Version info: Code for this page was tested in Stata 12.

Zero-truncated negative binomial regression is used to model count data for which the value zero
cannot occur and when there is evidence of over dispersion .

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-truncated negative binomial

Example 1.

A study of the length of hospital stay, in days, as a function of age, kind of health insurance and whether
or not the patient died while in the hospital.
Length of hospital stay is recorded as a minimum of at least one day.

Example 2.

A study of the number of journal articles published by tenured faculty as a function of
discipline (fine arts, science, social science, humanities, medical,
etc). To get tenure faculty must publish, i.e., there are no tenured faculty with
zero publications.

Example 3.

A study by the county traffic court on the number of tickets received by teenagers
as predicted by school performance, amount of driver training and gender. Only individuals
who have received at least one citation are in the traffic court files.

Description of the data

Let’s pursue Example 1 from above.

We have a hypothetical data file, ztp.dta with 1,493 observations.
The variable describing length of hospital visit is stay.
The variable age gives the age group from 1 to 9 which will be treated as
interval in this example.
The variables hmo and died are binary indicator variables
for HMO insured patients and patients who died while in hospital, respectively. These are the same
data as were used in the ztp example.

Let’s look at the data.

use https://stats.idre.ucla.edu/stat/stata/dae/ztp, clear

summarize stay

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
        stay |      1493    9.728734    8.132908          1         74

histogram stay, discretetab1 age hmo died

-> tabulation of age  

  Age Group |      Freq.     Percent        Cum.
------------+-----------------------------------
          1 |          6        0.40        0.40
          2 |         60        4.02        4.42
          3 |        163       10.92       15.34
          4 |        291       19.49       34.83
          5 |        317       21.23       56.06
          6 |        327       21.90       77.96
          7 |        190       12.73       90.69
          8 |         93        6.23       96.92
          9 |         46        3.08      100.00
------------+-----------------------------------
      Total |      1,493      100.00

-> tabulation of hmo  

        hmo |      Freq.     Percent        Cum.
------------+-----------------------------------
          0 |      1,254       83.99       83.99
          1 |        239       16.01      100.00
------------+-----------------------------------
      Total |      1,493      100.00

-> tabulation of died  

       died |      Freq.     Percent        Cum.
------------+-----------------------------------
          0 |        981       65.71       65.71
          1 |        512       34.29      100.00
------------+-----------------------------------
      Total |      1,493      100.00

Analysis methods you might consider

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

Zero-truncated negative binomial regression

The tnbreg command will analyze models that are left truncated on any
value not just zero. The ztnb command previously was used for
zero-truncated negative binomial regression, but is no longer supported in
Stata12 and has been superseded by tnbreg.

tnbreg stay age i.hmo i.died, ll(0)

Fitting truncated Poisson model:

Iteration 0:   log likelihood = -6908.7992  
Iteration 1:   log likelihood = -6908.7991  

Fitting constant-only model:

Iteration 0:   log likelihood =  -4817.852  
Iteration 1:   log likelihood = -4778.7604  
Iteration 2:   log likelihood = -4770.8734  
Iteration 3:   log likelihood =  -4770.848  
Iteration 4:   log likelihood =  -4770.848  

Fitting full model:

Iteration 0:   log likelihood = -4755.5912  
Iteration 1:   log likelihood = -4755.2798  
Iteration 2:   log likelihood = -4755.2796  

Truncated negative binomial regression            Number of obs   =       1493
Truncation point: 0                               LR chi2(3)      =      31.14
Dispersion     = mean                             Prob > chi2     =     0.0000
Log likelihood = -4755.2796                       Pseudo R2       =     0.0033

