Version info: Code for this page was tested in Mplus version 6.12.

Zero-inflated negative binomial regression is for modeling count variables
with excessive zeros and it is usually for
overdispersed count outcome variables. Furthermore, 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.

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 checking, verification of assumptions, model diagnostics or
potential follow-up analyses.

Examples of zero-inflated negative binomial regression

Example 1.

School administrators study the attendance behavior of high school juniors at two schools.
Predictors of the number of days of absence include gender of the student and standardized
test scores in math and language arts.

Example 2.

The state wildlife biologists want to model how many fish are being caught by fishermen
at a state park. Visitors are asked how long they stayed, 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 is 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

Before we show how you can analyze this with a zero-inflated negative binomial analysis, let’s
consider some other methods that you might use.

Zero-inflated negative binomial regression 

In the syntax below, we have indicated that count is a count
variable by using the count statement.  The (nbi) option is
used to indicate 2 things: that we are modeling our count variable with a
negative binomial distribution, and that we are specifying a zero-inflated model. 
Without the (nb) option we would be specifying a (zero-inflated) poisson
model, and without the (i) option, we would be estimating a negative
binomial 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. 
Two regression equations are specified in the model statement: the first
equation is the negative binomial 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(nbi);
    Usevariables are camper persons child count;
Analysis:
    estimator = ml;
Model:
    count on child camper;
    count#1 on persons;
    

MODEL RESULTS

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 COUNT      ON
    CHILD             -1.515      0.196     -7.747      0.000
    CAMPER             0.879      0.269      3.265      0.001

 COUNT#1    ON
    PERSONS           -1.666      0.679     -2.454      0.014

 Intercepts
    COUNT#1            1.603      0.836      1.916      0.055
    COUNT              1.371      0.256      5.353      0.000

 Dispersion
    COUNT              2.679      0.471      5.683      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
negative binomial 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 both child and camper are
significant predictor 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 1.515.  For each additional
person, the log odds of membership to the excess zero-generating process
decreases by1.666.

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(nbi);
    Usevariables are camper persons child count;
Model:
    count on child camper;
    count#1 on persons;
    
MODEL FIT INFORMATION

Number of Free Parameters                        6

Loglikelihood

          H0 Value                        -432.891
          H0 Scaling Correction Factor       1.762
            for MLR

Information Criteria

          Akaike (AIC)                     877.782
          Bayesian (BIC)                   898.911
          Sample-Size Adjusted BIC         879.890
            (n* = (n + 2) / 24)



MODEL RESULTS

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 COUNT      ON
    CHILD             -1.515      0.241     -6.280      0.000
    CAMPER             0.879      0.470      1.869      0.062

 COUNT#1    ON
    PERSONS           -1.666      0.431     -3.871      0.000

 Intercepts
    COUNT#1            1.603      0.665      2.410      0.016
    COUNT              1.371      0.389      3.520      0.000

 Dispersion
    COUNT              2.679      0.577      4.645      0.000

The robust standard errors attempt to adjust for heterogeneity in the model. Robust standard errors tend to be larger than “regular” standard errors for parameters in the negative binomial part of the model and smaller for parameters in the logit part of the model.  
We see that now camper is not statistically significant.

Things to consider

Here are some issues that you may want to consider in the course of your
research analysis.

References