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

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
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.

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.