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Zeroinflated poisson regression is used to model count data that has an excess of zero counts. Further, theory suggests that the excess zeros are generated by a separate process from the count values and that the excess zeros can be modeled independently. Thus, the zip model has two parts, a poisson count model and the logit model for predicting excess zeros. You may want to review these Data Analysis Example pages, Poisson Regression and Logit Regression.
Please Note: The purpose of this page is to show how to use various data analysis commands. It does not cover all aspects of the research process which researchers are expected to do. In particular, it does not cover data cleaning and verification, verification of assumptions, model diagnostics and potential followup analyses.
Examples of zeroinflated Poisson regression
Example 1.
School administrators study the attendance behavior of high school juniors over one semester at two schools. Attendance is measured by number of days of absent and is predicted by gender of the student and standardized test scores in math and language arts. Many students have no absences during the semester.
Example 2.
The state wildlife biologists want to model how many fish are being caught by fishermen at a state park. Visitors are asked whether or not they have a camper, how many people were in the group, were there children in the group and how many fish were caught. Some visitors do not fish, but there is no data on whether a person fished or not. Some visitors who did fish did not catch any fish so there are excess zeros in the data because of the people that did not fish.
Description of the data
Let’s pursue Example 2 from above.
We have data on 250 groups that went to a park. Each group was questioned about how many fish they caught (count), how many children were in the group (child), how many people were in the group (persons), and whether or not they brought a camper to the park (camper).
In addition to predicting the number of fish caught, there is interest in predicting the existence of excess zeros, i.e. the zeroes that were not simply a result of bad luck fishing. We will use the variables child, persons, and camper in our model. Let’s look at the data.
use http://www.statapress.com/data/r10/fish, clear summarize count child persons camper Variable  Obs Mean Std. Dev. Min Max + count  250 3.296 11.63503 0 149 child  250 .684 .8503153 0 3 persons  250 2.528 1.11273 1 4 camper  250 .588 .4931824 0 1 histogram count, discrete freq tab1 child persons camper > tabulation of child child  Freq. Percent Cum. + 0  132 52.80 52.80 1  75 30.00 82.80 2  33 13.20 96.00 3  10 4.00 100.00 + Total  250 100.00 > tabulation of persons persons  Freq. Percent Cum. + 1  57 22.80 22.80 2  70 28.00 50.80 3  57 22.80 73.60 4  66 26.40 100.00 + Total  250 100.00 > tabulation of camper camper  Freq. Percent Cum. + 0  103 41.20 41.20 1  147 58.80 100.00 + Total  250 100.00
Analysis methods you might consider
Below is a list of some analysis methods you may have encountered. Some of the methods listed are quite reasonable while others have either fallen out of favor or have limitations.
 Zeroinflated Poisson Regression – The focus of this web page.
 Zeroinflated Negative Binomial Regression – Negative binomial regression does better with over dispersed data, i.e. variance much larger than the mean.
 Ordinary Count Models – Poisson or negative binomial models might be more appropriate if there are no excess zeros.
 OLS Regression – You could try to analyze these data using OLS regression. However, count data are highly nonnormal and are not well estimated by OLS regression.
Zeroinflated Poisson regression
We will run the zip command with child and camper as predictors of the counts, persons as the predictor of the excess zeros.
zip count child camper, inflate(persons) Fitting constantonly model: Iteration 0: log likelihood = 1347.807 Iteration 1: log likelihood = 1315.5343 Iteration 2: log likelihood = 1126.3689 Iteration 3: log likelihood = 1125.5358 Iteration 4: log likelihood = 1125.5357 Iteration 5: log likelihood = 1125.5357 Fitting full model: Iteration 0: log likelihood = 1125.5357 Iteration 1: log likelihood = 1044.8553 Iteration 2: log likelihood = 1031.8733 Iteration 3: log likelihood = 1031.6089 Iteration 4: log likelihood = 1031.6084 Iteration 5: log likelihood = 1031.6084 Zeroinflated Poisson regression Number of obs = 250 Nonzero obs = 108 Zero obs = 142 Inflation model = logit LR chi2(2) = 187.85 Log likelihood = 1031.608 Prob > chi2 = 0.0000  count  Coef. Std. Err. z P>z [95% Conf. Interval] + count  child  1.042838 .0999883 10.43 0.000 1.238812 .846865 1.camper  .8340222 .0936268 8.91 0.000 .650517 1.017527 _cons  1.597889 .0855382 18.68 0.000 1.430237 1.76554 + inflate  persons  .5643472 .1629638 3.46 0.001 .8837503 .244944 _cons  1.297439 .3738522 3.47 0.001 .5647022 2.030176 
The output looks very much like the output from an OLS regression:

 Begins with the iteration log giving the values of the log likelihoods starting with a constantonly model that has no predictors for the count model and the intercept only sets to zero for the inflated model. Then the second part, fitting full model, starts with estimated parameters for the inflated model and intercept only model for the count model until iteration converges to estimation of the full model. The last value in the log is the final value of the log likelihood for the full model and is repeated below.
