Version info: Code for this page was tested in Mplus version 6.12.
Zero-inflated 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 follow-up analyses.
Examples of zero-inflated 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. The associated dataset can be found here.
We have data on 250 groups that went to a park. Each group was questioned before leaving the park 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). The outcome variable of interest will be the number of fish caught. Even though the question about the number of fish caught was asked to everyone, it does not mean that everyone went fishing. What would be the reason for someone to report a zero count? Was it because this person was unlucky and didn’t catch any fish, or was it because this person didn’t go fishing at all? If a person didn’t go fishing, the outcome would be always zero. Otherwise, if a person went to fishing, the count could be zero or non-zero. So we can see that there seemed to be two processes that would generate zero counts: unlucky in fishing or didn’t go fishing.
Let’s first look at the data. We will start with reading in the data and the descriptive statistics and plots. This helps us understand the data and give us some hint on how we should model the data.
Let’s look at the data.
Data:
File is C:fish.dat;
Variable:
Names are
nofish livebait camper persons child xb zg count;
Missing are all (-9999);
Usevariables are
camper persons child count;
Analysis:
type = basic;
Plot:
type = plot1;
ESTIMATED SAMPLE STATISTICS
Means
CAMPER PERSONS CHILD COUNT
________ ________ ________ ________
1 0.588 2.528 0.684 3.296
Covariances
CAMPER PERSONS CHILD COUNT
________ ________ ________ ________
CAMPER 0.242
PERSONS -0.026 1.233
CHILD -0.014 0.515 0.720
COUNT 0.730 2.856 -1.670 134.832
Correlations
CAMPER PERSONS CHILD COUNT
________ ________ ________ ________
CAMPER 1.000
PERSONS -0.048 1.000
CHILD -0.034 0.546 1.000
COUNT 0.128 0.221 -0.170 1.000
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.
- Zero-inflated Poisson Regression – The focus of this web page.
- Zero-inflated 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 non-normal and are not well estimated by OLS regression.
Zero-inflated poisson regression
In the syntax below, we have indicated that count is a count variable by using the count statement. The (i) option is used to indicate that we are specifying a zero-inflated poisson model. Without the (i) option, we would be estimating a poisson model without zero-inflation. Also, we use the usevariables statement to indicate that we are not using all of the variables in the data set in the current model. We have omitted the missing statement because we have no missing data in this data set. The default estimation method is MLR – maximum likelihood parameter estimates with standard errors and a chi-square test statistic that are robust to non-normality and non-independence of observations when used with type = complex. The MLR standard errors are computed using a sandwich estimator. This is what we generally call robust standard errors. To get the "regular" standard errors, we use the estimator = ml on the analysis statement. (In the next example, we will omit the analysis statement and obtain the robust standard errors.) Two regression equations are specified in the model statement: the first equation is the poisson model, predicting the count of fish using child and camper. The second equation is the logit model, indicated by count#1, predicting membership to the zero generating process using persons.
Data: File is C:fish.dat; Variable: Names are nofish livebait camper persons child xb zg count; Count is count(i); Usevariables are camper persons child count; Analysis: estimator = ml; Model: count on child camper; count#1 on persons; MODEL FIT INFORMATION Number of Free Parameters 5 Loglikelihood H0 Value -1031.608 Information Criteria Akaike (AIC) 2073.217 Bayesian (BIC) 2090.824 Sample-Size Adjusted BIC 2074.974 (n* = (n + 2) / 24) MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value COUNT ON CHILD -1.043 0.100 -10.430 0.000 CAMPER 0.834 0.094 8.908 0.000 COUNT#1 ON PERSONS -0.564 0.163 -3.463 0.001 Intercepts COUNT#1 1.297 0.374 3.470 0.001 COUNT 1.598 0.086 18.680 0.000
In the MODEL FIT INFORMATION portion of the output, you will find the log likelihood for the final model as well as a number of fit statistics. In the MODEL RESULTS section of the output you will find the poisson regression coefficients (estimates) for each of the variables, standard errors and the ratio of the estimate to its standard error. This can be used as a Z test, where values greater than 2 are considered to be statistically significant. Following these are logit coefficients for predicting excess zeros. In the above output, we see that child and camper are both significant predictors of count, and persons is a significant predictor in the logit model. Thus for each additional child, the log count of number of fish count decreases by 1.043. For each additional person, the log odds of membership to the excess zero-generating process decreases by 0.564.
Now let’s rerun the model without the analysis statement in order to obtain robust standard errors.
Data: File is C:fish.dat; Variable: Names are nofish livebait camper persons child xb zg count; Count is count(i); Missing are all (-9999); Usevariables are camper persons child count; Model: count on child camper; count#1 on persons;MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value COUNT ON CHILD -1.043 0.389 -2.684 0.007 CAMPER 0.834 0.407 2.050 0.040 COUNT#1 ON PERSONS -0.564 0.288 -1.957 0.050 Intercepts COUNT#1 1.297 0.493 2.632 0.008 COUNT 1.598 0.293 5.456 0.000
Robust standard errors tend to be larger than "regular" standard errors. Still we see that child, camper, and persons are still significant predictors within their respective models.
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 zero-inflated 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 zero-inflated poisson models be applied to small samples. What constitutes a small sample does not seem to be clearly defined in the literature.
- Pseudo-R-squared values differ from OLS R-squareds, please see FAQ: What are pseudo R-squareds? for a discussion on this issue.
See Also
References
- Long, J. S. 1997. Regression Models for Categorical and Limited Dependent Variables.
Thousand Oaks, CA: Sage Publications.
- Cameron, A. Colin and Trivedi, P.K. (2009) Microeconometrics using Stata. College Station, TX: Stata Press.