3. The Negative Binomial Regression Model (NBRM)
The SAS GENMODE procedure, STATA .nbreg command, and LIMDEP Negbin$ command estimate the negative binomial regression model (NBRM).
The GENMOD procedure estimates the NBRM using the /DIST=NEGBIN option. Note that the dispersion parameter is equivalent to the alpha in STATA and LIMDEP.
PROC GENMOD DATA = masil.accident;
MODEL accident=emps strict /DIST=NEGBIN LINK=LOG;
RUN;
The GENMOD Procedure
Model Information
Data Set COUNT.WASTE
Distribution Negative Binomial
Link Function Log
Dependent Variable Accident
Observations Used 778
Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance 775 589.7752 0.7610
Scaled Deviance 775 589.7752 0.7610
Pearson Chi-Square 775 845.6033 1.0911
Scaled Pearson X2 775 845.6033 1.0911
Log Likelihood 37.5628
Algorithm converged.
Analysis Of Parameter Estimates
Standard Wald 95% Confidence Chi-
Parameter DF Estimate Error Limits Square Pr > ChiSq
Intercept 1 0.3851 0.1278 0.1345 0.6357 9.07 0.0026
Emps 1 0.0052 0.0023 0.0008 0.0096 5.29 0.0214
Strict 1 -0.6703 0.1671 -0.9978 -0.3427 16.09 <.0001
Dispersion 1 3.9554 0.3501 3.3254 4.7048
NOTE: The negative binomial dispersion parameter was estimated by maximum likelihood.
Model Information
Data Set COUNT.WASTE
Distribution Negative Binomial
Link Function Log
Dependent Variable Accident
Observations Used 778
Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance 775 589.7752 0.7610
Scaled Deviance 775 589.7752 0.7610
Pearson Chi-Square 775 845.6033 1.0911
Scaled Pearson X2 775 845.6033 1.0911
Log Likelihood 37.5628
Algorithm converged.
Analysis Of Parameter Estimates
Standard Wald 95% Confidence Chi-
Parameter DF Estimate Error Limits Square Pr > ChiSq
Intercept 1 0.3851 0.1278 0.1345 0.6357 9.07 0.0026
Emps 1 0.0052 0.0023 0.0008 0.0096 5.29 0.0214
Strict 1 -0.6703 0.1671 -0.9978 -0.3427 16.09 <.0001
Dispersion 1 3.9554 0.3501 3.3254 4.7048
NOTE: The negative binomial dispersion parameter was estimated by maximum likelihood.
The restricted model produces a log likelihood of 28.8627. Thus, the likelihood ratio for goodness-of-fit is 17.4002 = 2 * (37.5628 – 28.8627) (p<.00017).
PROC GENMOD DATA = masil.accident;
MODEL accident= /DIST=NEGBIN LINK=LOG;
RUN;
The likelihood ratio for overdispersion is 1409.5838 = 2 * (37.5628 - (-667.2291)).
STATA has the .nbreg command for the NBRM. The command reports three log likelihood statistics: for the PRM, restricted NBRM (constant-only model), and unrestricted NBRM (full model), which make it easy to conduct likelihood ratio tests.
