Poisson and negative binomial regression
What question does this help answer?
“How does the expected event count or event rate differ across predictors?”
When to use it
Use count regression for outcomes such as number of visits, infections, falls, prescriptions, or events per person-time. Poisson regression is a common starting point; negative binomial regression is often useful when variability exceeds the Poisson mean-variance relationship.
When not to use it
Do not force a count model onto a continuous measurement merely because it is skewed. Excess zeros, recurrent-event processes, clustering, or time-varying rates may require extensions.
How it works in plain language
The model predicts a nonnegative event count on a log scale. With a log exposure-time offset, coefficients can be interpreted as rate ratios. Negative binomial regression adds flexibility for overdispersion.
Why Mallard may recommend it
Mallard may recommend these models when your outcome is an event count or incidence rate rather than a simple yes/no outcome.
Key assumptions
- The outcome and exposure-time definition match the scientific question.
- Overdispersion is assessed.
- Independence or clustering is handled appropriately.
- Structural excess zeros are considered when relevant.
Excess zeros
Some counts have more zeros than a Poisson or negative binomial model predicts, for example visits to a service that some people could never use. A zero-inflated model treats the data as a mixture of two groups: structural zeros, people who could not have had any event, modelled with a logistic part, and people at risk, whose counts follow a Poisson or negative binomial model that can itself produce zeros. The count part's rate ratio then applies to people at risk, not to the whole population, and the logistic part describes who is a structural zero. It suits data where a group that cannot experience the event is plausible on subject-matter grounds; excess zeros alone are not evidence of such a group, since a negative binomial model often accommodates them. A hurdle model is the alternative when every zero is taken to come from one process and every positive count from another. The two parts answer different questions, so say which one answers the study's.
What to watch out for
A Poisson model can underestimate uncertainty when the data are overdispersed. Choosing a negative binomial or other count model should be based on the data-generating structure, not only on which model produces smaller p-values.
Clinical example
Compare emergency-department visit rates per person-year between two care programs while adjusting for age and comorbidity.
How to interpret it
An incidence-rate ratio of 0.80 suggests an estimated 20% lower event rate for the exposed group, conditional on modeled covariates and the offset.
What adjusted rate ratios look like
- Too many zeros and/or highly skewed? A tutorial on modelling health behaviour as count data with Poisson and negative binomial regression (opens in a new tab) · Health Psychol Behav Med · doi 10.1080/21642850.2021.1920416
- Segmented Regression in an Interrupted Time Series Study Design (opens in a new tab) · Anesth Analg · doi 10.1213/ANE.0000000000005269