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Poisson regression
Regression analysis 

File:Linear regression.svg 
Models 
Estimation 
Background 
In statistics, Poisson regression is a form of regression analysis used to model count data and contingency tables. 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. A Poisson regression model is sometimes known as a loglinear model, especially when used to model contingency tables.
Poisson regression models are generalized linear models with the logarithm as the (canonical) link function, and the Poisson distribution function as the assumed probability distribution of the response.
Contents
Regression models
If <math>\mathbf{x} \in \mathbb{R}^n</math> is a vector of independent variables, then the model takes the form
 <math>\log (\operatorname{E}(Y\mid\mathbf{x}))=\alpha + \mathbf{\beta}' \mathbf{x},</math>
where <math>\alpha \in \mathbb{R}</math> and <math>\mathbf{\beta} \in \mathbb{R}^n</math>. Sometimes this is written more compactly as
 <math>\log (\operatorname{E}(Y\mid\mathbf{x}))=\boldsymbol{\theta}' \mathbf{x},\,</math>
where x is now an (n + 1)dimensional vector consisting of n independent variables concatenated to a vector of ones. Here θ is simply α concatenated to β.
Thus, when given a Poisson regression model θ and an input vector x, the predicted mean of the associated Poisson distribution is given by
 <math>\operatorname{E}(Y\mid\mathbf{x})=e^{\boldsymbol{\theta}' \mathbf{x}}.\,</math>
If Y_{i} are independent observations with corresponding values x_{i} of the predictor variables, then θ can be estimated by maximum likelihood. The maximumlikelihood estimates lack a closedform expression and must be found by numerical methods. The probability surface for maximumlikelihood Poisson regression is always concave, making Newton–Raphson or other gradientbased methods appropriate estimation techniques.
Maximum likelihoodbased parameter estimation
Given a set of parameters θ and an input vector x, the mean of the predicted Poisson distribution, as stated above, is given by
 <math>\operatorname{E}(Y\mid x)=e^{\theta' x}\,</math>,
and thus, the Poisson distribution's probability mass function is given by
 <math>p(y\mid x;\theta) = \frac{[\operatorname{E}(Y\mid x)]^y \times e^{\operatorname{E}(Y\mid x)}}{y!} = \frac{e^{y \theta' x} e^{e^{\theta' x}}}{y!}</math>
Now suppose we are given a data set consisting of m vectors <math>x_i \in \mathbb{R}^{n+1}, \, i = 1,\ldots,m</math>, along with a set of m values <math>y_1,\ldots,y_m \in \mathbb{R}</math>. Then, for a given set of parameters θ, the probability of attaining this particular set of data is given by
 <math>p(y_1,\ldots,y_m\mid x_1,\ldots,x_m;\theta) = \prod_{i=1}^m \frac{e^{y_i \theta' x_i} e^{e^{\theta' x_i}}}{y_i!}.</math>
By the method of maximum likelihood, we wish to find the set of parameters θ that makes this probability as large as possible. To do this, the equation is first rewritten as a likelihood function in terms of θ:
 <math>L(\theta\mid X,Y) = \prod_{i=1}^m \frac{e^{y_i \theta' x_i} e^{e^{\theta' x_i}}}{y_i!}</math>.
Note that the expression on the right hand side has not actually changed. A formula in this form is typically difficult to work with; instead, one uses the loglikelihood:
 <math>\ell(\theta\mid X,Y) = \log L(\theta\mid X,Y) = \sum_{i=1}^m \left( y_i \theta' x_i  e^{\theta' x_i}  \log(y_i!)\right)</math>.
Notice that the parameters θ only appear in the first two terms of each term in the summation. Therefore, given that we are only interested in finding the best value for θ we may drop the y_{i}! and simply write
 <math>\ell(\theta\mid X,Y) = \sum_{i=1}^m \left( y_i \theta' x_i  e^{\theta' x_i}\right)</math>.
To find a maximum, we need to solve an equation <math>\frac{\partial \ell(\theta\mid X,Y)}{\partial \theta} = 0 </math> which has no closedform solution. However, the negative loglikelihood, <math>\ell(\theta\mid X,Y)</math>, is a convex function, and so standard convex optimization techniques such as gradient descent can be applied to find the optimal value of θ.
Poisson regression in practice
Poisson regression may be appropriate when the dependent variable is a count, for instance of events such as the arrival of a telephone call at a call centre.^{[1]} The events must be independent in the sense that the arrival of one call will not make another more or less likely, but the probability per unit time of events is understood to be related to covariates such as time of day.
"Exposure" and offset
Poisson regression may also be appropriate for rate data, where the rate is a count of events occurring to a particular unit of observation, divided by some measure of that unit's exposure. For example, biologists may count the number of tree species in a forest, and the rate would be the number of species per square kilometre. Demographers may model death rates in geographic areas as the count of deaths divided by person−years. More generally, event rates can be calculated as events per unit time, which allows the observation window to vary for each unit. In these examples, exposure is respectively unit area, person−years and unit time. In Poisson regression this is handled as an offset, where the exposure variable enters on the righthand side of the equation, but with a parameter estimate (for log(exposure)) constrained to 1.
