11. Categorical choice and survival
models
•
•
•
•
11.1
11.2
11.3
11.4
Homogeneous models
Multinomial logit models with random effects
Transition (Markov) models
Survival models
11.1 Homogeneous models
•
•
•
•
•
•
11.1.1 Statistical inference
11.1.2 Generalized logit
11.1.3 Multinomial (conditional) logit
11.1.4 Random utility interpretation
11.1.5 Nested logit
11.1.6 Generalized extreme-value distributions
11.1.1 Statistical inference
• We are now trying to understand a response that is an
unordered categorical variable.
• Let’s assume that the response may take on values 1, 2, ..., c
corresponding to c categories.
• Examples include
– employment choice, such as Valletta (1999E),
– mode of transportation choice, such as the classic work
by McFadden (1978E),
– choice of political party affiliation, such as Brader and
Tucker (2001O)
– marketing brand choice, such as Jain et al. (1994O).
Likelihood
• Denote the probability of choosing the jth category as πit,j =
Prob(yit = j),
– so that πit,1 + … + πit,c = 1.
• The likelihood for the ith subject at the tth time point is:
 π 
π it1
π it 2


π itc
c
yitj
itj
j 1
if yit  1
if yit  2

if yit  c
– where yitj is an indicator of the event yit= j.
• Assuming independence, the total log-likelihood is
L
c
 y
it
j 1
it , j
ln π it , j
11.1.2 Generalized logit
• Define the linear combinations of explanatory variables
• And probabilities
Vit, j  xit β j
Prob( yit  j )   it , j



 expV 
exp Vit , j
c
k 1
it ,k
• Here βj may depend on the alternative j, whereas the
explanatory variables xit do not.
• A convenient normalization for this model is βc = 0.
• We interpret j as the proportional change in the odds ratio.
 Prob( yit  j ) 
  Vit, j  Vit,c  xit β j
ln 
 Prob( yit  c) 
Example 11.2 Job security
•
•
•
Valetta (1999E) - determinants of job turnover
There are has c = 5 categories: dismissal, left job because of plant
closures, “quit,” changed jobs for other reasons and no change in
employment.
Table 11.1 shows that turnover declines as tenure increases.
– To illustrate, consider a typical man in the 1992 sample where we
have time = 16 and focus on dismissal probabilities.
– For this value of time, the coefficient associated with tenure for
dismissal is -0.221 + 16  (0.008) = -0.093 (due to the interaction
term).
– From this, we interpret an additional year of tenure to imply that
the dismissal probability is exp(-0.093) = 91% of what it would be
otherwise, representing a decline of 9%.
Variable
Table 11.1 Turnover Generalized Logit and Probit Regression Estimates
Probit
Generalized Logit Model
Regression
Dismissed
Plant
Other
Model
Closed
Reason
Tenure
Time Trend
Tenure  (Time Trend)
-0.084
(0.010)
-0.002
(0.005)
0.003
(0.001)
0.094
(0.057)
-0.020
(0.009)
-0.221
(0.025)
-0.008
(0.011)
0.008
(0.002)
0.286
(0.123)
-0.061
(0.023)
-0.086
(0.019)
-0.024
(0.016)
0.004
(0.001)
0.459
(0.189)
-0.053
(0.025)
-0.068
(0.020)
0.011
(0.013)
-0.005
(0.002)
-0.022
(0.158)
-0.005
(0.025)
Quit
-0.127
(0.012)
-0.022
(0.007)
0.006
(0.001)
0.333
(0.082)
-0.027
(0.012)
Change in Logarithmic Sector
Employment
Tenure  (Change in Logarithmic
Sector Employment)
Notes: Standard errors in parentheses. Omitted category is no change in employment. Other variables
controlled for consisted of education, marital status, number of children, race, years of full-time work
experience and its square, union membership, government employment, logarithmic wage, the U.S.
employment rate and location.
11.1.3 Multinomial (conditional)
logit
• Use the linear combination of explanatory variables
Vit , j  xit , j β
• Here, xit,j is a vector of explanatory variables that depends
on the jth alternative, whereas the parameters β do not.
• The total log-likelihood is
L(β) 
c
 y
it
j 1
it , j
ln π it , j 
c

