The COUNTREG Procedure

Getting Started: COUNTREG Procedure

The COUNTREG procedure is similar in use to other SAS regression model procedures. For example, the following statements are used to estimate a Poisson regression model:

   proc countreg data=one ;
      model y = x / dist=poisson ;
   run;

The response variable $y$ is numeric and has nonnegative integer values. To allow for variance greater than the mean, specify the DIST=NEGBIN option to fit the negative binomial model instead of the Poisson.

The following example illustrates the use of PROC COUNTREG. The data are taken from Long (1997) and can be found in the SAS/ETS Sample Library. This study examines how factors such as gender (fem), marital status (mar), number of young children (kid5), prestige of the graduate program (phd), and number of articles published by the mentor (ment) of a doctoral candidate in science affect the number of articles (art) published by the scientist.

The first 10 observations are shown in Figure 11.1.

Figure 11.1: Article Count Data

Obs	art	mar	kid5	phd	ment
1	3	1	2	1.38000	8.0000
2	0	0	0	4.29000	7.0000
3	4	0	0	3.85000	47.0000
4	1	1	1	3.59000	19.0000
5	1	1	0	1.81000	0.0000
6	1	1	1	3.59000	6.0000
7	0	1	1	2.12000	10.0000
8	0	1	0	4.29000	2.0000
9	3	1	2	2.58000	2.0000
10	3	1	1	1.80000	4.0000

The following SAS statements estimate the Poisson regression model:

proc countreg data=long97data;
   model art = fem mar kid5 phd ment / dist=poisson;
run;

The “Model Fit Summary” table, shown in Figure 11.2, lists several details about the model. By default, the COUNTREG procedure uses the Newton-Raphson optimization technique. The maximum log-likelihood value is shown, in addition to two information measures, Akaike’s information criterion (AIC) and Schwarz’s Bayesian information criterion (SBC), which can be used to compare competing Poisson models. Smaller values of these criteria indicate better models.

Figure 11.2: Estimation Summary Table for a Poisson Regression

The COUNTREG Procedure

Model Fit Summary
Dependent Variable	art
Number of Observations	915
Data Set	WORK.LONG97DATA
Model	Poisson
Log Likelihood	-1651
Maximum Absolute Gradient	3.57454E-9
Number of Iterations	5
Optimization Method	Newton-Raphson
AIC	3314
SBC	3343
Number of Threads	4

The parameter estimates of the model and their standard errors are shown in Figure 11.3. All covariates are significant predictors of the number of articles, except for the prestige of the program (phd), which has a $p$ -value of 0.6271.

Figure 11.3: Parameter Estimates of Poisson Regression

Parameter Estimates
Parameter	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|
Intercept	1	0.304617	0.102982	2.96	0.0031
fem	1	-0.224594	0.054614	-4.11	<.0001
mar	1	0.155243	0.061375	2.53	0.0114
kid5	1	-0.184883	0.040127	-4.61	<.0001
phd	1	0.012823	0.026397	0.49	0.6271
ment	1	0.025543	0.002006	12.73	<.0001

The following statements fit the negative binomial model. Although the Poisson model requires that the conditional mean equal the conditional variance, the negative binomial model allows for overdispersion; that is, the conditional variance can exceed the conditional mean.

proc countreg data=long97data;
   model art = fem mar kid5 phd ment / dist=negbin(p=2) method=qn;
run;

The fit summary is shown in Figure 11.4, and parameter estimates are listed in Figure 11.5.

Figure 11.4: Estimation Summary Table for a Negative Binomial Regression

The COUNTREG Procedure

Model Fit Summary
Dependent Variable	art
Number of Observations	915
Data Set	WORK.LONG97DATA
Model	NegBin(p=2)
Log Likelihood	-1561
Maximum Absolute Gradient	5.72129E-7
Number of Iterations	16
Optimization Method	Quasi-Newton
AIC	3136
SBC	3170
Number of Threads	4

Figure 11.5: Parameter Estimates of Negative Binomial Regression

Parameter Estimates
Parameter	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|
Intercept	1	0.256144	0.138560	1.85	0.0645
fem	1	-0.216418	0.072672	-2.98	0.0029
mar	1	0.150489	0.082106	1.83	0.0668
kid5	1	-0.176415	0.053060	-3.32	0.0009
phd	1	0.015271	0.036040	0.42	0.6718
ment	1	0.029082	0.003470	8.38	<.0001
_Alpha	1	0.441620	0.052967	8.34	<.0001

The parameter estimate for _Alpha of $0.4416$ is an estimate of the dispersion parameter in the negative binomial distribution. A $t$ test for the hypothesis $H_0: \alpha =0$ is provided. It is highly significant, indicating overdispersion ( $p<0.0001$ ).

The null hypothesis $H_0: \alpha =0$ can be also tested against the alternative $\alpha > 0$ by using the likelihood ratio test, as described by Cameron and Trivedi (1998, pp. 45, 77–78). The likelihood ratio test statistic is equal to $-2(\mathcal{L}_{P}-\mathcal{L}_{NB})=-2(-1651+1561)=180$ , where $\mathcal{L}_{P}$ and $\mathcal{L}_{NB}$ are the log likelihoods for the Poisson and negative binomial models, respectively. The likelihood ratio test is highly significant, providing strong evidence of overdispersion.