Poisson Regression in Biostatistics

Core Idea

Poisson regression models count outcomes (number of infections, hospital admissions, deaths) by relating the log of the expected count to a linear combination of predictors: log(mu) = beta_0 + beta_1*x_1 + ... + beta_k*x_k. The log link ensures predicted counts are always positive. When subjects contribute different amounts of observation time, an offset term log(person-time) is included, effectively modeling incidence rates rather than raw counts. Exponentiated coefficients represent incidence rate ratios. The key assumption is equidispersion — that the variance equals the mean — which is frequently violated in practice (overdispersion), requiring extensions like negative binomial regression or robust standard errors.

Explainer

Many outcomes in biostatistics are counts: the number of asthma attacks per year, the number of infections in a hospital ward per month, the number of cancer cases in a population. These outcomes are non-negative integers with a right-skewed distribution that cannot be modeled well with ordinary linear regression. Poisson regression is the generalized linear model designed for count data, using a log link function and assuming the outcome follows a Poisson distribution.

The model specifies that the log of the expected count is a linear function of predictors: log(E[Y|X]) = beta_0 + beta_1*x_1 + ... This means that exp(beta_j) gives the rate ratio — the multiplicative change in the expected count for a one-unit increase in x_j. If exp(beta_1) = 1.3, the expected count is 30% higher for each additional unit of x_1. The log link ensures predicted counts are always positive (you cannot have negative asthma attacks), and the multiplicative interpretation is natural for biological processes where risk factors scale rates proportionally.

When observations contribute different amounts of person-time (patients followed for different durations, populations of different sizes), raw counts are not comparable. A hospital that follows 1,000 patients for a year will have more infections than one following 100 patients for a month, even if the rate is identical. The offset term handles this by including log(person-time) as a predictor with a fixed coefficient of 1. Algebraically, this converts the model from log(expected count) = Xβ to log(expected count / person-time) = Xβ, which models rates rather than counts. The offset is essential whenever the denominator (time at risk or population size) varies across observations.

The critical assumption of Poisson regression is equidispersion: the variance of the outcome equals its mean. In practice, this assumption is frequently violated — real count data often exhibit overdispersion (variance > mean) due to unobserved heterogeneity, clustering, or excess zeros. When overdispersion is present, the model's standard errors are too small, producing artificially narrow confidence intervals and inflated significance. Diagnostics include comparing the residual deviance to the degrees of freedom (a ratio much greater than 1 suggests overdispersion). Solutions include quasi-Poisson estimation (which scales standard errors by a dispersion parameter), negative binomial regression (which adds a parameter for overdispersion), or zero-inflated models when the excess variance comes specifically from too many zeros.