 c







 yit , j x it , j β  ln  expx it ,k β 

 k 1

 j 1


it

• Drawback: independence of irrelevant alternatives
– The ratio, it,1 / it,2 = exp(Vit,1 ) / exp(Vit,2 ) ,
does not depend on the underlying values of the other
alternatives, Vit,j, for j=3, ..., c.
Multinomial vs generalized logit
• The generalized logit model is a special case of the
multinomial logit model.
– To see this, consider explanatory variables xit and
parameters βj, each of dimension K  1.
 β1 
 
β2 
β 

 
β 
 c
– With this specification, we have xit,j β = xit βj .
x it, j
0
 
  
0
 
  x it 
 
0
  
 
0
• Thus, a statistical package that performs multinomial logit
estimation can also perform generalized logit estimation
• Further, some authors use the descriptor multinomial logit
when referring to the Section 11.1.2 generalized logit model.
Ex. 11.3 Choice of yogurt brands
•
•
•
•
•
Scanner data - they are obtained from optical scanning of grocery
purchases at check-out.
The subjects consist of n =100 households in Springfield, Missouri.
The response of interest is the type of yogurt purchased, consisting of
four brands: Yoplait, Dannon, Weight Watchers and Hiland.
The households were monitored over a two-year period with the
number of purchases ranging from 4 to 185; the total number of
purchases is N=2,412.
The two marketing variables of interest are PRICE and FEATURES.
– For the brand purchased, PRICE is recorded as price paid, that is,
the shelf price net of the value of coupons redeemed. For other
brands, PRICE is the shelf price.
– FEATURES is a binary variable, defined to be one if there was a
newspaper feature advertising the brand at time of purchase, and
zero otherwise.
– Note that the explanatory variables vary by alternative, suggesting
the use of a multinomial (conditional) logit model.
Multinomial
logit model
fit
•
Table 11.4 Yogurt Multinomial Logit Model Estimates
Variable
Parameter t-statistic
estimate
4.450
23.78
Yoplait
3.716
25.55
Dannon
3.074
21.15
Weight Watchers
0.491
4.09
FEATURES
-36.658
-15.04
PRICE
10,138
-2 Log Likelihood
10,148
AIC
A multinomial logit model was fit to the data, using
Vit, j   j  1PRICEit, j   2 FEATUREit, j
•
•
using Hiland as the omitted alternative.
Results:
– Each parameter is statistically significantly different from zero.
– For the FEATURES coefficient, a consumer is exp(0.4914) = 1.634
times more likely to purchase a product that is featured in a
newspaper ad compared to one that is not.
– For the PRICE coefficient, a one cent decrease in price suggests
that a consumer is exp(0.3666) = 1.443 times more likely to
purchase a brand of yogurt.
11.1.4 Random utility interpretation
• In economic applications, we think of an individual
choosing among c categories.
– Preferences among categories are indexed by an
unobserved utility function.
– For the ith individual at the tth time period, let’s use Uitj for
the utility of the jth choice.
• To illustrate, we assume that the individual chooses the first
category if
– Uit1 > Uitj for j=2, ..., c
– and denote this choice as yit = 1.
Random utility
• We model utility as an underlying value plus random noise,
that is, Uitj = Vitj + eitj .
• The value function is an unknown linear combination of
explanatory variables.
• The noise variable is assumed to have an extreme-value, or
Gumbel, distribution of the form F(eitj  a) = exp ( e-a) .
• This is mainly for computational convenience. With this,
one can show
1
Prob( y  1) 
1
 expV

c
j 2
• Our multinomial logit form
j
 V1
expV1 
  expV 
c
j
j 1
11.1.5 Nested logit
•
•
We now introduce a type of hierarchical model known as a nested logit
model.
In the first stage, one chooses an alternative (say the first type) with
probability
expVit ,1 
 it ,1  Prob( yit  1) 

