Poisson Regression for Count Data

Count outcomes are nonnegative integers, such as visits, support requests or equipment failures. Poisson regression relates predictors to the conditional mean count. With a log link, log(mu) = x beta, so mu = exp(x beta). The exponentiation guarantees a positive expected value, while allowing the mean to vary multiplicatively across predictor settings. A predicted mean of 2.4 is valid even though an observed count must be an integer: it represents an average across comparable opportunities. A basic Poisson distribution assumes that, conditional on the predictors, the variance equals the mean. This is a modeling assumption, not a property automatically established by having count data. Real counts may show extra variation from unobserved heterogeneity, clustering, dependence, omitted predictors or excess zeros. If overdispersion is ignored, standard errors can be too small and tests overly confident. Inspect residual and deviance diagnostics, compare observed variability with model expectations, and consider whether the sampling process was represented appropriately. Exposure matters when cases have different time at risk. In a hypothetical incident model, one site observed for 20 hours had more opportunity to register events than one observed for 5 hours. A log exposure offset lets expected counts scale with exposure while estimating predictor effects on a rate. The offset should correspond to the actual opportunity measure and should not be added as though it were a freely estimated feature without considering the question. A negative-binomial model allows variance to exceed the mean under its chosen parameterization and can be a reasonable alternative when overdispersion is supported. It is not a universal cure: check fit, assumptions and prediction performance on appropriately held-out data. A zero-inflated or hurdle model may be relevant when a separate process generates structural zeros, but needs a defensible data-generating rationale. Select the model based on the count process and intended use, and communicate whether estimates describe counts, rates or relative changes in expected counts.