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In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. [1] Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters.
The Poisson distribution is a special case of the discrete compound Poisson distribution (or stuttering Poisson distribution) with only a parameter. [33] [34] The discrete compound Poisson distribution can be deduced from the limiting distribution of univariate multinomial distribution. It is also a special case of a compound Poisson distribution.
Hilbe [3] notes that "Poisson regression is traditionally conceived of as the basic count model upon which a variety of other count models are based." In a Poisson model, "… the random variable y {\displaystyle y} is the count response and parameter λ {\displaystyle \lambda } (lambda) is the mean.
In addition it yields standard errors for the regression parameters (via the Fisher Information matrix) compared to the full posterior distributions obtainable via the Bayesian formulation. It also provides a statistical test for the level of dispersion compared to a Poisson model. Code for fitting a CMP regression, testing for dispersion, and ...
Generalized linear models were formulated by John Nelder and Robert Wedderburn as a way of unifying various other statistical models, including linear regression, logistic regression and Poisson regression. [1] They proposed an iteratively reweighted least squares method for maximum likelihood estimation (MLE) of the model parameters. MLE ...
In linear regression, the model specification is that the dependent variable, is a linear combination of the parameters (but need not be linear in the independent variables). For example, in simple linear regression for modeling n {\displaystyle n} data points there is one independent variable: x i {\displaystyle x_{i}} , and two parameters, β ...
Linear panel data models use the linear additivity of the fixed effects to difference them out and circumvent the incidental parameter problem. Even though Poisson models are inherently nonlinear, the use of the linear index and the exponential link function lead to multiplicative separability, more specifically [2] E[y it ∨ x i1...
in which the f i (X) are quantities that are functions of the variable X, in general a vector of values, while c and the w i stand for the model parameters. The term may specifically be used for: A log-linear plot or graph, which is a type of semi-log plot. Poisson regression for contingency tables, a type of generalized linear model.