4. The Zero-inflated Poission Regression Model (ZIP)
STATA and LIMDEP have commands for the zero-inflated Poisson regression model (ZIP).
STATA has the .zip command to estimate the ZIP. The inflate() option specifies a list of variables that determines whether the observed count is zero. The vuong option computes the Vuong statistic to compare the ZIP and PRM.
. zip accident emps strict, inflate(emps strict) vuong
Fitting
constant-only model:
Iteration 0: log likelihood = -1627.0779
Iteration 1: log likelihood = -1309.5825
Iteration 2: log likelihood = -1272.433
Iteration 3: log likelihood = -1270.9543
Iteration 4: log likelihood = -1270.9523
Iteration 5: log likelihood = -1270.9523
Fitting full model:
Iteration 0: log likelihood = -1270.9523
Iteration 1: log likelihood = -1269.7219
Iteration 2: log likelihood = -1269.7206
Iteration 3: log likelihood = -1269.7206
Zero-inflated Poisson regression Number of obs = 778
Nonzero obs = 280
Zero obs = 498
Inflation model = logit LR chi2(2) = 2.46
Log likelihood = -1269.721 Prob > chi2 = 0.2918
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
accident |
emps | -.000277 .0008633 -0.32 0.748 -.001969 .001415
strict | -.0923911 .0729023 -1.27 0.205 -.2352771 .0504948
_cons | 1.361978 .0493222 27.61 0.000 1.265308 1.458647
-------------+----------------------------------------------------------------
inflate |
emps | -.0109897 .0022678 -4.85 0.000 -.0154344 -.006545
strict | 1.057031 .1767509 5.98 0.000 .7106059 1.403457
_cons | .488656 .1211099 4.03 0.000 .2512849 .726027
------------------------------------------------------------------------------
Vuong test of zip vs. standard Poisson: z = 8.40 Pr>z = 0.0000
Iteration 0: log likelihood = -1627.0779
Iteration 1: log likelihood = -1309.5825
Iteration 2: log likelihood = -1272.433
Iteration 3: log likelihood = -1270.9543
Iteration 4: log likelihood = -1270.9523
Iteration 5: log likelihood = -1270.9523
Fitting full model:
Iteration 0: log likelihood = -1270.9523
Iteration 1: log likelihood = -1269.7219
Iteration 2: log likelihood = -1269.7206
Iteration 3: log likelihood = -1269.7206
Zero-inflated Poisson regression Number of obs = 778
Nonzero obs = 280
Zero obs = 498
Inflation model = logit LR chi2(2) = 2.46
Log likelihood = -1269.721 Prob > chi2 = 0.2918
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
accident |
emps | -.000277 .0008633 -0.32 0.748 -.001969 .001415
strict | -.0923911 .0729023 -1.27 0.205 -.2352771 .0504948
_cons | 1.361978 .0493222 27.61 0.000 1.265308 1.458647
-------------+----------------------------------------------------------------
inflate |
emps | -.0109897 .0022678 -4.85 0.000 -.0154344 -.006545
strict | 1.057031 .1767509 5.98 0.000 .7106059 1.403457
_cons | .488656 .1211099 4.03 0.000 .2512849 .726027
------------------------------------------------------------------------------
Vuong test of zip vs. standard Poisson: z = 8.40 Pr>z = 0.0000
The restricted model is estimated with the intercept only.
. zip accident, inflate(emps strict)
The Vuong statistic at the bottom compares the ZIP and PRM. Since the V 8.40 is greater than 1.96, we conclude that the ZIP is preferred to the PRM.
The LIMDEP Poisson$ command needs to have the Zip and Rh2 subcommands. The Rh2 is equivalent to the inflate() option in STATA. The Alg=Newton$ subcommand is needed to use the Newton-Raphson algorithm because the default Broyden algorithm failed to converge.
POISSON;
Lhs=ACCIDENT;
Rhs=ONE,EMPS,STRICT;
Rh2=ONE,EMPS,STRICT;
ZIP; Alg=Newton$
+---------------------------------------------+
| Poisson Regression |
| Maximum Likelihood Estimates |
| Model estimated: Sep 06, 2005 at 00:25:07PM.|
| Dependent variable ACCIDENT |
| Weighting variable None |
| Number of observations 778 |
| Iterations completed 8 |
| Log likelihood function -1821.510 |
| Restricted log likelihood -1883.921 |
| Chi squared 124.8218 |
| Degrees of freedom 2 |
| Prob[ChiSqd > value] = .0000000 |
| Chi- squared = 4944.94781 RsqP= -.0051 |
| G - squared = 2827.20794 RsqD= .0423 |
| Overdispersion tests: g=mu(i) : 4.720 |
| Overdispersion tests: g=mu(i)^2: 4.253 |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Constant .3900961420 .46678663E-01 8.357 .0000
EMPS .5418599057E-02 .74341923E-03 7.289 .0000 42.012853
STRICT -.7041663804 .66761926E-01 -10.547 .0000 .50771208
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
Normal exit from iterations. Exit status=0.