c
expVit,1   k 2 expVit ,k /  


•
Conditional on not choosing the first alternative, the probability of
choosing any one of the other alternatives follows a multinomial logit
model with probabilities
 it, j
expVit , j /  
 Prob( yit  j | yit  1)  c
1   it ,1
expVit ,k /  

k 2
•
•
•
The value of ρ = 1 reduces to the multinomial logit model.
Interpret Prob(yit = 1) to be a weighted average of values from the first
choice and the others.
Conditional on not choosing the first category, the form of Prob(yit = j|
yit 1) has the same form as the multinomial logit.
Nested logit
• The nested logit generalizes the multinomial logit model we no longer have the problem of independence of
irrelevant alternatives.
• A disadvantageis that only one choice is observed;
– thus, we do not know which category belongs in the first
stage of the nesting without additional theory regarding
choice behavior.
• One may view the nested logit as a robust alternative to the
multinomial logit - examine each one of the categories in
the first stage of the nesting.
11.1.6 Generalized extreme value
distribution
• The nested logit model can also be given a random utility
interpretation.
• McFadden (1978E) introduced the generalized extremevalue distribution of the form:
 
Fa1 ,..., ac   exp  G e  a1 ,..., e  ac

• Under regularity conditions on G, McFadden (1978E)
showed that this yields

Prob( y  j )  Prob U j  U k for k  1,..., c, k  j
• where
G j ( x1 ,..., xc ) 

G( x1 ,..., xc )
x j

e j G j (e V1 ,..., e Vc )
V

G( e V1 ,..., e Vc )
11.2 Multinomial logit models with
random effects
• We now consider linear combinations of explanatory
variables
Vit , j  z it , j α i  xit , j β
• A typical parameterization for the value function is of the
form: Vit,j = ij + xit j.
– That is, the slopes may depend on the choice j and the
intercepts may depend on the individual i and the choice j.
Parameterizing the value function
• Consider: Vit,j = ij + xit j.
• We need to impose additional restrictions on the model.
– To see this, replace ij by ij*= ij + 6 to get that
 
*
itj

exp  ij*  xit β j
 exp
k
*
ik

 xit β k


exp  ij  6  xit β j
  exp
k
ik

 6  xit β k 


exp  ij  xit β j
 exp
k
ik

 xit β k 
• A convenient normalization is to take i1 = 0 and 1 = 0 so
that Vit,1 = 0 and
it1 = 1/( 1 + Sk=2 exp(ik + xit k) )
itj = exp(ik + xit k) /(1+Sk=2 exp(ik + xit k)) , for j=2,...,c.
  itj
Likelihood
• The conditional (on the heterogeneity) probability that the
ith subject at time t chooses the jth alternative is
π itj
•


expα  x β 
α  

 expV   expα  x β 
exp Vit , j
i
ij
c
it , j
c
it ,k
k 1
ik
k 1
To avoid parameter redundancies, we take ic = 0.
it ,k
• The conditional likelihood for the ith subject is
Ly i | α i  
 π α 
Ti
c
it , j
t 1
yit , j
i
j 1

Ti

t 1
 exp







it , j β 
y
α

x
it
,
j
ij

j 1

c
expαik  xit ,k β  
k 1

c

• We assume that {αi} is i.i.d. with distribution function Gα,
– typically taken to be multivariate normal.
• The (unconditional) likelihood for the ith subject
Ly i   Ly i | adG α (a)

Relation with nonlinear random effects
Poisson model
• Few statistical packages will estimate multinomial logit
models with random effects.
– Statistical packages for nonlinear Poisson models are readily
available;
– they can be used to estimate parameters of the multinomial logit
model with random effects.
• Instruct a statistical package to “assume” that
– the binary random variables yit,j are Poisson distributed with
conditional means πit,j and,
– conditional on the heterogeneity terms, are independent.
– Not valid assumptions
• Binary varibles can’t be Possion
• Certainly not independent
Relation with nonlinear random effects
Poisson model
• With this “assumption,” the conditional likelihood
interpreted by the statistical package is
Ly i | α i  
Ti
c

