For dispersion parameter \(\lambda\), Gamma means that random effects are distributed as \(u ~\)Gamma(shape=1/\(\lambda\),scale=\(\lambda\)), so \(u\) has mean 1 and variance \(\lambda\). Both the log (\(v=log(u)\)) and identity (\(v=u\)) links are possible, though in the latter case the variance of \(u\) is constrained below 1 (otherwise Laplace approximations fail).
The two-parameter inverse Gamma distribution is the distribution of the reciprocal of a variable distributed according to the Gamma distribution Gamma with the same shape and scale parameters. inverse.Gamma implements the one-parameter inverse Gamma family with shape=1+1/\(\lambda\) and rate=1/\(\lambda\)) (rate=1/scale). It is used to model the distribution of random effects. Its mean=1; and its variance =\(\lambda/(1-\lambda))\) if \(\lambda<1\), otherwise infinite. The default link is "-1/mu", in which case v=-1/u is “-Gamma”-distributed with the same shape and rate, hence with mean \(-(\lambda+1)\) and variance \(\lambda(\lambda+1)\), which is a different one-parameter Gamma family than the above-described Gamma. The other possible link is v=log(u) in which case
\(v ~ -\log(X~\)Gamma\((1+1/\lambda,1/\lambda))\), with mean \(-(\log(1/\lambda)+\)digamma\((1+1/\lambda))\) and variance trigamma(\(1+1/\lambda\)).
inverse.Gamma(link = "-1/mu")
# Gamma(link = "inverse") using stats::Gamma
For Gamma, allowed links are log and identity (the default link from Gamma, "inverse", cannot be used for the random effect specification). For inverse.Gamma, allowed links are "-1/mu" (default) and log.
# see help("HLfit") for fits using the inverse.Gamma distribution.
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