. nbreg accident emps strict
Fitting
comparison Poisson model:
Iteration 0: log likelihood = -1821.5112
Iteration 1: log likelihood = -1821.5101
Iteration 2: log likelihood = -1821.5101
Fitting constant-only model:
Iteration 0: log likelihood = -1256.6761
Iteration 1: log likelihood = -1152.6155
Iteration 2: log likelihood = -1125.6643
Iteration 3: log likelihood = -1125.4183
Iteration 4: log likelihood = -1125.4183
Fitting full model:
Iteration 0: log likelihood = -1117.1731
Iteration 1: log likelihood = -1116.7201
Iteration 2: log likelihood = -1116.7182
Iteration 3: log likelihood = -1116.7182
Negative binomial regression Number of obs = 778
LR chi2(2) = 17.40
Prob > chi2 = 0.0002
Log likelihood = -1116.7182 Pseudo R2 = 0.0077
------------------------------------------------------------------------------
accident | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
emps | .0051981 .0022595 2.30 0.021 .0007694 .0096267
strict | -.6702548 .1671191 -4.01 0.000 -.9978021 -.3427074
_cons | .3851111 .1278468 3.01 0.003 .134536 .6356861
-------------+----------------------------------------------------------------
/lnalpha | 1.37509 .0885176 1.201599 1.548582
-------------+----------------------------------------------------------------
alpha | 3.955434 .3501257 3.32543 4.704793
------------------------------------------------------------------------------
Likelihood ratio test of alpha=0: chibar2(01) = 1409.58 Prob>=chibar2 = 0.000
Iteration 0: log likelihood = -1821.5112
Iteration 1: log likelihood = -1821.5101
Iteration 2: log likelihood = -1821.5101
Fitting constant-only model:
Iteration 0: log likelihood = -1256.6761
Iteration 1: log likelihood = -1152.6155
Iteration 2: log likelihood = -1125.6643
Iteration 3: log likelihood = -1125.4183
Iteration 4: log likelihood = -1125.4183
Fitting full model:
Iteration 0: log likelihood = -1117.1731
Iteration 1: log likelihood = -1116.7201
Iteration 2: log likelihood = -1116.7182
Iteration 3: log likelihood = -1116.7182
Negative binomial regression Number of obs = 778
LR chi2(2) = 17.40
Prob > chi2 = 0.0002
Log likelihood = -1116.7182 Pseudo R2 = 0.0077
------------------------------------------------------------------------------
accident | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
emps | .0051981 .0022595 2.30 0.021 .0007694 .0096267
strict | -.6702548 .1671191 -4.01 0.000 -.9978021 -.3427074
_cons | .3851111 .1278468 3.01 0.003 .134536 .6356861
-------------+----------------------------------------------------------------
/lnalpha | 1.37509 .0885176 1.201599 1.548582
-------------+----------------------------------------------------------------
alpha | 3.955434 .3501257 3.32543 4.704793
------------------------------------------------------------------------------
Likelihood ratio test of alpha=0: chibar2(01) = 1409.58 Prob>=chibar2 = 0.000
The restricted model or “constant-only model” gives us a log likelihood -1125.4183. Thus, the likelihood ratio for goodness-of-fit is 17.4002 = 2 * [-1116.7182 - (-1125.4183)] (p<.00017). The p-value is computed as follows (Note the .disp or .di is an abbreviation of the .display).
. disp chi2tail(2, 17.4002)
.00016657
The likelihood ratio test for overdispersion results in a chi-squared of 1409.5838 (p<.0000) and rejects the null hypothesis of alpha=0. The statistically significant evidence of overdispersion indicates that the NBRM is preferred to the PRM.
. di 2 * (-1116.7182 - (-1821.5101))
1409.5838
The p-value of the likelihood ratio for overdispersion is computed as,
. di chi2tail(1, 1409.5838)
1.74e-308
Now, let us calculate marginal effects (or changes) at the means of independent variables. You should the read the discrete change labeled “0->1” of a binary variable strict, since its marginal change at the mean (.5077) is meaningless.
. prchange
nbreg:
Changes in Predicted Rate for accident
min->max 0->1 -+1/2 -+sd/2 MargEfct
emps 1.5326 0.0055 0.0068 0.2585 0.0068
strict -0.8931 -0.8931 -0.8885 -0.4383 -0.8721
exp(xb): 1.3011
emps strict
x= 42.0129 .507712
sd(x)= 38.1548 .500262
min->max 0->1 -+1/2 -+sd/2 MargEfct
emps 1.5326 0.0055 0.0068 0.2585 0.0068
strict -0.8931 -0.8931 -0.8885 -0.4383 -0.8721
exp(xb): 1.3011
emps strict
x= 42.0129 .507712
sd(x)= 38.1548 .500262
LIMDEP has the Negbin$ command for the NBRM that reports the PRM as well. Note that the standard errors of parameter estimates are slightly different from those of SAS and STATA. The Marginal Effects$ and the Means$ subcommands compute marginal effects at the mean of independent variables. You may not omit the Means$ subcommand.
