Poisson regression

Poisson regression is a generalized linear model for count outcomes that models the conditional mean as a function of explanatory variables, typically using a logarithmic link.

Poisson distribution: A discrete probability distribution for nonnegative integer counts with one parameter governing both its mean and variance. It supplies the standard response distribution and variance assumption for the model.
Generalized linear model: A regression framework combining a response distribution, linear predictor, and link function. Poisson regression is one member of this broader modeling framework.
Incidence rate: The occurrence of new events divided by the population or person-time at risk. A log exposure offset lets fitted means represent incidence rates.
Negative binomial regression: A count regression model that allows conditional variance to exceed the conditional mean through an additional dispersion parameter. It is a common alternative when Poisson variance assumptions understate count variability.
Log link function: A link function that maps a positive mean to its natural logarithm. It keeps fitted conditional means positive while making log means linear in predictors.
Linear predictor: A weighted sum of model predictors and coefficients, often including an intercept. Its value is transformed by the link to determine the conditional count mean.
Accident frequency modeling: Statistical modeling of how often accidents occur across locations, periods, or conditions. Poisson regression can relate accident counts to traffic, road, and exposure variables.
Quasi-Poisson regression: A count regression approach that retains a Poisson mean structure while scaling variance by a dispersion parameter. It adjusts uncertainty estimates for overdispersion without specifying a full count distribution.
Maximum likelihood estimation: A method that estimates parameters by maximizing the probability of observed data under a statistical model. Poisson regression coefficients are commonly estimated by maximizing the count-data likelihood.
Conditional expectation: The expected value of a random variable given specified information or covariates. The model targets the expected count conditional on the explanatory variables.

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