Annotated Mplus Output
Zero Inflated Poisson Regression

This page shows an example of zero inflated poisson regression with footnotes explaining the output. First an example is shown using Stata, and then an example is shown using Mplus, to help you relate the output you are likely to be familiar with (Stata) to output that may be new to you (Mplus). We suggest that you view this page using two web browsers so you can show the page side by side showing the Stata output in one browser and the corresponding Mplus output in the other browser. 

This example is from the Mplus User's Guide (example 3.8) and we suggest that you see the Mplus User's Guide for more details about this example. We thank the kind people at Muthén & Muthén for permission to use examples from their manual.

Example Using Stata

Here is a logit regression example using Stata with two continuous predictors x1 and x3 used to predict a binary outcome variable, u1. These same predictors are used to predict the zero inflation in u1 as well.

infile u1 x1 x3 using http://www.ats.ucla.edu/stat/mplus/output/ex3.8.dat, clear

zip u1 x1 x3, inflate(x1 x3)

<some output omitted>

Zero-inflated Poisson regression                  Number of obs   =        500
                                                  Nonzero obs     =        282
                                                  Zero obs        =        218

Inflation model = logit                           LR chi2(2)      =     209.64
Log likelihood  =  -758.855                       Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
u1           |
          x1 |   .5478153C   .0398304    13.75   0.000     .4697492    .6258814
          x3 |   .3087748C   .0416855     7.41   0.000     .2270728    .3904768
       _cons |    1.06087D   .0380128    27.91   0.000     .9863659    1.135373
-------------+----------------------------------------------------------------
inflate      |
          x1 |   1.629311E   .2163157     7.53   0.000      1.20534    2.053282
          x3 |   1.054722E   .1717856     6.14   0.000      .718028    1.391415
       _cons |  -.9500156F   .1674828    -5.67   0.000    -1.278276   -.6217554
------------------------------------------------------------------------------

estat ic

------------------------------------------------------------------------------
       Model |    Obs    ll(null)   ll(model)     df          AIC         BIC
-------------+----------------------------------------------------------------
           . |    500   -863.6735    -758.855A      6      1529.71B    1554.998B
------------------------------------------------------------------------------

The output is labeled with superscripts to help you relate the later Mplus output to this Stata output. To summarize the output, both predictors in this model, x1 and x3, are significantly related to the outcome variable, u1, and both predictors are related to the zero inflation in u1. The estat ic command produces fit indices for the model including the log likelihood for the empty (null) model, the log likelihood for the model, as well as the AIC and BIC fit indices.

---

Mplus Example #1

Here is the same example illustrated in Mplus based on the ex3.8.dat data file. The output has been edited and condensed to save space.

TITLE:
  this is an example of a zero-inflated
  Poisson regression for a count dependent
  variable with two covariates
DATA:
  FILE IS ex3.8.dat;
VARIABLE:
  NAMES ARE u1 x1 x3;
  COUNT IS u1 (i);
MODEL:
  u1 ON x1 x3;
  u1#1 ON x1 x3;

SUMMARY OF ANALYSIS
Number of observations                                         500

TESTS OF MODEL FIT

Loglikelihood

          H0 Value                        -758.855A

Information Criteria

          Number of Free Parameters              6
          Akaike (AIC)                    1529.710B
          Bayesian (BIC)                  1554.998B
          Sample-Size Adjusted BIC        1535.953
            (n* = (n + 2) / 24)

MODEL RESULTS
                   Estimates     S.E.  Est./S.E.

 U1         ON
    X1                 0.548C    0.041     13.308
    X3                 0.309C    0.041      7.619

 U1#1       ON
    X1                 1.629E    0.237      6.871
    X3                 1.055E    0.181      5.827

 Intercepts
    U1#1              -0.950F    0.170     -5.596
    U1                 1.061D    0.041     26.115

1. This is the log likelihood value associated with the model (see the ll(model) from the estat ic command in Stata.
2. These are the AIC and BIC values, see the AIC and BIC values from the estat ic command in Stata.
3. These are the coefficients for the poisson model expressing the relationship between x1 x3 and u1, the same as those from the Stata zip command.
4. This is the intercept for the poisson model, the same as that from the Stata poisson command.
5. These are the coefficients for the zero inflation part of the model model expressing the relationship between x1 x3 and the zero inflation in u1, the same as those from the inflate part of the model from the Stata zip command.
6. This is the intercept for the zero inflation part of the model, the same as the intercept from the inflate part of the model from the Stata zip command.