Poisson Regression

Regression & Modelling

poisson

count-regression

offset

rate

Log-linear count regression with rates and offsets

Published

April 17, 2026

Introduction

Poisson regression models count data with a log link, making it the canonical GLM for events per unit exposure: hospital admissions per month, mutations per kilobase, falls per resident-year. The log-linear structure means coefficients are interpretable as log-rate ratios, and the model accommodates varying exposures naturally through an offset term. The defining assumption — equidispersion, \(\mathrm{Var}(Y) = \mathbb E[Y]\) — is also its main limitation: real count data are typically overdispersed, and either quasi-Poisson, negative binomial, or zero-inflated extensions are then required.

Prerequisites

A working understanding of the Poisson distribution, generalised linear models, the log link, and the role of an offset for modelling rates rather than counts.

Theory

The model is \(Y \sim \mathrm{Poisson}(\mu)\) with \(\log \mu = X^\top \beta\). Exponentiated coefficients are rate ratios: \(\exp(\beta_j)\) multiplies the expected count per unit increase in \(X_j\), all else equal. For rate-per-exposure problems, an offset of \(\log(\text{exposure})\) converts the model into a rate model:

\[\log \mathbb E[Y] = \log(\text{exposure}) + X^\top \beta.\]

The offset’s coefficient is fixed at 1, so the remaining \(\beta\) represent log-rate ratios per unit covariate change.

Equidispersion is the central assumption; if the variance exceeds the mean, standard errors are underestimated and inference is anti-conservative. Pearson chi-squared divided by residual degrees of freedom is the standard dispersion check; values much greater than 1 motivate quasi-Poisson or negative binomial.

Assumptions

Non-negative integer outcomes, independent observations, equidispersion (variance equal to mean), and a linear relationship between covariates and the log-rate.

R Implementation

library(broom)

# Simulated: event counts per patient-year
set.seed(2026)
d <- data.frame(
  age = rnorm(300, 60, 12),
  smoker = rbinom(300, 1, 0.3),
  exposure_years = runif(300, 0.5, 5)
)
d$events <- rpois(nrow(d),
  lambda = d$exposure_years * exp(-2 + 0.02 * d$age + 0.6 * d$smoker))

fit <- glm(events ~ age + smoker, data = d, family = poisson,
           offset = log(exposure_years))
tidy(fit, conf.int = TRUE, exponentiate = TRUE)
glance(fit)

# Check overdispersion
dispersion <- sum(resid(fit, type = "pearson")^2) / fit$df.residual
dispersion

Output & Results

tidy(fit, exponentiate = TRUE) returns rate ratios and confidence intervals; glance() provides AIC, deviance, and other model-level summaries. The dispersion ratio close to 1 confirms Poisson is appropriate; values \(\gg 1\) suggest overdispersion that needs different family handling.

Interpretation

A reporting sentence: “After adjusting for age (rate ratio 1.02 per year, 95 % CI 1.01 to 1.03), smokers had an event rate 1.81 times that of non-smokers (95 % CI 1.40 to 2.34, \(p < 0.001\)); the dispersion ratio of 1.05 is consistent with the Poisson assumption.” Always report rate ratios on the multiplicative scale and check dispersion.

Practical Tips

Check the dispersion statistic immediately after fitting; if \(\gg 1\), switch to quasi-Poisson (family = quasipoisson) for inflated SEs or to negative binomial (MASS::glm.nb) for explicit overdispersion modelling.
Include an offset whenever exposure varies across observations; without it, coefficients absorb the exposure variation and are biased.
Coefficients are on the log-rate scale; exponentiate for rate ratios in publication-friendly units.
For excessive zero counts beyond what Poisson predicts, fit a zero-inflated or hurdle model (pscl::zeroinfl, glmmTMB).
For count data with hierarchical structure (repeated measures, clustered observations), use Poisson mixed models (glmer(family = poisson) or glmmTMB(family = poisson)).
Pearson residuals should look approximately uniformly scattered; systematic patterns suggest model misspecification (missing covariates, non-linear effects, overdispersion).

R Packages Used

Base R glm() with family = poisson; MASS::glm.nb() for negative binomial; pscl::zeroinfl() for zero-inflated Poisson; glmmTMB for mixed-effects Poisson, NB, and zero-inflated count models with cleaner overdispersion handling; broom::tidy() and glance() for tidy summaries.