+----------------------------------------------------------------------+
| Zero Altered Poisson Regression Model |
| Logistic distribution used for splitting model. |
| ZAP term in probability is F[tau x Z(i) ] |
| Comparison of estimated models |
| Pr[0|means] Number of zeros Log-likelihood |
| Poisson .27329 Act.= 498 Prd.= 212.6 -1821.51007 |
| Z.I.Poisson .64642 Act.= 498 Prd.= 502.9 -1259.88568 |
| Note, the ZIP log-likelihood is not directly comparable. |
| ZIP model with nonzero Q does not encompass the others. |
| Vuong statistic for testing ZIP vs. unaltered model is 9.5740 |
| Distributed as standard normal. A value greater than |
| +1.96 favors the zero altered Z.I.Poisson model. |
| A value less than -1.96 rejects the ZIP model. |
+----------------------------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Poisson/NB/Gamma regression model
Constant 1.361977491 .23944641E-01 56.880 .0000
EMPS -.2770010575E-03 .37770090E-03 -.733 .4633 42.012853
STRICT -.9239125073E-01 .33326502E-01 -2.772 .0056 .50771208
Zero inflation model
Constant .4886559537 .12210013 4.002 .0001
EMPS -.1098971050E-01 .22152492E-02 -4.961 .0000 42.012853
STRICT 1.057031399 .17715551 5.967 .0000 .50771208
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
| Poisson Regression |
| Maximum Likelihood Estimates |
| Model estimated: Sep 06, 2005 at 00:25:07PM.|
| Dependent variable ACCIDENT |
| Weighting variable None |
| Number of observations 778 |
| Iterations completed 8 |
| Log likelihood function -1821.510 |
| Restricted log likelihood -1883.921 |
| Chi squared 124.8218 |
| Degrees of freedom 2 |
| Prob[ChiSqd > value] = .0000000 |
| Chi- squared = 4944.94781 RsqP= -.0051 |
| G - squared = 2827.20794 RsqD= .0423 |
| Overdispersion tests: g=mu(i) : 4.720 |
| Overdispersion tests: g=mu(i)^2: 4.253 |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Constant .3900961420 .46678663E-01 8.357 .0000
EMPS .5418599057E-02 .74341923E-03 7.289 .0000 42.012853
STRICT -.7041663804 .66761926E-01 -10.547 .0000 .50771208
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
Normal exit from iterations. Exit status=0.
+----------------------------------------------------------------------+
| Zero Altered Poisson Regression Model |
| Logistic distribution used for splitting model. |
| ZAP term in probability is F[tau x Z(i) ] |
| Comparison of estimated models |
| Pr[0|means] Number of zeros Log-likelihood |
| Poisson .27329 Act.= 498 Prd.= 212.6 -1821.51007 |
| Z.I.Poisson .64642 Act.= 498 Prd.= 502.9 -1259.88568 |
| Note, the ZIP log-likelihood is not directly comparable. |
| ZIP model with nonzero Q does not encompass the others. |
| Vuong statistic for testing ZIP vs. unaltered model is 9.5740 |
| Distributed as standard normal. A value greater than |
| +1.96 favors the zero altered Z.I.Poisson model. |
| A value less than -1.96 rejects the ZIP model. |
+----------------------------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Poisson/NB/Gamma regression model
Constant 1.361977491 .23944641E-01 56.880 .0000
EMPS -.2770010575E-03 .37770090E-03 -.733 .4633 42.012853
STRICT -.9239125073E-01 .33326502E-01 -2.772 .0056 .50771208
Zero inflation model
Constant .4886559537 .12210013 4.002 .0001
EMPS -.1098971050E-01 .22152492E-02 -4.961 .0000 42.012853
STRICT 1.057031399 .17715551 5.967 .0000 .50771208
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
In order to estimate the restricted model, run the following command with the ONE only in the Lhs$ subcommand. The Rh2$ subcommand remains unchanged.
POISSON;
Lhs=ACCIDENT;
Rhs=ONE;
Rh2=ONE,EMPS,STRICT;
ZIP; Alg=Newton$
STATA and LIMDEP report the same parameter estimates, but they produce different standard errors and log likelihoods. In particular, LIMDEP returned a suspicious log likelihood for the restricted model, and thus ended up with the “unlikely” likelihood ratio of -.0304. In addition, the Vuong statistics in STATA and LIMDEP are different.
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