t 1 j 1
π α 
it , j
i
y it , j


exp - πit, j α i 
yit, j !
c



πit , j α i 

t 1 
 j 1
Ti
 
• this is the same conditional likelihood as above

y it , j

 1
e


Example 11.3 Choice of yogurt brands
•
•
•
Random intercepts for Yoplait, Dannon and Weight Watchers were
assumed to follow a multivariate normal distribution with an
unstructured covariance matrix.
Here, we see that the coefficients for FEATURES and PRICE are
qualitatively similar to the model without random effects.
Overall, the AIC statistic suggests that the model with random effects is
the preferred model.
Table 11.5 Yogurt Multinomial Logit Model Estimates
Without Random
With Random Effects
Effects
Variable
Parameter t-statistic
Parameter
t-statistic
estimate
estimate
4.450
23.78
5.622
7.29
Yoplait
3.716
25.55
4.772
6.55
Dannon
3.074
21.15
1.896
2.09
Weight Watchers
0.491
4.09
0.849
4.53
FEATURES
-36.658
-15.04
-44.406
-11.08
PRICE
10,138
7,301.4
-2 Log Likelihood
10,148
7,323.4
AIC
11.3 Transition (Markov) models
• Another way of accounting for heterogeneity
– trace the development of a dependent variable over time
– represent the distribution of its current value as a
function of its history.
• Define Hit to be the history of the ith subject up to time t.
– If the explanatory variables are assumed to be nonstochastic, then we might use Hit = {yi1, …, yi,t-1}.
– Thus, the likelihood for the ith subject is
Ti
f ( y
Ly i   f ( yi1 )
t 2
it
| H it )
Special Case: Generalized linear
model (GLM)
• The conditional distribution is
 yit it  b( it )

f ( yit | H it )  exp
 S( yit , ) 



• where
E( yit | H it )  b( it )
Var ( yit | H it )   b( it )
• Assuming a canonical link, for the systematic component,
one could use
 it  gE( yit | H it )   xit β   j  j yi ,t  j
•
See Diggle, Heagerty, Liang and Zeger (2002S, Chapter 10) for further
applications of and references to general transition GLMs.
Unordered categorical response
• First consider Markov models of order 1 so that the history
Hit need only contain yi,t-1.
• Thus,
 it, jk  Probyit  k | yi ,t 1  j   Probyit  k | yi ,t 1  j, yi ,t 2 ,..., yi ,1
• Organize the set of transition probabilities πit,jk as a matrix
  it ,11  it ,12

  it , 21  it , 22
Π it  




 it ,c1  it ,c 2
  it ,1c 

  it , 2c 

 

  it ,cc 
• Here, each row sums to one.
• We call the row identifier, j, the state of origin and the
column identifier, k, the destination state.
Pension plan example
• c = 4 categories
– 1 denotes active continuation in the pension plan,
– 2 denotes retirement from the pension plan,
– 3 denotes withdrawal from the pension plan and
– 4 denotes death.
• In a given problem, one assumes that a certain subset of
transition probabilities are zero.
1
2
1 = active membership
1
2
3
4
2 = retirement
3 = withdrawal
4 = death
3
4
Multinomial logit
• For each state of origin, use
π it , jk 

exp Vit , jk

 expV 
c
it , jh
h 1
• where Vit, jk  xit, jk β j
• The conditional likelihood is
f ( yit | yi ,t 1 ) 
 π 
c
c
yit , jk
it , jk
j 1 k 1
• where
yit , jk
1 if yit  k and yi ,t 1  j

0 otherwise
• When πit,jk = 0 (by assumption), we have that yit,jk = 0.
– Use the convention that 00 = 1.
Partial log-likelihood
•
•
To simplify matters, assume that the initial state distribution, Prob(yi1),
is described by a different set of parameters than the transition
distribution, f(yit | yi,t-1).
Thus, to estimate this latter set of parameters, one only needs to
maximize the partial log-likelihood
LP 
•
Ti
 ln f ( y
i
it
| yi ,t 1 )
t 2
Special case: specify separate components for each alternative
c

 c







(
β
)

ln
f
(
y
|
y

j
)

y
x
β

ln
exp(
x
β
)
•L PIdea:
we
can
split
up
the
data
according
to
each
(lagged)
choice
(j)

