# ExponentialFamily

is an option for GeneralizedLinearModelFit that specifies the exponential family for the model.

# Details  • ExponentialFamily specifies the assumed distribution for the independent observations modeled by .
• The density function for an exponential family can be written in the form for functions , , , , and , random variable , canonical parameter , and dispersion parameter .
• Possible parametric distributions include: "Binomial", "Poisson", "Gamma", "Gaussian", "InverseGaussian".
• The observed responses are restricted to the domains of parametric distributions as follows:
•  "Binomial" "Gamma" "Gaussian" "InverseGaussian" "Poisson" • The setting ExponentialFamily->"QuasiLikelihood", defines a quasi-likelihood function, used for a maximum likelihood fit.
• The log quasi-likelihood function for the response and prediction is given by , where is the dispersion parameter and is the variance function. The dispersion parameter is estimated from input data and can be controlled through the option DispersionEstimatorFunction.
• The setting ExponentialFamily->{"QuasiLikelihood",opts} allows the following quasi-likelihood suboptions to be specified:
•  "ResponseDomain" Function[y,y>0] domain for responses "VarianceFunction" Function[μ,1] variance as function of mean
• The parametric distributions can be emulated with quasi-likelihood structures by using the following "VarianceFunction" and "ResponseDomain" suboption settings:
•  "Binomial"  "Gamma"  "Gaussian"  "InverseGaussian"  "Poisson"  • "QuasiLikelihood" variants of "Binomial" and "Poisson" families can be used to model overdispersed ( ) or underdispersed ( ) data, different from the theoretical dispersion ( ).
• Common variance functions, response domains, and uses include:
•  power models, actuarial science, meteorology, etc.  probability models, binomial related, etc.  counting models, Poisson related, etc.

# Examples

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## Basic Examples(1)

Fit data to a simple linear regression model:

Fit to a canonical gamma regression model:

Fit to a canonical inverse Gaussian regression model:

## Scope(2)

Use the "Binomial" family for logit models of probabilities:

Use "Poisson" for loglinear models of count data:

## Properties & Relations(3)

The default ExponentialFamily and LinkFunction match LinearModelFit:

The default "Binomial" model matches LogitModelFit:

Fit a "Gamma" model and the "QuasiLikelihood" analog:

The models differ from named analogs by a constant in the "LogLikelihood":

Fitted parameters agree:

Results based on differences of log-likelihoods agree: