Poisson regression

Poisson regression is a form of statistical modeling used to model count data and contingency tables. It assumes that the dependent variable, Y, follows a Poisson distribution and the logarithm of its expected value can be modeled by a linear combination of unknown parameters. It is a type of generalized linear model (GLM).

Overview

Poisson regression is used when the data being analyzed are counts, which are non-negative integers (0, 1, 2, ...). The counts typically represent the number of times an event occurred in a fixed interval of time or space. The model provides a framework for understanding how the expected counts change in response to changes in the explanatory variables.

Mathematical Formulation

The basic form of Poisson regression is:

\[ \log(\mathbb{E}(Y|X)) = \beta_0 + \beta_1X_1 + \cdots + \beta_pX_p \]

where:

• \( Y \) is the count of events,
• \( X_1, \ldots, X_p \) are the explanatory variables,
• \( \beta_0, \beta_1, \ldots, \beta_p \) are the parameters to be estimated,
• \( \mathbb{E}(Y|X) \) is the expected value of \( Y \) given \( X \).

Assumptions

The key assumptions of Poisson regression include:

• The mean and variance of the dependent variable (Y) are equal.
• Observations are independent of each other.
• The logarithm of the expected value of the response variable can be modeled by a linear combination of input variables.

Applications

Poisson regression is widely used in various fields such as:

• Epidemiology for analyzing the rates of diseases,
• Insurance for claim count modeling,
• Transportation for traffic flow analysis,
• Sociology for crime or accident counts.

Model Fitting

The parameters of a Poisson regression model are typically estimated using maximum likelihood estimation (MLE). Software packages like R, SAS, and SPSS provide functions to fit Poisson regression models to data.

Extensions

Several extensions of Poisson regression exist to handle overdispersion and excess zeros, which are common issues in count data. These include:

• Negative binomial regression,
• Zero-inflated Poisson models,
• Hurdle models.

See Also

• Log-linear analysis
• Negative binomial regression
• Generalized linear model