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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.
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.
It is a main ingredient in the generalized linear model framework and a tool used in non-parametric regression, [1] semiparametric regression [1] and functional data analysis. [2] In parametric modeling, variance functions take on a parametric form and explicitly describe the relationship between the variance and the mean of a random quantity.
In probability theory and statistics, the Poisson distribution (/ ˈ p w ɑː s ɒ n /; French pronunciation:) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. [1]
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 ...
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 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.
Commonly used models in the GLM family include binary logistic regression [5] for binary or dichotomous outcomes, Poisson regression [6] for count outcomes, and linear regression for continuous, normally distributed outcomes. This means that GLM may be spoken of as a general family of statistical models or as specific models for specific ...