 <math>\log{(\operatorname{E}(Y\mid x))} = \log{(\text{exposure})} + \theta' x</math>
which implies
 <math>\log{(\operatorname{E}(Y\mid x))}  \log{(\text{exposure})} =
\log{\left(\frac{\operatorname{E}(Y\mid x)}{\text{exposure}}\right)} = \theta' x</math>
Offset in the case of a GLM in R can be achieved using the offset() function:
glm(y ~ offset(log(exposure)) + x, family=poisson(link=log) )
Overdispersion and zero inflation
A characteristic of the Poisson distribution is that its mean is equal to its variance. In certain circumstances, it will be found that the observed variance is greater than the mean; this is known as overdispersion and indicates that the model is not appropriate. A common reason is the omission of relevant explanatory variables, or dependent observations. Under some circumstances, the problem of overdispersion can be solved by using quasilikelihood estimation or a negative binomial distribution instead.^{[2]}^{[3]}
Another common problem with Poisson regression is excess zeros: if there are two processes at work, one determining whether there are zero events or any events, and a Poisson process determining how many events there are, there will be more zeros than a Poisson regression would predict. An example would be the distribution of cigarettes smoked in an hour by members of a group where some individuals are nonsmokers.
Other generalized linear models such as the negative binomial model or zeroinflated model may function better in these cases.
Use in survival analysis
Poisson regression creates proportional hazards models, one class of survival analysis: see proportional hazards models for descriptions of Cox models.
Extensions
Regularized Poisson regression
When estimating the parameters for Poisson regression, one typically tries to find values for θ that maximize the likelihood of an expression of the form
 <math>\sum_{i=1}^m \log(p(y_i;e^{\theta' x})),</math>
where m is the number of examples in the data set, and <math>p(y_i;e^{\theta' x})</math> is the probability mass function of the Poisson distribution with the mean set to <math>e^{\theta' x}</math>. Regularization can be added to this optimization problem by instead maximizing
 <math>\sum_{i=1}^m \log(p(y_i;e^{\theta' x}))  \lambda \left\\theta\right\_2^2,</math>
for some positive constant <math>\lambda</math>. This technique, similar to ridge regression, can reduce overfitting.
Implementations
Some statistics packages include implementations of Poisson regression.
 GenStat: Poisson regression is a standard option of the regression section, using the "MODEL", "FIT" and associated commands; it is also available in the "Stats > Regression Analysis > Generalized Linear Models" menu.
 MATLAB Statistics Toolbox: Poisson regression can be performed using the "glmfit" and "glmval" functions.^{[4]}
 Microsoft Excel: Excel is not capable of doing Poisson regression by default. One of the Excel Addins for Poisson regression is XPost
 mPlus: mPlus allows for Poisson regression using the command COUNT IS when specifying the data
 R: The function for fitting a generalized linear model in R is glm(), and can be used for Poisson Regression
 SAS: Poisson regression in SAS is done by using GENMOD, HPGENSELECT, COUNTREG, GLIMMIX, and NLMIXED
 SPSS: In SPSS, Poisson regression is done by using the GENLIN command
 Stata: Stata has a procedure for Poisson regression named
poisson
,^{[5]} and for panel dataxtpoisson
.^{[6]}  CrimeStat: CrimeStat has Poisson, Poisson NB1, PoissonGamma(negative binomial), and PoissonLognormal regression models.
See also
References
 ^ Greene, William H. (2003). Econometric Analysis (Fifth ed.). PrenticeHall. pp. 740–752. ISBN 0130661899.
 ^ Paternoster R, Brame R (1997). "Multiple routes to delinquency? A test of developmental and general theories of crime". Criminology 35: 45–84. doi:10.1111/j.17459125.1997.tb00870.x.
 ^ Berk R, MacDonald J (2008). "Overdispersion and Poisson regression" (PDF). Journal of Quantitative Criminology 24 (3): 269–284. doi:10.1007/s1094000890484.
 ^ http://www.mathworks.com/help/toolbox/stats/glmfit.html
 ^ Poisson regression
 ^ Fixedeffects, randomeffects, and populationaveraged Poisson models
Further reading
 Cameron, A. C.; Trivedi, P. K. (1998). Regression analysis of count data. Cambridge University Press. ISBN 0521632013.
 Christensen, Ronald (1997). Loglinear models and logistic regression. Springer Texts in Statistics (Second ed.). New York: SpringerVerlag. pp. xvi+483. ISBN 0387982477. MR 1633357.
 Greene, William H. (2008). "Models for Event Counts and Duration". Econometric Analysis (8th ed.). Upper Saddle River: Prentice Hall. pp. 906–944. ISBN 9780136003830.
 Hilbe, J. M. (2007). Negative Binomial Regression. Cambridge University Press. ISBN 9780521857727.

