inverse.Gamma {spaMM}

Distribution families for Gamma and inverse Gamma-distributed random effects

Description

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).

Usage

inverse.Gamma(link = "-1/mu")
# Gamma(link = "inverse") using stats::Gamma 

Arguments

link

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.

Examples

# see help("HLfit") for fits using the inverse.Gamma distribution.