# Poisson regression

**PoissonDiscrete RegressionNegative binomial regressionPoisson modelPoisson regression test**

In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables.wikipedia

59 Related Articles

### Generalized linear model

**generalized linear modelslink functiongeneralised linear model**

In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. Offset in the case of a GLM in R can be achieved using the offset function:

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.

### Regression analysis

**regressionmultiple regressionregression model**

In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables.

If the variable is positive with low values and represents the repetition of the occurrence of an event, then count models like the Poisson regression or the negative binomial model may be used.

### Count data

**countcount type response variable datacounting statistics**

In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables.

The Poisson distribution can form the basis for some analyses of count data and in this case Poisson regression may be used.

### Poisson distribution

**PoissonPoisson-distributedPoissonian**

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.

Poisson regression and negative binomial regression are useful for analyses where the dependent (response) variable is the count (0, 1, 2, ...) of the number of events or occurrences in an interval.

### Log-linear model

**log-linearLog-linear modelingLog-linear regression**

A Poisson regression model is sometimes known as a log-linear model, especially when used to model contingency tables.

### Overdispersion

**underdispersionover-dispersionoverdispersed**

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.

For example, Poisson regression analysis is commonly used to model count data.

### Negative binomial distribution

**negative binomialGamma-Poisson distributioninverse binomial distribution**

Under some circumstances, the problem of overdispersion can be solved by using quasi-likelihood estimation or a negative binomial distribution instead.

### Fixed-effect Poisson model

In statistics, fixed-effect Poisson models are used for static panel data when the outcome variable is count data.

### Proportional hazards model

**proportional hazards modelsCox proportional hazards modelCox model**

Poisson regression creates proportional hazards models, one class of survival analysis: see proportional hazards models for descriptions of Cox models.

There is a relationship between proportional hazards models and Poisson regression models which is sometimes used to fit approximate proportional hazards models in software for Poisson regression.

### Statistics

**statisticalstatistical analysisstatistician**

In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables.

### Contingency table

**cross tabulationcontingency tablescrosstab**

### Logarithm

**logarithmsloglogarithmic function**

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.

### Expected value

**expectationexpectedmean**

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.

### Parameter

**parametersparametricargument**

### Dependent and independent variables

**dependent variableindependent variableexplanatory variable**

If is a vector of independent variables, then the model takes the form

### Independence (probability theory)

**independentstatistically independentindependence**

If Y i are independent observations with corresponding values x i of the predictor variables, then θ can be estimated by maximum likelihood.

### Maximum likelihood estimation

**maximum likelihoodmaximum likelihood estimatormaximum likelihood estimate**

If Y i are independent observations with corresponding values x i of the predictor variables, then θ can be estimated by maximum likelihood.

### Closed-form expression

**closed formclosed-formanalytical solution**

The maximum-likelihood estimates lack a closed-form expression and must be found by numerical methods.

### Probability mass function

**mass functionprobability massmass**

and thus, the Poisson distribution's probability mass function is given by

### Likelihood function

**likelihoodlikelihood ratiolog-likelihood**

To do this, the equation is first rewritten as a likelihood function in terms of θ:

### Sides of an equation

**left-hand sideright hand sideLHS**

Note that the expression on the right hand side has not actually changed.

### Convex optimization

**convex minimizationconvexconvex programming**

However, the negative log-likelihood, is a convex function, and so standard convex optimization techniques such as gradient descent can be applied to find the optimal value of θ.

### Gradient descent

**steepest descentgradient ascentgradient**

However, the negative log-likelihood, is a convex function, and so standard convex optimization techniques such as gradient descent can be applied to find the optimal value of θ.

### R (programming language)

**RR programming languageCRAN**

Offset in the case of a GLM in R can be achieved using the offset function:

### Variance

**sample variancepopulation variancevariability**

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.