Poissonmodel

Poissonmodel refers to a statistical framework for modeling count data where the response variable Y_i follows a Poisson distribution with mean λ_i. In its regression form, the mean is modeled as a function of explanatory variables X_i, often via a log link: log(λ_i) = X_iβ. When an exposure or observation window t_i is present, an offset term log(t_i) can be added so that λ_i = t_i exp(X_iβ), representing the expected count per unit exposure.

Assumptions include independence of observations and that the variance equals the mean (var(Y_i) = λ_i). The likelihood for observed counts {y_i} is L(β) = ∏ e^{-λ_i} λ_i^{y_i} / y_i!, with λ_i = exp(X_iβ) (or λ_i = t_i exp(X_iβ)). The coefficients β are estimated by maximum likelihood, and exp(β_j) is interpreted as a rate ratio associated with a one-unit change in the corresponding covariate (holding others constant).

Model checking uses residuals and goodness-of-fit measures such as deviance and Pearson chi-square. A common issue is overdispersion, where var(Y_i) > E[Y_i], which can arise from unobserved heterogeneity or clustering. If overdispersion is present, quasi-Poisson models or negative binomial regression provide more flexible variance structures.

Extensions include zero-inflated Poisson and hurdle models for excess zeros. Poissonmodels are widely used in epidemiology, ecology, insurance, criminology, and other fields to analyze counts of rare events, incidents, or arrivals, with applications ranging from rate comparisons to exposure-adjusted risk assessment.