 Next comes the header information. On the righthand side the number of observations used (250), number of nonzero observations (108) are given along with the likelihood ratio chisquared. This compares the full model to a model without count predictors, giving a difference of two degrees of freedom. This is followed by the pvalue for the chisquare. The model, as a whole, is statistically significant.
 Below the header you will find the Poisson regression coefficients for each of the count predicting variables along with standard errors, zscores, pvalues and 95% confidence intervals for the coefficients.
 Following these are logit coefficients for the variable predicting excess zeros along with its standard errors, zscores, pvalues and confidence intervals.
Cameron and Trivedi (2009) recommend robust standard errors for Poisson models. We will rerun the model with the vce(robust) option.
zip count child i.camper, inflate(persons) vce(robust) Fitting constantonly model: Iteration 0: log pseudolikelihood = 1347.807 Iteration 1: log pseudolikelihood = 1315.5343 Iteration 2: log pseudolikelihood = 1126.3689 Iteration 3: log pseudolikelihood = 1125.5358 Iteration 4: log pseudolikelihood = 1125.5357 Iteration 5: log pseudolikelihood = 1125.5357 Fitting full model: Iteration 0: log pseudolikelihood = 1125.5357 Iteration 1: log pseudolikelihood = 1044.8553 Iteration 2: log pseudolikelihood = 1031.8733 Iteration 3: log pseudolikelihood = 1031.6089 Iteration 4: log pseudolikelihood = 1031.6084 Iteration 5: log pseudolikelihood = 1031.6084 Zeroinflated Poisson regression Number of obs = 250 Nonzero obs = 108 Zero obs = 142 Inflation model = logit Wald chi2(2) = 7.25 Log pseudolikelihood = 1031.608 Prob > chi2 = 0.0266   Robust count  Coef. Std. Err. z P>z [95% Conf. Interval] + count  child  1.042838 .3893772 2.68 0.007 1.806004 .2796731 1.camper  .8340222 .4076029 2.05 0.041 .0351352 1.632909 _cons  1.597889 .2934631 5.44 0.000 1.022711 2.173066 + inflate  persons  .5643472 .2888849 1.95 0.051 1.130551 .0018567 _cons  1.297439 .493986 2.63 0.009 .3292445 2.265634 
Now we can move on to the specifics of the individual results.
 Using the robust option has resulted in a fairly large change in the model chisquare, which is now a Wald chisquare. This statistic is based on log pseudolikelihoods instead of loglikelihoods.
 The coefficients for child and camper can be interpreted as follows:
 For each unit increase of child the expected log count of the response variable decreases by 1.043.
 Being a camper increases the expected log count by .834.
 The inflate coefficient for persons suggests that for each unit increase in person the log odds of an inflated zero decrease by .564.
We can use the margins to help understand our model. We will first compute the expected counts for the categorical variable camper while holding the continuous variable child at its mean value using the atmeans option.
margins camper, atmeans Adjusted predictions Number of obs = 250 Model VCE : Robust Expression : Predicted number of events, predict() at : child = .684 (mean) 0.camper = .412 (mean) 1.camper = .588 (mean) persons = 2.528 (mean)   Deltamethod  Margin Std. Err. z P>z [95% Conf. Interval] + camper  0  1.289132 .4393168 2.93 0.003 .4280866 2.150177 1  2.968305 .619339 4.79 0.000 1.754423 4.182187 
The expected count for the number of fish caught by noncampers is 1.289 while for campers it is 2.968 at the means of child and persons.
