of Maximum Likelihood Estimates
Parameter
Intercept 5
Intercept 4
Intercept 3
Intercept 2
Age
gender
race
edu
south
DF
1
1
1
1
1
1
1
1
1
Estimate
-1.446
0.1255
1.6139
3.138
-0.0313
0.00989
-0.2122
0.1553
-0.7989
Standard
Error
0.2473
0.2463
0.2479
0.2539
0.00262
0.0605
0.0669
0.0114
0.1072
Chi-Square
34.1904
0.2598
42.3953
152.7003
143.3251
0.0267
10.0676
184.097
55.5218
Pr > Chisq
<.0001
0.6103
<.0001
<.0001
<.0001
0.8701
0.0015
<.0001
<.0001
..contd
Odds Ratio Estimate
Point
Effect Estimate
Age
0.969
95% Wald
Confidence
Limit
0.964
0.897
1.137
race
0.709
0.922
edu
1.168
south 0.45
1.142
0.365
Percent
Concordant 65.8
Somers'D 0.322
Percent
Discordant 33.6
Gamma
0.324
Percent
Tied
0.6
Tau-a
0.244
Pairs
522179
9
c
0.974
gender1.01
0.809
Association of Predicted
Probabilities and Observed
Responses
1.194
0.555
0.661
Probability estimate for ith individual
(-1.4460+β’Xi)
1
(-1.4460+β’Xi)
2
(0.1225+β’Xi)
(0.1225+β’Xi)
(1.6139+β’Xi)
3
(1.6139+β’Xi)
(3.138+β’Xi)
4
(3.138+β’Xi)
5
1
(-1.4460+β’Xi)
(-1.4460+β’Xi)
(0.1225+β’Xi)
(0.1225+β’Xi)
(1.6139+β’Xi)
(1.6139+β’Xi)
(3.138+β’Xi)
(3.138+β’Xi)
INFERENCE (ORDERED LOGIT)
One
additional
year of age
results in a
3.13%
decreases in
odds ratio of
higher self
rating.
The impact
of gender is
almost
negligible.
Blacks are
19.12% less
likely than
whites to rate
their health
at higher
response
values
An additional
year of
schooling
leads to
16.80%
increase in
odds ratio
higher self
rating
The Southern
residents in
each district
are 55% less
likely than
the northern
to rate their
health at
higher
response
values.
There are
522179 pairs
of
observations
Of these
65.8% are
concordant
pairs while
33.6% are
discordant
pairs.
SEQUENTIAL LOGIT MODEL
 Choices/Responses follow a sequence, so we need
(m-1) latent variables to characterize (m) unordered
choices.
 Self-rated health measure can be considered as a
purely cardinal variable following a sequence instead
of some natural ordering. This allows us to perform
discrete choice analysis using (non-ordered)
sequential logit model.
SEQUENTIAL LOGIT MODEL
Framework
•Five choices, and hence we have 4 latent
variables to describe the choices.
•Choices in each step are independent of
the previous step.
Root
(Sample)
Poor (1)
Fair (2)
Fair+++
(2 or 3 or 4 or 5)
Good++ (3 or 4 or 5)
Good (3)
VeryGood+
(4 or 5)
Very
Good (4)
Excellent
(5)
Probability Computation Example
P (Yi = 2) = P [Yi ≠ 1 and Yi = 2 |Yi ≠ 1]
= P [Yi ≠ 1] P [Yi = 2|Yi ≠ 1 ]
Therefore, for an individual i the conditional
probability that his self-rated health measure
will have a value j є {1,2,3,4,5} will be given by :
Pij= P (Yi = j |Xi )
and so on till j = 5
ESTIMATION IN SEQUENTIAL LOGIT
MODEL
One-shot
joint
optimization
with
Independent
Examples
Maximum
Likelihood
Estimation
•Thus, the parameter β1 can be estimated by dividing the entire sample into two
groups
Poor
Fair OR Good OR Very Good OR Excellent
•β2 can be estimated by first taking the sub-sample of those did not report poor into
two groups
Fair
Good OR Very Good OR Excellent
•β3 can be estimated by taking the sub-sample of those who didn’t report poor or fair
into two groups
Good
Repeated
Optimization
Very Good OR Excellent
•β4 can be estimated by taking the sub-sample of those who didn’t report poor or fair
or good into two groups
Very Good
In each case the binary models can be estimated by logit using MLE.
Excellent
SEQUENTIAL LOGIT MODEL
Implementation in SAS
 data seqlogit;

















Among those who report fair or
good or very good or excellent
health, the odds of reporting
fair (rather than good++) are
64% lower among residents
south of baseline than
residents north of baseline of
the same age, gender,
education and race.
set seqlogit;
fairplus = (shm>1);
fair = (shm=2);
if fairplus = 1;
run;
proc format;
value shm 1='poor' 2-5='fair+++';
value gender 0='male' 1='female';
value race 0='white' 1='black';
value resid 0='north' 1='south';
run;
proc qlim data=seqlogit; *covest=qml;
class race resid gender;
endogenous fair ~
discrete(dist=logistic order=formatted);
model fair = age gender race edu resid;
format gender gender. race race. resid resid.;
run;
The QLIM Procedure
Parameter Estimates
Standard
Estimate Error
Parameter
Intercept
Age
t
Value
Pr > |t|
-0.9028
0.40898
-2.21
0.0273
0.031085
0.004264
7.29
<.0001
-0.03239
0.098606
-0.33
0.7426
Gender
female
Gender
male
0
.
.
.
Race
black
0.12122
0.10717
1.13
0.258
Race
white
0
.
.
.
-0.15498
0.018192
-8.52
<.0001
Edu
Resid
south
-1.03592
0.142367
-7.28
<.0001
Resid
north
0
.
.
.
CONCLUSION
Ordered
Logit
Model
Sequential
Logit
Model
• Age, race, education (in terms of number of
years of schooling ), and having residence in
southern part of the district have a significant
impact on self rated health.
• Gender doesn’t have a significant impact.
• Age, education ( in terms of schooling) and
having residence in southern part of the
district have a significant impact on self rated
health.
• Gender and race don’t have significant impact.
REFERENCES
• Agresti A. Categorical Data Analysis, Second
edition. New York: John Wiley & Sons; 2002
• Gardiner J C. , Luo Z. Logit Models in Practice: B, C,
E, G, M, N, O… SAS Institute Inc. ; 2011