------------------------------------------------------------------------------
        stay |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |  -.0156929    .013107    -1.20   0.231    -.0413822    .0099964
       1.hmo |  -.1470576   .0592161    -2.48   0.013     -.263119   -.0309962
      1.died |  -.2177714   .0461605    -4.72   0.000    -.3082442   -.1272985
       _cons |   2.408328    .071982    33.46   0.000     2.267245     2.54941
-------------+----------------------------------------------------------------
    /lnalpha |  -.5686389   .0551506                     -.6767321   -.4605457
-------------+----------------------------------------------------------------
       alpha |   .5662957   .0312316                      .5082753    .6309393
------------------------------------------------------------------------------
Likelihood-ratio test of alpha=0:  chibar2(01) = 4307.04 Prob>=chibar2 = 0.000

The output looks very much like the output from an OLS regression:

Looking through the results we see the following:

We can also use the margins command to help understand our model. We will first
compute the expected counts for the categorical variable hmo while holding the continuous
variables age and died at their mean values using the atmeans option. 
Please note that the unit for stay is days and not log days for the
margins command.

margins hmo, atmeans

Adjusted predictions                              Number of obs   =       1493
Model VCE    : OIM

Expression   : Predicted number of events, predict()
at           : age             =    5.233758 (mean)
               0.hmo           =    .8399196 (mean)
               1.hmo           =    .1600804 (mean)
               0.died          =    .6570663 (mean)
               1.died          =    .3429337 (mean)

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         hmo |
          0  |   9.502109   .2258589    42.07   0.000     9.059433    9.944784
          1  |   8.202641   .4478629    18.32   0.000     7.324845    9.080436
------------------------------------------------------------------------------

The expected stay for non-HMO patients was 9.502, days while it was 8.203 days for HMO patients.

Using the dydx option computes the difference in expected counts between HMO and non-HMO
patients while still holding the other variables at their mean value.

margins, dydx(hmo) atmeans

Conditional marginal effects                      Number of obs   =       1493
Model VCE    : OIM

Expression   : Predicted number of events, predict()
dy/dx w.r.t. : 1.hmo
at           : age             =    5.233758 (mean)
               0.hmo           =    .8399196 (mean)
               1.hmo           =    .1600804 (mean)
               0.died          =    .6570663 (mean)
               1.died          =    .3429337 (mean)

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       1.hmo |  -1.299468   .4985062    -2.61   0.009    -2.276522   -.3224139
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

As shown above, HMO patients spend 1.299 days less in the hospital than non-HMO patients when the
other variables are held at their mean levels.

One last margins command will give the expected counts for values of age variable from one
through nine while averaging across the two levels of hmo and died. We will
show these results even though age was not statistically significant.

margins, at(age=(1(1)9)) vsquish

Predictive margins                                Number of obs   =       1493
Model VCE    : OIM

Expression   : Predicted number of events, predict()
1._at        : age             =           1
2._at        : age             =           2
3._at        : age             =           3
4._at        : age             =           4
5._at        : age             =           5
6._at        : age             =           6
7._at        : age             =           7
8._at        : age             =           8
9._at        : age             =           9

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         _at |
          1  |   9.984497   .5918896    16.87   0.000     8.824414    11.14458
          2  |   9.829034   .4654886    21.12   0.000     8.916693    10.74138
          3  |   9.675992   .3508834    27.58   0.000     8.988273    10.36371
          4  |   9.525333   .2575035    36.99   0.000     9.020636    10.03003
          5  |    9.37702   .2076088    45.17   0.000     8.970114    9.783926
          6  |   9.231016   .2248183    41.06   0.000      8.79038    9.671652
          7  |   9.087286   .2930141    31.01   0.000     8.512989    9.661583
          8  |   8.945793    .382671    23.38   0.000     8.195772    9.695815
          9  |   8.806504   .4794145    18.37   0.000     7.866868    9.746139
------------------------------------------------------------------------------

A number of model fit indicators are available using the estat ic
command.

estat ic


-----------------------------------------------------------------------------
       Model |    Obs    ll(null)   ll(model)     df          AIC         BIC
-------------+---------------------------------------------------------------
           . |   1493   -4770.848    -4755.28      5     9520.559    9547.102
-----------------------------------------------------------------------------
               Note:  N=Obs used in calculating BIC; see [R] BIC note

Things to consider

See Also

References