Table of Contents
Zero-truncated Poisson regression is a statistical technique that can be applied to Stata data analysis to model count data with no zero values. This method is suitable for analyzing data with a high proportion of zero values, which are common in many real-world datasets. By modeling the remaining non-zero counts, this approach allows for more accurate estimation and prediction of the data. In Stata, this can be achieved through the use of specialized commands and functions, which can handle the unique characteristics of zero-truncated data. By incorporating this technique into Stata data analysis, researchers can gain a better understanding of their data and make more informed decisions based on the results.
Zero-Truncated Poisson Regression | Stata Data Analysis Examples
Version info: Code for this page was tested in Stata 12.
Zero-truncated poisson regression is used to model count data for which the value zero cannot occur.
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 Poisson regression
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, therefore, 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 https://stats.idre.ucla.edu/stat/data/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 | 1,493 100.00Analysis methods you might consider
Below is a list of some analysis methods you may have encountered.
Some of the methods listed are quite reasonable while others have either fallen out of favor or
have limitations.
Zero-truncated Poisson regression
You
can use the tpoisson command for zero-truncated poisson regression. The tpoisson
command will analyze models that are left truncated on any value not just zero. Additionally, since Cameron and Trivedi (2009)
recommend robust standard errors for poisson models we will include the vce(robust) option.
tpoisson stay age i.hmo i.died, ll(0) vce(robust)
Iteration 0: log pseudolikelihood = -6908.7992
Iteration 1: log pseudolikelihood = -6908.7991
Truncated Poisson regression Number of obs = 1493
Truncation point: 0 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
1.hmo | -.1359033 .0520484 -2.61 0.009 -.2379163 -.0338902
1.died | -.2037709 .0491608 -4.14 0.000 -.3001242 -.1074175
_cons | 2.435808 .0708745 34.37 0.000 2.296897 2.57472
------------------------------------------------------------------------------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.
For example we can find the expected number of days spent at the hospital across
age groups for the two hmo statuses and for the two died
statuses.
margins hmo, at(age=(1(1)9)) vsquish
Predictive margins Number of obs = 1493
Model VCE : Robust
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#hmo |
1 0 | 10.5493 .6310057 16.72 0.000 9.312549 11.78605
1 1 | 9.208768 .6261728 14.71 0.000 7.981491 10.43604
2 0 | 10.39804 .5078432 20.47 0.000 9.402685 11.39339
2 1 | 9.07673 .541332 16.77 0.000 8.015739 10.13772
3 0 | 10.24895 .3956085 25.91 0.000 9.473572 11.02433
3 1 | 8.946586 .4723194 18.94 0.000 8.020857 9.872315
4 0 | 10.102 .3016365 33.49 0.000 9.510801 10.69319
4 1 | 8.818307 .4242343 20.79 0.000 7.986823 9.649792
5 0 | 9.957153 .2419017 41.16 0.000 9.483034 10.43127
5 1 | 8.691868 .4019681 21.62 0.000 7.904025 9.479712
6 0 | 9.814385 .2375591 41.31 0.000 9.348778 10.27999
6 1 | 8.567242 .4072901 21.03 0.000 7.768969 9.365516
7 0 | 9.673664 .2867397 33.74 0.000 9.111665 10.23566
7 1 | 8.444403 .4370317 19.32 0.000 7.587837 9.30097
8 0 | 9.534961 .3653709 26.10 0.000 8.818847 10.25107
8 1 | 8.323325 .4848934 17.17 0.000 7.372952 9.273699
9 0 | 9.398246 .4560941 20.61 0.000 8.504318 10.29217
9 1 | 8.203984 .5445834 15.06 0.000 7.13662 9.271347
------------------------------------------------------------------------------
We can see that the number of days spent tends to decrease as we move up age groups
(the left column under _at#hmo) and that patients enrolled in an hmo (the right
column under _at#hmo) tend to spend fewer days at the hospital as well than
those not in hmos. For example, we expect that a non-hmo patient in age
group 1 to stay for 10.5493 days whereas an hmo patient in age group 1 is
expected to stay 9.2088 days. We can plot the number of days predicted by
age group and hmo status using the marginsplot command.
