Poisson and Negative Binomial Regression
Model count outcomes such as visits, patents or accidents. Fit Poisson and negative binomial (NB2) regressions, read incidence rate ratios, and test for overdispersion to choose between them.
Updated · By Bibin XavierData 500 × 11
| IDnumeric | Log_Wagenumeric | Educationnumeric | Experiencenumeric | Femalenumeric · coded | Unionnumeric · coded | Distance_Collegenumeric | Father_Educnumeric | Employednumeric · coded | Doctor_Visitsnumeric | Agenumeric |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2.47 | 12 | 0 | 0 | 0 | 5.7 | 9 | 0 | 1 | 18 |
| 2 | 1.83 | 14 | 21 | 0 | 1 | 0 | 9 | 1 | 0 | 41 |
| 3 | 2.56 | 11 | 12 | 0 | 0 | 0 | 12 | 0 | 0 | 29 |
| 4 | 2.02 | 10 | 14 | 0 | 0 | 27.2 | 13 | 0 | 0 | 30 |
| 5 | 1.75 | 12 | 21 | 1 | 1 | 21.6 | 9 | 1 | 1 | 39 |
| 6 | 2.73 | 10 | 10 | 0 | 0 | 0 | 9 | 1 | 0 | 26 |
| 7 | 1.54 | 11 | 14 | 1 | 0 | 13.8 | 10 | 0 | 0 | 31 |
| 8 | 1.46 | 11 | 9 | 1 | 1 | 7.2 | 8 | 1 | 2 | 26 |
Variables and options
Results
500 rows · example dataPoisson regression
| Variable | Coefficient | SE | z | p | IRR | 95% CI for IRR |
|---|---|---|---|---|---|---|
| Constant | -0.1069 | 0.2560 | -0.417 | .676 | 0.899 | [0.544, 1.484] |
| Age | 0.0222 | 0.0049 | 4.535 | < .001 | 1.022 | [1.013, 1.032] |
| Female | 0.3274 | 0.0747 | 4.385 | < .001 | 1.387 | [1.198, 1.606] |
| Education | -0.0259 | 0.0205 | -1.263 | .207 | 0.974 | [0.936, 1.014] |
| Employed | -0.1555 | 0.0783 | -1.986 | .047 | 0.856 | [0.734, 0.998] |
Note. Log-likelihood = -814.71, AIC = 1639.4, BIC = 1660.5.
Negative binomial (NB2) regression
| Variable | Coefficient | SE | z | p | IRR | 95% CI for IRR |
|---|---|---|---|---|---|---|
| Constant | -0.1198 | 0.2972 | -0.403 | .687 | 0.887 | [0.495, 1.588] |
| Age | 0.0222 | 0.0058 | 3.824 | < .001 | 1.022 | [1.011, 1.034] |
| Female | 0.3273 | 0.0876 | 3.735 | < .001 | 1.387 | [1.168, 1.647] |
| Education | -0.0247 | 0.0239 | -1.031 | .303 | 0.976 | [0.931, 1.022] |
| Employed | -0.1608 | 0.0926 | -1.737 | .082 | 0.851 | [0.710, 1.021] |
Note. Log-likelihood = -801.65, AIC = 1615.3, BIC = 1640.6, α = 0.254 (SE 0.065).
Overdispersion tests
| Test | Statistic | p | Conclusion |
|---|---|---|---|
| Cameron-Trivedi (α in Var = μ + αμ²) | α = 0.235, t = 3.90 | < .001 | Overdispersion |
| Likelihood ratio (α = 0) | χ̄² = 26.12 | < .001 | Prefer negative binomial |
Interpretation
The data are overdispersed, so the negative binomial model is preferred. In that model, each one-unit increase in Age multiplies the expected count by 1.02 (+2%); each one-unit increase in Female multiplies the expected count by 1.39 (+39%).
How to cite this tool
Using these results in a thesis, journal article, report or class assignment? Please cite this free tool so others can find it. It keeps the site free.
APA 7
Xavier, B. (2026). Poisson and Negative Binomial Regression [Web application]. FreeStatisticalTools.com. https://freestatisticaltools.com/poisson-negative-binomial-regression/
BibTeX
@software{xavier_poissonnegativebinomialregression_2026,
author = {Xavier, Bibin},
title = {Poisson and Negative Binomial Regression},
publisher = {FreeStatisticalTools.com},
year = {2026},
url = {https://freestatisticaltools.com/poisson-negative-binomial-regression/},
note = {Web application}
}Suggested sentence for your methods section: “The analysis was carried out with the free Poisson and Negative Binomial Regression tool on FreeStatisticalTools.com (Xavier, 2026).”
What is Poisson and negative binomial regression?
Count outcomes, such as the number of doctor visits or patents, call for count regression. Poisson regression assumes the variance equals the mean; negative binomial regression adds a dispersion parameter for the common case where the variance is larger.
When to use Poisson and negative binomial regression
Use count models when the outcome is a non-negative integer. Negative binomial is preferred when the variance exceeds the mean, which is common.
How to use this tool
- Enter your data. Paste columns from Excel, Google Sheets or SPSS with variable names in the first row, or upload a CSV or Excel file. The example data shows the expected layout.
- Choose the variables. Select the count outcome and regressors.
- Set the options. Fill in model.
- Read the results. Results update automatically. Read the plain-English interpretation, then use Copy table to paste APA-style tables into Word and Copy for the results sentence.
How to report Poisson and negative binomial regression in APA style
Report incidence rate ratios with confidence intervals, the dispersion parameter and the test used to choose the model.
Frequently asked questions
How do I interpret an incidence rate ratio?
An IRR of 1.20 means a one-unit increase in the regressor multiplies the expected count by 1.20, a 20% increase.
How do I detect overdispersion?
Compare the variance with the mean, check the Pearson dispersion statistic, or use the Cameron-Trivedi test; a significant negative binomial α also signals overdispersion.
What about many zeros?
Excess zeros may need a zero-inflated or hurdle model; the negative binomial often handles moderate excess zeros well.
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