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Stata Data Analysis Examples
Zero-Truncated Negative Binomial

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 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 hospital, respectively. These are the same data as were used in the ztp example.

Let's look at the data.

Some Strategies You Might Be Tempted To Try

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.

Stata Zero-Truncated Negative Binomial Analysis

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 (0.0033).

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

In addition to the negative binomial coefficients, Stata also estimates the log of the over dispersion parameter alpha. Below lnalpha, Stata displays alpha. If alpha equals zero then there is no over dispersion. The likelihood ratio chi-square tests alpha. If the test is significant, as in this case, then zero-truncated negative binomial is preferred over zero-truncated poisson.

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.

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 robust standard errors attempt to adjust for heterogeneity in the model.

We again see the log alpha and alpha, but there is no likelihood ratio test of alpha because we are using log pseudolikelihoods. That's okay because we can use the likelihood ratio test from the model without the robust option.

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

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 ztnb command. The estout command (findit estout by Ben Jann of ETH Zurich), will get us close to what we want. With a little bit of manual editing we can produce an acceptable table of the output. I also manually added the likelihood ratio test for alpha from the non-robust ztnb. The zero-truncated negative binomial regression model predicting length of hospital stay from age, hmo membership and death during the hospital stay was statistically significant (chi-squared = 25.29, df = 3, p<.001). The likelihood ratio test for alpha, the over dispersion parameter, was significant (chi-squared = 4307.04, df = 1, p<.001) indicating that the zero-truncated negative binomial model is preferred over a zero-truncated poisson model. 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 -0.15 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 -0.22 or almost two days.

Cautions, Flies in the Ointment

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

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