Using the dydx option computes the difference in expected counts between camper = 0 and camper = 1 while still holding child at its mean of .684 and persons at its mean of 2.528.
margins, dydx(camper) atmeans Conditional marginal effects Number of obs = 250 Model VCE : Robust Expression : Predicted number of events, predict() dy/dx w.r.t. : 1.camper at : child = .684 (mean) 0.camper = .412 (mean) 1.camper = .588 (mean) persons = 2.528 (mean)   Deltamethod  dy/dx Std. Err. z P>z [95% Conf. Interval] + 1.camper  1.679173 .7754611 2.17 0.030 .1592975 3.199049  Note: dy/dx for factor levels is the discrete change from the base level.
The difference in the number of fish caught by campers and noncampers is 1.679, which is statistically significant.
One last margins command will give the expected counts for values of child from zero to three at both levels of camper.
margins, at(child=(0(1)3) camper=(0/1)) vsquish Predictive margins Number of obs = 250 Model VCE : Robust Expression : Predicted number of events, predict() 1._at : child = 0 camper = 0 2._at : child = 0 camper = 1 3._at : child = 1 camper = 0 4._at : child = 1 camper = 1 5._at : child = 2 camper = 0 6._at : child = 2 camper = 1 7._at : child = 3 camper = 0 8._at : child = 3 camper = 1   Deltamethod  Margin Std. Err. z P>z [95% Conf. Interval] + _at  1  2.616441 .6470522 4.04 0.000 1.348242 3.88464 2  6.024516 2.159288 2.79 0.005 1.79239 10.25664 3  .922172 .4142303 2.23 0.026 .1102954 1.734048 4  2.123358 .4771534 4.45 0.000 1.188154 3.058561 5  .3250221 .2611556 1.24 0.213 .1868335 .8368777 6  .7483834 .3929987 1.90 0.057 .0218798 1.518647 7  .114555 .1351887 0.85 0.397 .1504101 .37952 8  .2637699 .2365495 1.12 0.265 .1998587 .7273984  marginsplot
The expected number of fish caught goes down as the number of children goes up for both people with and without campers.
A number of model fit indicators are available using the fitstat command, which is part of the spostado utilities by J. Scott Long and Jeremy Freese (search spostado).
fitstat Measures of Fit for zip of count LogLik Intercept Only: 1127.023 LogLik Full Model: 1031.608 D(244): 2063.217 LR(4): 190.829 Prob > LR: 0.000 McFadden's R2: 0.085 McFadden's Adj R2: 0.079 ML (CoxSnell) R2: 0.534 CraggUhler(Nagelkerke) R2: 0.534 AIC: 8.301 AIC*n: 2075.217 BIC: 715.980 BIC': 168.743 BIC used by Stata: 2090.824 AIC used by Stata: 2073.217
Things to consider
 Since zip has both a count model and a logit model, each of the two models should have good predictors. The two models do not necessarily need to use the same predictors.
 Problems of perfect prediction, separation or partial separation can occur in the logistic part of the zeroinflated model.
 Count data often use exposure variables to indicate the number of times the event could have happened. You can incorporate exposure into your model by using the exposure() option.
 It is not recommended that zeroinflated Poisson models be applied to small samples. What constitutes a small sample does not seem to be clearly defined in the literature.
 PseudoRsquared values differ from OLS Rsquareds, please see FAQ: What are pseudo Rsquareds? for a discussion on this issue.
 In times past, the Vuong test had been used to test whether a zeroinflated Poisson model or a Poisson model (without the zeroinflation) was a better fit for the data. However, this test is no longer considered valid. Please see The Misuse of The Vuong Test For NonNested Models to Test for ZeroInflation by Paul Wilson for further information.
See also
References
 Cameron, A. Colin and Trivedi, P.K. (2009) Microeconometrics using stata. College Station, TX: Stata Press.
 Long, J. Scott, & Freese, Jeremy (2006). Regression Models for Categorical Dependent Variables Using Stata (Second Edition). College Station, TX: Stata Press.
 Long, J. Scott (1997). Regression Models for Categorical and Limited Dependent Variables. Thousand Oaks, CA: Sage Publications.