The COUNTREG Procedure

## Example 10.2 ZIP and ZINB Models for Data Exhibiting Extra Zeros

In the study by Long (1997) of the number of published articles by scientists (see the section Getting Started: COUNTREG Procedure), the observed proportion of scientists publishing no articles is 0.3005. The following statements use PROC FREQ to compute the proportion of scientists publishing each observed number of articles. Output 10.2.1 shows the results.

proc freq data=long97data;
table art / out=obs;
run;

Output 10.2.1 Proportion of Scientists Publishing a Certain Number of Articles
The FREQ Procedure

art Frequency Percent Cumulative
Frequency
Cumulative
Percent
0 275 30.05 275 30.05
1 246 26.89 521 56.94
2 178 19.45 699 76.39
3 84 9.18 783 85.57
4 67 7.32 850 92.90
5 27 2.95 877 95.85
6 17 1.86 894 97.70
7 12 1.31 906 99.02
8 1 0.11 907 99.13
9 2 0.22 909 99.34
10 1 0.11 910 99.45
11 1 0.11 911 99.56
12 2 0.22 913 99.78
16 1 0.11 914 99.89
19 1 0.11 915 100.00

PROC COUNTREG is then used to fit Poisson and negative binomial models to the data. For each model, the PROBCOUNTS macro computes the probability that the number of published articles is , where is a value in a list of nonnegative integers specified in the COUNTS= option. The computations require the parameter estimates of the fitted model. These are saved using the ODS OUTPUT statement as shown and passed to the PROBCOUNTS macro by using the INMODEL= option. Variables containing the probabilities are created with names beginning with the PREFIX= string followed by the COUNTS= values and are saved in the OUT= data set. For the Poisson model, the variables poi0, poi1, , poi10 are created and saved in the data set predpoi, which also contains all of the variables in the DATA= data set. The PROBCOUNTS macro is available from the Samples section at http://support.sas.com. The following statements compute the estimates for Poisson and negative binomial models.

/*-- Poisson Model --*/
proc countreg data=long97data;
model art=fem mar kid5 phd ment / dist=poisson;
ods output ParameterEstimates=pe;
run;

%include probcounts;
%probcounts(data=long97data,
inmodel=pe,
counts=0 to 10,
prefix=poi, out=predpoi)

/*-- Negative Binomial Model --*/
proc countreg data=long97data;
model art=fem mar kid5 phd ment / dist=negbin(p=2);
ods output ParameterEstimates=pe;
run;

%probcounts(data=predpoi,
inmodel=pe,
counts=0 to 10,
prefix=nb, out=prednb)

Parameter estimates for these two models are shown in the section Getting Started: COUNTREG Procedure. For each model, the predicted proportion of zero articles can be calculated as the average predicted probability of zero articles across all scientists as shown in the macro probcounts in the following program. Under the Poisson model, the predicted proportion of zero articles is 0.2092, which considerably underestimates the observed proportion. The negative binomial more closely estimates the proportion of zeros (0.3036). Also, the test of the dispersion parameter, _Alpha, in the negative binomial model indicates significant overdispersion (). As a result, the negative binomial model is preferred to the Poisson model.

Another way to account for the large number of zeros in this data set is to fit a zero-inflated Poisson (ZIP) or a zero-inflated negative binomial (ZINB) model. In the following statements, DIST=ZIP requests the ZIP model. In the ZEROMODEL statement, you can specify the predictors, , for the process that generated the additional zeros. The ZEROMODEL statement also specifies the model for the probability . By default, a logistic model is used for . The default can be changed using the LINK= option. In this particular ZIP model, all variables used to model the article counts are also used to model .

proc countreg data=long97data;
model art = fem mar kid5 phd ment / dist=zip;
zeromodel art ~ fem mar kid5 phd ment;
ods output ParameterEstimates=pe;
run;

%probcounts(data=prednb,
inmodel=pe,
counts=0 to 10,
prefix=zip, out=predzip)

The parameters of the ZIP model are displayed in Output 10.2.2. The first set of parameters gives the estimates of in the model for the Poisson process mean. Parameters with the prefix "Inf_" are the estimates of in the logistic model for .

Output 10.2.2 ZIP Model Estimation
The COUNTREG Procedure

Model Fit Summary
Dependent Variable art
Number of Observations 915
Data Set WORK.LONG97DATA
Model ZIP
Log Likelihood -1605
Number of Iterations 16
Optimization Method Newton-Raphson
AIC 3234
SBC 3291

 Algorithm converged.