,j
j
it
i ,t 1
it , jk it , jk j
it , jk j 

i t 2
i t 2 
 k 1

 k 1

Ti

Ti
 

– determine MLEs for each alternative (at time t-1), in isolation of
the others.
Example 11.3 Choice of yogurt
brands - Continued
• Ignore the length of time between purchases
– Sometimes referred to as “operational time.”
– One’s most recent choice of a brand of yogurt has the
same effect on today’s choice, regardless as to whether
the prior choice was one day or one month ago.
1
2
1 = Yoplait
2 = Dannon
3 = Weight Watchers
4 = Hiland
3
4
Choice of yogurt brands
• Table 11.6a
– Only 2,312 observations under consideration
– Initial values from each of 100 subjects are not available
for the transition analysis.
– “Brand loyalty” is exhibited by most observation pairs
• the current choice of the brand of yogurt is the same as chosen
most recently.
– Other observation pairs can be described as “switchers.”
• Table 11.6b
– Rescale counts by row totals to give (rough) empirical
transition probabilities.
– Brand loyalty and switching behavior is more apparent
– Customers of Yoplait, Dannon and Weight Watchers
exhibit more brand loyalty compared to those of Hiland
who are more prone to switching.
Choice of yogurt brands
Origin State
Yoplait
Dannon
Weight Watchers
Hiland
Total
Table 11.6a Yogurt Transition Counts
Destination State
Yoplait
Dannon
Weight
Watchers
654
71
44
14
783
65
822
18
17
922
41
19
473
6
539
Hiland
Total
17
16
5
30
68
777
928
540
67
2,312
Table 11.6b Yogurt Transition Empirical Probabilities, in Percent
Destination State
Yoplait
Dannon
Weight
Hiland
Origin State
Watchers
84.2
8.4
5.3
2.2
Yoplait
7.7
88.6
2.0
1.7
Dannon
8.1
3.3
87.6
0.9
Weight Watchers
20.9
25.4
9.0
44.8
Hiland
Total
100.0
100.0
100.0
100.0
Choice of yogurt brands
Variable
Yoplait
Dannon
Weight Watchers
FEATURES
PRICE
-2 Log Likelihood
•
•
•
Table 11.7 Yogurt Transition Model Estimates
State of Origin
Yoplait
Dannon
Weight Watchers
Estimate
t-stat Estimate
t-stat Estimate
t-stat
Hiland
Estimate
t-stat
5.952
12.75
2.529
7.56
1.986
5.81
0.593
2.07
-41.257
-6.28
2,397.8
0.215
0.32
0.210
0.42
-1.105
-1.93
1.820
3.27
-13.840
-1.21
281.5
4.125
9.43
5.458
16.45
1.522
3.91
0.907
2.89
-48.989
-8.01
2,608.8
4.266
6.83
2.401
4.35
5.699
11.19
0.913
2.39
-37.412
-5.09
1,562.2
Purchase probabilities for customers of Dannon, Weight Watchers and
Hiland are more responsive to a newspaper ad than Yoplait customers.
Compared to the other three brands, Hiland customers are not price
sensitive in that changes in PRICE have relatively impact on the
purchase probability (it is not even statistically significant).
The total (minus two times the partial) log-likelihood is 2,379.8 + … +
281.5 = 6,850.3.
– Without origin state, the corresponding value of 9,741.2.
– The likelihood ratio test statistic is LRT = 2,890.9.
– This corroborates the intuition that the most recent type of purchase
has a strong influence in the current brand choice.
Example 11.4 Income tax payments
and tax preparers
•
•
•
•
Yogurt - the explanatory variables FEATURES and PRICE depend on
the alternative.
The choice of a professional tax preparer - financial and demographic
characteristics do not depend on the alternative.
97 tax filers never used a preparer in the five years under consideration
and 89 always did, out of 258 subjects.
