Download Alternative III Zero-inflated Poisson Regression

Count Data Models in SAS
May 25, 2017
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Introduction
 A comprehensive survey of models for count data in SAS
 Why? Gaining popularity since 1980
=> Insurance: # of auto/medical insurance claims
=> Banking: # of delinquencies / missed payments
=> Marketing: # of responses / purchases
 5 Models to be covered:
poisson regression, negative binomial regression,
hurdle poisson regression, zero-inflated poisson regression,
finite mixture (latent class) poisson regression
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SAS Capability
Procedures
GENMOD
GLIMMIX
NLIN
NLMIXED
COUNTREG
MODEL
Poisson
Regression
NB
Regression
Hurdle
Regression
ZIP
Regression
LC Poisson
Regression
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Count Data
 Nature of count data
nonnegative, discrete, skewed distribution
high proportion of zero outcomes
potential problems: over-dispersion (variance >> mean) , excess
zeroes
 Why OLS won’t work?
counts are heteroskedastic (variance dependent on mean)
predicted has to be nonnegative (log transformation won’t work)
 A case study: model # of hospital stays
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Data Summary
Classical data for count models:
- 4406 elderly respondents sampled from National Medical
Expenditure Survey (NMES) in 1987
- Information included: 7 health, demo, and socio-econ variables
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Starting Point
100%
Observations:
1) 80% zeroes ==> excess zeroes
2) Variance = 2 * Mean ==> possible over-dispersion
3) Poor fit with univariate Poisson
80%
60%
40%
20%
0%
0
1
2
Observed Probability
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Univariate Poisson Probability
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Baseline Model
 Probability Function of Poisson Regression
Exp ui   ui
f Yi | X i  
Yi !
Yi
proc nlmixed data = data;
params b0 = 0 b1 = 0 b2 = 0 ... ...;
mu = exp(b0 + b1 * x1 + b2 * x2...);
p = exp(-mu) * mu ** y / fact(y);
ll = log(p);
Identical to
Prob. Function
model y ~ general(ll);
Run;
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Result of Poisson Model
100%
Observations:
1) Improvement by including observed heterogeneity
2) Significantly under-fit at zeroes
80%
What's wrong? ==> Over-Dispersion
60%
40%
20%
0%
0
1
2
Observed Probability
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Predicted Probability of Poisson Regerssion
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Test for Over-Dispersion
 Auxiliary OLS regression (Cameron, 1996):
 yi  ui 2  yi
ui
 ui  ei
data ols_tmp;
set poi_out;
dep = ((y - yhat) ** 2 - y) / yhat;
run;
proc reg data = ols_tmp;
model dep = yhat / noint;
run;
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significant yhat
indicates
over-dispersion
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Alternative I
 Most common alternative: Negative Binomial Regression
 NB can be considered a generalized Poisson by including a
dispersion parameter.
ui  Exp X i   ei   Exp X i  Expei 

where Expei  ~ Gamma  1 ,  1

s.t. E Expei   1 and V Expei   
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Alternative I
 Probability Function of Negative Binomial Regression
f Yi | X i  


 Yi  
Yi  1  1
1
 
 
 1
   ui
1
 1



 ui
 1
   ui



Yi
proc nlmixed data = data;
params b0 = 0 b1 = 0 b2 = 0 ... ...;
mu = exp(b0 + b1 * x1 + b2 * x2 ... ...);
p = gamma(y + 1/alpha) / (gamma(y + 1) *
gamma(1/alpha)) * ((1/alpha) / (1/alpha + mu)) **
(1/alpha) * (mu / (1/alpha + mu)) ** y;
ll = log(p);
model y ~ general(ll);
Run;
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Result of NB Model
100%
Observations:
1) Significant Improvement by including unobserved
heterogeneity
80%
Comparison with Poisson model:
Likelihood Ratio = 2 * (LL_poi - LL_nb)
= 2 * (-3048 - -2857)
= 378
60%
40%
20%
0%
0
1
2
Observed Probability
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Predicted Probability of NB Regerssion
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Alternative II
 Hurdle Regression (Mullahy, 1986)
Two Parts:
- zero outcomes: Logistic regression
- positive outcomes: Truncated Poisson regression
 Probability Function of Hurdle Regression
 i

