Counts such as accidents, inquiries, and visits require attention to the mean, variance, and exposure.
Accidents, customer inquiries, and hospital visits are nonnegative integer counts. Their variance often grows with the mean, and organizations may be observed for different lengths of time or serve different population sizes.
Count regression models expected counts or rates while respecting these features.
Key question
How should we model an event count when variability and opportunity for exposure differ across observations?
What does Poisson regression model?
Poisson mean model
log(E[Y|X]) = Xβ
Exponentiated coefficients are incidence-rate ratios. An IRR above 1 indicates a higher expected rate; an IRR below 1 indicates a lower expected rate, holding other predictors constant.
Counts are whole and never negative
0, 1, 2, …The outcome piles up on small integers and has a hard floor at zero.
The log link
exp(Xβ)Exponentiating keeps the expected count positive and makes exp(β) a multiplier.
Overdispersion
σ² > μWhen the spread beats the mean, Poisson standard errors come out too small.
Why is overdispersion a problem?
Real data often vary more than this because of unobserved heterogeneity, clustering, or excess zeros. This is overdispersion.
Why it matters
If overdispersion is ignored, standard errors can be too small and effects can appear more certain than they are.
What does negative-binomial regression change?
| Model | Variance pattern | Typical use |
|---|---|---|
| Poisson | variance roughly equals the mean | counts with limited extra variation |
| Negative binomial | variance exceeds the mean | overdispersed counts |
| Zero-inflated / hurdle | more zeros than a single count process explains | data generated by separate zero and positive-count mechanisms |
Why are exposure and offsets needed?
Accidents per operating hour
A plant observed for 2,000 hours has more opportunity for accidents than one observed for 500 hours. Include log(exposure) as an offset so the model compares accident rates rather than raw counts.
Exposure must match the question
Time, population, distance, or number of transactions can serve as exposure only when it represents the amount of opportunity for events to occur.
Change the mean, variance, and exposure
Increase overdispersion and compare Poisson with negative-binomial behavior. Then change exposure and observe the difference between an expected count and an expected rate.
Expected count
2.72
Variance
2.72
Expected rate
2.72
Expected count equals expected rate multiplied by exposure. Changing exposure does not change the rate itself.
What to watch
Two observations can have different expected counts but the same expected rate when their exposure differs.