Table 11.8 shows the relationship between the current and most recent
choice of preparer.
– 258 initial observations are not available for the transition analysis,
reducing our sample size to 1,290 – 258 = 1,032.
Table 11.8 Tax Preparers Transition Empirical
Probabilities, in Percent
Origin State
Destination State
Count
PREP = 0
PREP = 1
546
89.4
10.6
PREP = 0
486
8.4
91.6
PREP = 1
Tax preparers
Table 11.9 Tax Preparers Transition Model Estimates
State of Origin
PREP = 0
PREP = 1
Variable
Estimate
t-stat Estimate
t-stat
-10.704 -3.06
0.208
0.18
Intercept
1.024
2.50
0.104
0.73
LNTPI
-0.072 -2.37
0.047
2.25
MR
0.352
0.85
0.750
1.56
EMP
361.5
264.6
-2 Log Likelihood
•
•
•
•
A likelihood ratio test provides strong evidence that most recent choice
is an important predictor of the current choice.
To interpret the regression coefficients, consider a “typical” tax filer
with LNTPI = 10, MR = 23 and EMP = 0.
If this tax filer had not previously chosen to use a preparer, then
– V = -10.704 + 1.024(10) – 0.072(23) + 0.352(0) = -2.12.
– Estimated prob = exp(-2.12)/(1+exp(-2.12)) = 0.107.
If this tax filer had chosen to use a preparer, then the estimated
probability is 0.911.
– This example underscores the importance of the intercept.
Higher order Markov models
• It is customary in Markov modeling to simply expand the
state space to handle higher order time relationships.
• Define a new categorical response, yit* = {yit, yi,t-1}.
– The order 1 transition probability, f(yit* | yi,t-1*), is
equivalent to an order 2 transition probability of the
original response,
f(yit | yi,t-1, yi,t-2).
– The conditioning events are the same,
yi,t-1* = {yi,t-1, yi,t-2}
– yi,t-1 is completely determined by the conditioning event
yi,t-1*.
• Expansions to higher orders can be readily accomplished in
a similar fashion.
Tax preparers
Table 11.10 Tax Preparers Order 2 Transition Empirical Probabilities,
in Percent
Origin State
Destination State
Lag
Lag 2
Count
PREP = 0
PREP = 1
PREP
PREP
376
89.1
10.9
0
0
28
67.9
32.1
0
1
43
25.6
74.4
1
0
327
6.1
93.9
1
1
•
Table 11.10 suggests that there are important differences in the
transition probabilities for each lag 1 origin state (yi,t-1 = Lag PREP)
between levels of the lag two origin state (yi,t-2 = Lag 2 PREP).
Tax preparers
Variable
Intercept
LNTPI
MR
EMP
-2 Log Likelihood
•
•
Table 11.11 Tax Preparers Order 2 Transition Model Estimates
State of Origin
Lag PREP = 0,
Lag PREP = 0,
Lag PREP = 1,
Lag PREP = 1,
Lag 2 PREP = 0
Lag 2 PREP = 1
Lag 2 PREP = 0
Lag 2 PREP = 1
Estimate
t-stat Estimate
t-stat Estimate
t-stat Estimate
t-stat
-9.866
0.923
-0.066
0.406
254.2
-2.30
1.84
-1.79
0.84
-7.331
-0.81
0.675
0.63
-0.001
-0.01
0.050
0.04
33.4
1.629
-0.210
0.065
0.25
-0.27
0.89
NA
NA
42.7
-0.251
-0.19
0.197
1.21
0.040
1.42
1.406
1.69
139.1
The total (minus two times the partial) log-likelihood is 469.4.
– With no origin state, the corresponding value of 1,067.7.
– Thus, the likelihood ratio test statistic is LRT = 567.3.
– With 12 degrees of freedom, this is statistically significant.
With one lag, the log-likelihood of 490.2.
– Thus, the likelihood ratio test statistic is LRT = 20.8. With 8 degrees of
freedom, this is statistically significant.
– Thus, the lag two component is statistically significant contribution to
the model.
11.4 Survival Models