Mplus Data Analysis Examples
Zero-inflated Poisson Regression

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

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

See Also

References