({{, (RefLink[LinkFunction, paclet:ref/LinkFunction]) 
is an option for (RefLink[GeneralizedLinearModelFit, paclet:ref/GeneralizedLinearModelFit]) that specifies the link function for the generalized linear model.}})

OpenersNotesSection(Spacer1)Details

The link function is an invertible function g in the generalized linear model (OverHat[y])=(g^(-1))((Subscript[β, 0])+(Subscript[β, 1]) (Subscript[f, 1])+(Subscript[β, 2]) (Subscript[f, 2])+⋯).

Possible settings for (RefLink[LinkFunction, paclet:ref/LinkFunction]) include:

The default value (RefLink[Automatic, paclet:ref/Automatic]) uses the canonical link for the (RefLink[ExponentialFamily, paclet:ref/ExponentialFamily]) associated with the model.

The canonical link functions are as follows:

For "QuasiLikelihood" models, "IdentityLink" is used by default.

Other common link functions for binomial data include:

Other common link functions for count data include:

Other common link functions for positive real-valued data include:

For "OddsPowerLink", "NegativeBinomialLink", and "PowerLink", the additional parameter α can be given by (RefLink[LinkFunction, paclet:ref/LinkFunction])->{linkname,"LinkParameter"->α}. The parameter α can be any real value for "OddsPowerLink" and "PowerLink" and any positive value for "NegativeBinomialLink".

With setting (RefLink[LinkFunction, paclet:ref/LinkFunction])->g, g can be any pure function that is real-valued and invertible on the response domain for the model.

OpenersPrimaryExamplesSection(Spacer1)Examples  (1)

OpenersExampleSection0(Spacer1)Basic Examples  (1)

data={{0,1},{1,0},{3,2},{5,4}};

Fit a Poisson model with canonical (RefLink[Log, paclet:ref/Log]) link:

GeneralizedLinearModelFit[data,x,x,ExponentialFamily->"Poisson"]//Normal

Use a named link:

GeneralizedLinearModelFit[data,x,x,ExponentialFamily->"Poisson",LinkFunction->"IdentityLink"]//Normal

Use a pure function for a shifted (RefLink[Sqrt, paclet:ref/Sqrt]) link:

GeneralizedLinearModelFit[data,x,x,ExponentialFamily->"Poisson",LinkFunction->(Sqrt[#1+1]&)]//Normal

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[GeneralizedLinearModelFit, paclet:ref/GeneralizedLinearModelFit])  ▪  (RefLink[ExponentialFamily, paclet:ref/ExponentialFamily])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Statistical Model Analysis, paclet:guide/StatisticalModelAnalysis])}})}})