Y
f Yi | X i    1   i   Exp ui   ui i
 1  Exp u   Y !
i
i

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for Yi  0
for Yi  0
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Alternative II
proc nlmixed data = data;
params b0 = 0 b1 = 0 ... a0 = 0 a1 = 0 ...;
xb = b0 + b1 * x1 + b2 * x2 ... ...);
mu = exp(b0 + b1 * x1 + b2 * x2...);
xa = a0 + a1 * x1 + a2 * x2 ... ...);
if y = 0 then p = exp(xa) / (1 + exp(xa));
else p = (1 - exp(xa) / (1 + exp(xa))) / (1 exp(-mu)) * (exp(-mu) * mu ** y / fact(y));
ll = log(p);
Prob function
for zeroes
Prob function
for positive
model y ~ general(ll);
Run;
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Result of Hurdle Model
100%
80%
Observations:
1) Significant Improvement by modeling zeroes
separatedly
60%
How to compare with Poisson model?
AIC, BIC, & Vuong statistic
40%
20%
0%
0
1
2
Observed Probability
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Predicted Probability of Hurdle Regerssion
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Alternative III
 Zero-inflated Poisson Regression (Lambert, 1992)
Two sources of zeroes
- a point mass of zeroes
- zeroes from standard Poisson distribution
 Probability Function of Hurdle Regression
 i  1  i   Exp ui 

Y
Exp ui   ui i
f Yi | X i   

1  i 

Yi !

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for Yi  0
for Yi  0
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Alternative III
proc nlmixed data = data;
params b0 = 0 b1 = 0 ... a0 = 0 a1 = 0 ...;
xb = b0 + b1 * x1 + b2 * x2 ... ...);
mu = exp(b0 + b1 * x1 + b2 * x2...);
xa = a0 + a1 * x1 + a2 * x2 ... ...);
if y = 0 then p = exp(xa) / (1 + exp(xa)) +
(1 - exp(xa) / (1 + exp(xa)) * exp(-mu);
Prob function
for zeroes
else p = (1 - exp(xa) / (1 + exp(xa))) *
(exp(-mu) * mu ** y / fact(y));
Prob function
for zeroes
ll = log(p);
model y ~ general(ll);
Run;
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Result of ZIP Model
100%
80%
Observations:
1) Significant Improvement by assuming 2 sources of
zeroes
60%
How to compare with other models?
AIC, BIC, & Vuong statistic
40%
20%
0%
0
1
2
Observed Probability
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Predicted Probability of ZIP Regerssion
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Alternative IV
 Latent Class Poisson Regression (Wedel, 1993):
- Existence of S >= 2 classes of latent segments in the data
- Each latent segment is poisson with different parameter
- Each case drawn from such latent segments with certain probs.
- Interesting in marketing: segment and model at the same time
 Probability Function of LC Poisson Regression
S
f Yi | X i    ps
s 1
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Exp ui |s  ui |s
Yi
Yi !
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Alternative IV
proc nlmixed data = data;
params a0 = 0 ... b0 = 1 ... c0 = 2 ...
prior1 = 0 to 1 by 0.1 prior2 = 0 to 1 by 0.1;
xa = a0 + a1 * x1 + a2 * x2 ... ...);
ma = exp(xa);
pa = exp(-ma) * ma ** y / fact(y);
xb = b0 + b1 * x1 + b2 * x2 ... ...);
mb = exp(xb);
pb = exp(-mb) * mb ** y / fact(y);
xc = c0 + c1 * x1 + c2 * x2 ... ...);
mc = exp(xc);
pc = exp(-mc) * mc ** y / fact(y);
p = prior1 * pa + prior2 * pb + (1 - prior1 - prior2) * pc;
ll = log(p);
... ...
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Result of LC Poisson
100%
80%
Observations:
1) Significant Improvement by assuming 3 latent
classes with different sets of parameter
60%
How to compare with other models?
AIC, BIC, & Vuong statistic
40%
20%
0%
0
1
Observed Probability
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Predicted Probability of LC Poisson Regerssion
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Models Prediction
1) Poisson cannot give adequate fit for the data.
2) Hurdle and ZIP are better to model excess zeroes.
3) NB and LC are better to handle heterogeneity.
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Models Comparison
1) AIC & BIC is convenient and easy to compute for model comparison, good
enough for practitioners. BIC tends to select a more parsimonious model.
2) Vuong test is good but computationally tedious (code available in the
paper), recommended for researchers.
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Conclusion
 In practice, Poisson model usually is not sufficient for overdispersed data but useful as a baseline model.
(Rule of Thumb for Over-Dispersion: Variance ≥ 2 * Mean)
 It is important to identify the reason for over-dispersion, long tail,
excess zeroes, or … … ?
(Excess zeroes might be the most common reason)
 Statistics shouldn’t be the only consideration for model selection.
Examples:
1) Both Hurdle and ZIP suggest positive effect of private insurance
on hospital stays, which makes perfect sense.
2) LC provides a possibility to segment population, which is
invaluable in marketing, insurance, and credit risk.
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