Description

For dispersion parameter λ, Gamma means that random effects are distributed as u ~Gamma(shape=1/λ,scale=λ), so u has mean 1 and variance λ. 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/λ and rate=1/λ) (rate=1/scale). It is used to model the distribution of random effects. Its mean=1; and its variance =λ/(1-λ)) if λ<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 -(λ+1) and variance λ(λ+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/λ,1/λ)), with mean -(log(1/λ)+digamma(1+1/λ)) and variance trigamma(1+1/λ).

Usage

Arguments

Examples