marginsplot, recast(line) recastci(rline) ciopts(lpattern(dash))margins died, at(age=(1(1)9)) vsquish Predictive margins Number of obs = 1493 Model VCE : Robust 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#died | 1 0 | 11.03216 .6419426 17.19 0.000 9.773975 12.29034 1 1 | 8.998372 .6434904 13.98 0.000 7.737154 10.25959 2 0 | 10.87398 .5155445 21.09 0.000 9.863529 11.88443 2 1 | 8.869352 .5506018 16.11 0.000 7.790192 9.948511 3 0 | 10.71806 .4019963 26.66 0.000 9.930166 11.50596 3 1 | 8.742181 .4700277 18.60 0.000 7.820943 9.663418 4 0 | 10.56439 .3102963 34.05 0.000 9.956216 11.17256 4 1 | 8.616833 .4064251 21.20 0.000 7.820255 9.413412 5 0 | 10.41291 .2583831 40.30 0.000 9.906489 10.91933 5 1 | 8.493283 .3658669 23.21 0.000 7.776197 9.210369 6 0 | 10.26361 .2648261 38.76 0.000 9.744559 10.78266 6 1 | 8.371504 .3535566 23.68 0.000 7.678546 9.064462 7 0 | 10.11645 .321958 31.42 0.000 9.48542 10.74747 7 1 | 8.251472 .3698185 22.31 0.000 7.526641 8.976303 8 0 | 9.971394 .4058928 24.57 0.000 9.175859 10.76693 8 1 | 8.13316 .4091532 19.88 0.000 7.331234 8.935086 9 0 | 9.828422 .5009702 19.62 0.000 8.846538 10.81031 9 1 | 8.016545 .463983 17.28 0.000 7.107155 8.925935 ------------------------------------------------------------------------------
We can see that the number of days spent tends to decrease as we move up age groups
again
(the left column under _at#hmo) and that patients died (the right
column under _at#hmo) tend to spend fewer days at the hospital than those that
did not die (died = 0). For example, we expect that a patient who
died in age
group 1 to stay for 8.998372 days whereas a patient who lived in age group 1 is
expected to stay 11.03216 days. We can plot the number of days predicted by
age group and died status using the marginsplot command.
marginsplot, recast(line) recastci(rline) ciopts(lpattern(dash))
The AIC and BIC are useful for model comparisons. You can look at these criteria
using the estat ic command.
estat ic
-----------------------------------------------------------------------------
Model | Obs ll(null) ll(model) df AIC BIC
-------------+---------------------------------------------------------------
. | 1493 -6999.365 -6908.799 4 13825.6 13846.83
-----------------------------------------------------------------------------
Note: N=Obs used in calculating BIC; see [R] BIC note
Things to consider
See Also
References
Cite this article
stats writer (2024). How can the concept of zero-truncated Poisson regression be applied to Stata data analysis?. PSYCHOLOGICAL SCALES. Retrieved from https://scales.arabpsychology.com/stats/how-can-the-concept-of-zero-truncated-poisson-regression-be-applied-to-stata-data-analysis/
stats writer. "How can the concept of zero-truncated Poisson regression be applied to Stata data analysis?." PSYCHOLOGICAL SCALES, 29 Jun. 2024, https://scales.arabpsychology.com/stats/how-can-the-concept-of-zero-truncated-poisson-regression-be-applied-to-stata-data-analysis/.
stats writer. "How can the concept of zero-truncated Poisson regression be applied to Stata data analysis?." PSYCHOLOGICAL SCALES, 2024. https://scales.arabpsychology.com/stats/how-can-the-concept-of-zero-truncated-poisson-regression-be-applied-to-stata-data-analysis/.
stats writer (2024) 'How can the concept of zero-truncated Poisson regression be applied to Stata data analysis?', PSYCHOLOGICAL SCALES. Available at: https://scales.arabpsychology.com/stats/how-can-the-concept-of-zero-truncated-poisson-regression-be-applied-to-stata-data-analysis/.
[1] stats writer, "How can the concept of zero-truncated Poisson regression be applied to Stata data analysis?," PSYCHOLOGICAL SCALES, vol. X, no. Y, ص Z-Z, June, 2024.
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