Choosing gamma priors for variance components as per Fong et al. 2010 "Bayesian inference for generalized linear mixed models" Biostatistics 11

gamma-priors-var-components.R

#Visualise priors for a variance component selected using the method of Fong et al. 2010 (see page 400 of that paper)
R <- log(10) #+-range. Use log for GLMs that have log-link functions(e.g. poisson) or no logs for a gaussian model with identity link
d <- 1 #Degrees of freedom for student t
quant <-0.95 #quantile you want random effect estimates to vary between

#Parameters of Gamma
a1 <- d/2 
qtemp <- 1-(1-quant)/2
a2 <- (d/2) * (R^2)/(qt(qtemp, df = 1)^2)

#Plot it
qlwr <- (1-quant)/2
x <- seq(0.01, 100, length.out = 10000)
tau <- rgamma(50000, a1, a2)
rgaus <- rnorm(50000, mean = 0, sd = 1/sqrt(tau))
quantile(rgaus, c(qlwr, quant+qlwr)) #should be approx +- log(10) = 2.30
hist(rgaus)