LogPoisson

LogPoisson refers to the Poisson regression framework in which the logarithm of the Poisson mean is modeled as a linear function of covariates. It is a standard generalized linear model for count data, using the Poisson distribution with a log link as the canonical combination.

In the model, a count variable Y_i is assumed to follow a Poisson distribution with mean λ_i, and the link is specified as log(λ_i) = X_i^T β. Equivalently, λ_i = exp(X_i^T β), so the expected count is a multiplicative function of the covariates. The regression coefficients β are typically estimated by maximum likelihood, and standard errors are derived from the observed Fisher information or via robust methods if overdispersion is present.

Interpretation of the coefficients is on the multiplicative scale: exp(β_j) represents the factor by which the mean count changes with a one-unit increase in the corresponding predictor, holding other variables constant. When an offset term is included (for example, log(exposure time)), the model can be used to analyze rates rather than raw counts.

Applications and considerations: logPoisson models are widely used for count data in fields such as epidemiology and public health, and they are also employed to estimate relative risks in certain study designs. A common issue is overdispersion, where the variance exceeds the mean; in such cases, quasi-Poisson or negative binomial approaches may be preferred. Convergence and interpretability considerations arise in some contexts, particularly with rare outcomes or when alternative link functions are contemplated.

Related concepts include Poisson regression with a log link, generalized linear models, and the log-binomial model.