NEGBIN;
Lhs=ACCIDENT;
Rhs=ONE,EMPS,STRICT;
Marginal Effects;
Means$
+---------------------------------------------+
| Poisson Regression |
| Maximum Likelihood Estimates |
| Model estimated: Sep 08, 2005 at 09:35:36AM.|
| 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.
+---------------------------------------------+
| Negative Binomial Regression |
| Maximum Likelihood Estimates |
| Model estimated: Sep 08, 2005 at 09:35:36AM.|
| Dependent variable ACCIDENT |
| Weighting variable None |
| Number of observations 778 |
| Iterations completed 8 |
| Log likelihood function -1116.718 |
| Restricted log likelihood -1821.510 |
| Chi squared 1409.584 |
| Degrees of freedom 1 |
| Prob[ChiSqd > value] = .0000000 |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Constant .3851110699 .12855240 2.996 .0027
EMPS .5198057234E-02 .22602075E-02 2.300 .0215 42.012853
STRICT -.6702547660 .16729839 -4.006 .0001 .50771208
Dispersion parameter for count data model
Alpha 3.955434012 .35680876 11.086 .0000
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
+-------------------------------------------+
| Partial derivatives of expected val. with |
| respect to the vector of characteristics. |
| They are computed at the means of the Xs. |
| Observations used for means are All Obs. |
| Conditional Mean at Sample Point 1.3011 |
| Scale Factor for Marginal Effects 1.3011 |
+-------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Constant .5010628939 .19396434 2.583 .0098
EMPS .6763123170E-02 .29746591E-02 2.274 .0230 42.012853
STRICT -.8720595665 .22469308 -3.881 .0001 .50771208
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
| Poisson Regression |
| Maximum Likelihood Estimates |
| Model estimated: Sep 08, 2005 at 09:35:36AM.|
| 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.
+---------------------------------------------+
| Negative Binomial Regression |
| Maximum Likelihood Estimates |
| Model estimated: Sep 08, 2005 at 09:35:36AM.|
| Dependent variable ACCIDENT |
| Weighting variable None |
| Number of observations 778 |
| Iterations completed 8 |
| Log likelihood function -1116.718 |
| Restricted log likelihood -1821.510 |
| Chi squared 1409.584 |
| Degrees of freedom 1 |
| Prob[ChiSqd > value] = .0000000 |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Constant .3851110699 .12855240 2.996 .0027
EMPS .5198057234E-02 .22602075E-02 2.300 .0215 42.012853
STRICT -.6702547660 .16729839 -4.006 .0001 .50771208
Dispersion parameter for count data model
Alpha 3.955434012 .35680876 11.086 .0000
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
+-------------------------------------------+
| Partial derivatives of expected val. with |
| respect to the vector of characteristics. |
| They are computed at the means of the Xs. |
| Observations used for means are All Obs. |
| Conditional Mean at Sample Point 1.3011 |
| Scale Factor for Marginal Effects 1.3011 |
+-------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Constant .5010628939 .19396434 2.583 .0098
EMPS .6763123170E-02 .29746591E-02 2.274 .0230 42.012853
STRICT -.8720595665 .22469308 -3.881 .0001 .50771208
(Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
Read the coefficients (.0068 and -.8721) to confirm that they are identical to the corresponding marginal effects calculated in STATA.
SAS, STATA, and LIMDEP produce almost the same parameter estimates and goodness-of-fit statistics (Table 4). Note that SAS reports different log likelihoods, but the same likelihood ratio.
Figure 3 compares the PRM and NBRM. Look at the predictions for zero counts of the two models. As the likelihood ratio test indicates, the NBRM seems to fit these data better than PRM.
Figure 3. Comparison of the Poisson and Negative Binomial Regression Models
Up: Table of Contents
Next: The Zero-Inflated Poisson Regression Model
Prev: The Poisson Regression Model