Parameter Estimates
Parameter DF Estimate Standard Error t Value Approx
Pr > |t|
Intercept 1 0.640838 0.121306 5.28 <.0001
fem 1 -0.209145 0.063405 -3.30 0.0010
mar 1 0.103751 0.071111 1.46 0.1446
kid5 1 -0.143320 0.047429 -3.02 0.0025
phd 1 -0.006166 0.031008 -0.20 0.8424
ment 1 0.018098 0.002295 7.89 <.0001
Inf_Intercept 1 -0.577060 0.509383 -1.13 0.2573
Inf_fem 1 0.109747 0.280082 0.39 0.6952
Inf_mar 1 -0.354013 0.317611 -1.11 0.2650
Inf_kid5 1 0.217101 0.196481 1.10 0.2692
Inf_phd 1 0.001272 0.145262 0.01 0.9930
Inf_ment 1 -0.134114 0.045244 -2.96 0.0030

The proportion of zeros predicted by the ZIP model is 0.2986, which is much closer to the observed proportion than the Poisson model. But Output 10.2.4 shows that both models deviate from the observed proportions at one, two, and three articles.

The ZINB model is specified by the DIST=ZINB option. All variables are again used to model both the number of articles and . The METHOD=QN option specifies that the quasi-Newton method be used to fit the model rather than the default Newton-Raphson method. These options are implemented in the following program.

proc countreg data=long97data;
model art=fem mar kid5 phd ment / dist=zinb method=qn;
zeromodel art ~ fem mar kid5 phd ment;
ods output ParameterEstimates=pe;
run;

%probcounts(data=predzip,
inmodel=pe,
counts=0 to 10,
prefix=zinb, out=predzinb)

The estimated parameters of the ZINB model are shown in Output 10.2.3. The test for overdispersion again indicates a preference for the negative binomial version of the zero-inflated model (). The ZINB model also does a good job of estimating the proportion of zeros (0.3119), and it follows the observed proportions well, though possibly not as well as the negative binomial model.

Output 10.2.3 ZINB Model Estimation
The COUNTREG Procedure

Model Fit Summary
Dependent Variable art
Number of Observations 915
Data Set WORK.LONG97DATA
Model ZINB
Log Likelihood -1550
Number of Iterations 81
Optimization Method Quasi-Newton
AIC 3126
SBC 3189

 Algorithm converged.

Parameter Estimates
Parameter DF Estimate Standard Error t Value Approx
Pr > |t|
Intercept 1 0.416747 0.143596 2.90 0.0037
fem 1 -0.195507 0.075592 -2.59 0.0097
mar 1 0.097582 0.084452 1.16 0.2479
kid5 1 -0.151732 0.054206 -2.80 0.0051
phd 1 -0.000700 0.036270 -0.02 0.9846
ment 1 0.024786 0.003493 7.10 <.0001
Inf_Intercept 1 -0.191684 1.322807 -0.14 0.8848
Inf_fem 1 0.635928 0.848911 0.75 0.4538
Inf_mar 1 -1.499470 0.938661 -1.60 0.1102
Inf_kid5 1 0.628427 0.442780 1.42 0.1558
Inf_phd 1 -0.037715 0.308005 -0.12 0.9025
Inf_ment 1 -0.882291 0.316223 -2.79 0.0053
_Alpha 1 0.376681 0.051029 7.38 <.0001

The following statements compute the average predicted count probability across all scientists for each count 0, 1, , 10. The averages for each model, along with the observed proportions, are then arranged for plotting by PROC SGPLOT.

proc summary data=predzinb;
var poi0-poi10 nb0-nb10 zip0-zip10 zinb0-zinb10;
output out=mnpoi  mean(poi0-poi10)  =mn0-mn10;
output out=mnnb   mean(nb0-nb10)    =mn0-mn10;
output out=mnzip  mean(zip0-zip10)  =mn0-mn10;
output out=mnzinb mean(zinb0-zinb10)=mn0-mn10;
run;

data means;
set mnpoi mnnb mnzip mnzinb;
drop _type_ _freq_;
run;

proc transpose data=means out=tmeans;
run;

data allpred;
merge obs(where=(art<=10)) tmeans;
obs=percent/100;
run;

proc sgplot;
yaxis label='Probability';
xaxis label='Number of Articles';
series y=obs  x=art / name='obs' legendlabel='Observed'
lineattrs=(color=black thickness=4px);
series y=col1 x=art / name='poi' legendlabel='Poisson'
lineattrs=(color=blue);
series y=col2 x=art/ name='nb' legendlabel='Negative Binomial'
lineattrs=(color=red);
series y=col3 x=art/ name='zip' legendlabel='ZIP'
lineattrs=(color=blue pattern=2);
series y=col4 x=art/ name='zinb' legendlabel='ZINB'
lineattrs=(color=red pattern=2);
discretelegend 'poi' 'zip' 'nb' 'zinb' 'obs' / title='Models:'
location=inside position=ne across=2 down=3;
run;

For each of the four fitted models, Output 10.2.4 shows the average predicted count probability for each article count across all scientists. The Poisson model clearly underestimates the proportion of zero articles published, while the other three models are quite accurate at zero. All of the models do well at the larger numbers of articles.

Output 10.2.4 Average Predicted Count Probability

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