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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 .
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]
As the examples above show, zero-inflated data can arise as a mixture of two distributions. The first distribution generates zeros. The second distribution, which may be a Poisson distribution, a negative binomial distribution or other count distribution, generates counts, some of which may be zeros.
Another example of generalized linear models includes Poisson regression which models count data using the Poisson distribution. The link is typically the logarithm, the canonical link. The variance function is proportional to the mean =,
The earliest regression form was seen in Isaac Newton's work in 1700 while studying equinoxes, being credited with introducing "an embryonic linear aggression analysis" as "Not only did he perform the averaging of a set of data, 50 years before Tobias Mayer, but summing the residuals to zero he forced the regression line to pass through the ...
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...
But the VGLM framework readily handles models such as zero-inflated Poisson regression, zero-altered Poisson (hurdle) regression, positive-Poisson regression, and negative binomial regression. As another example, for the linear model, the variance of a normal distribution is relegated as a scale parameter and it is treated often as a nuisance ...
Hierarchical linear models (or multilevel regression) organizes the data into a hierarchy of regressions, for example where A is regressed on B, and B is regressed on C. It is often used where the variables of interest have a natural hierarchical structure such as in educational statistics, where students are nested in classrooms, classrooms ...