Stata Data Analysis Examples: Zero-Truncated Poisson



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Stata Data Analysis Examples
Zero-Truncated Poisson

Examples of Zero-Truncated Poisson

Example 1. A study of 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 length of hospital stay variable 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 the hospital, respectively.

Let's look at the data.

use http://www.ats.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, discrete



tab1 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

Some Strategies You Might Be Tempted To Try

Before we show how you can analyze this with a zero-truncated Poisson analysis, let's consider some other methods that you might use.

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.
Poisson Regression - Ordinary Poisson regression will have difficulty with zero-truncated data. It will try to predict zero counts even though there are no zero values.
Zero-truncated Negative Binomial Regression - This might actually be a good choice especially if the data are over-dispersed. See the Data Analysis Example for ztnb.

Stata Zero-Truncated Poisson Analysis

ztp stay age hmo died

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

Zero-truncated Poisson regression                 Number of obs   =       1493
                                                  LR chi2(3)      =     181.13
                                                  Prob > chi2     =     0.0000
Log likelihood = -6908.7991                       Pseudo R2       =     0.0129

------------------------------------------------------------------------------
        stay |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   -.014442   .0050347    -2.87   0.004    -.0243099   -.0045742
         hmo |  -.1359033   .0237419    -5.72   0.000    -.1824365   -.0893701
        died |  -.2037709   .0183728   -11.09   0.000     -.239781   -.1677608
       _cons |   2.435808   .0273324    89.12   0.000     2.382238    2.489379
------------------------------------------------------------------------------

The output looks very much like the output from an OLS regression. The output begins the iteration log giving the values of the log likelihoods starting with a model that has no predictors. The last value in the log is the final value of the log likelihood for the full model and is repeated below.

Next comes the header information. On the right-hand side the number of observations used (1493) is given along with the likelihood ratio chi-squared with three degrees of freedom for the full model, followed by the p-value for the chi-square. The model, as a whole, is statistically significant. The header also includes a pseudo-R² which is very low in this example (0.0129).

Below the header you will find the zero-truncated poisson coefficients for each of the variables along with standard errors, z-scores, p-values and 95% confidence intervals for the coefficients.

Now, just to be on the safe side, let's rerun the ztp command with the robust option in order to obtain robust standard errors for the zero-truncated Poisson coefficients.

ztp stay age hmo died, robust

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

Zero-truncated Poisson regression                 Number of obs   =       1493
                                                  Wald chi2(3)    =      25.65
                                                  Prob > chi2     =     0.0000
Log pseudolikelihood = -6908.7991                 Pseudo R2       =     0.0129

------------------------------------------------------------------------------
             |               Robust
        stay |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   -.014442   .0121867    -1.19   0.236    -.0383276    .0094436
         hmo |  -.1359033   .0520484    -2.61   0.009    -.2379163   -.0338902
        died |  -.2037709   .0491608    -4.14   0.000    -.3001242   -.1074175
       _cons |   2.435808   .0708745    34.37   0.000     2.296897     2.57472
------------------------------------------------------------------------------

Using the robust option has resulted in a fairly large change in the model chi-square, which is now a Wald chi-square, based on log pseudolikelihoods, instead of a likelihood ratio chi-square.

In the main body of the output are the zero-truncated poisson coefficients, robust standard errors, z-scores, p-values and 95% confidence intervals for the coefficients. The variable age was significant without the robust option and is not significant with it. The robust standard errors attempt to adjust for heterogeneity in the model.

Zero-truncated poisson models can also display results as incidence rate ratios using the irr option.

ztp, irr

Zero-truncated Poisson regression                 Number of obs   =       1493
                                                  Wald chi2(3)    =      25.65
                                                  Prob > chi2     =     0.0000
Log pseudolikelihood = -6908.7991                 Pseudo R2       =     0.0129

------------------------------------------------------------------------------
             |               Robust
        stay |        IRR   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .9856618    .012012    -1.19   0.236     .9623976    1.009488
         hmo |   .8729271   .0454345    -2.61   0.009     .7882687    .9666776
        died |   .8156492   .0400979    -4.14   0.000     .7407262    .8981506
------------------------------------------------------------------------------

Sample Write-Up of the Analysis

Before we begin the sample write-up we need to get the output into a form more acceptable for publication. In order to go back to the non-exponentiated version of the coefficients we will quietly rerun the ztp command. The estout command (findit estout by Ben Jann of ETH Zurich), will get us close to what we want.

quietly ztp

estout, cells(b(star fmt(%8.2f)) se(par fmt(%8.2f))) stats(ll chi2, fmt(%8.2f))

        b/se
stay    
age     -0.01
        (0.01)
hmo     -0.14**
        (0.05)
died    -0.20***
        (0.05)
_cons   2.44***
        (0.07)
ll      -6908.80
chi2    25.65

With a little bit of manual editing we can produce an acceptable table of the output.

model        
stay              -0.01
                  (0.01)
hmo               -0.14**
                  (0.05)
died              -0.20***
                  (0.05)
constant           2.44***
                  (0.07)            

log psuedo-
   likelihood  -6908.80
chi-squared       25.65

legend: coefficient/(standard error)  *** p<0.001

The zero-truncated Poisson regression model predicting length of hospital stay from age, hmo membership and death during the hospital stay was statistically significant (chi-squared = 25.65, df = 3, p<.001). The predictors hmo and died were each statically significant. The effect of age was not significant at the .05 level. For these data the expected log count for those enrolled in an hmo was -0.14 that of those not so enrolled. This amounts to a difference of about 1.25 days. Patients who died during the hospital stay had an expected log count difference of -0.20 or about 1.9 days.

Cautions, Flies in the Ointment

It is not recommended that zero-truncated poisson models be applied to small samples. What constitutes a small sample does not seem to be clearly defined in the literature.

Stata Data Analysis Examples Zero-Truncated Poisson