In julianfaraway/brinla: Bayesian Regression with INLA

library(knitr)
opts_chunk$set(cache=FALSE,comment=NA, fig.path="figs/", warning=FALSE, message=FALSE, pngquant='--speed=1 --quality=0-50')
options(digits=5,show.signif.stars=FALSE,width=120)
knit_hooks$set(pngquant = hook_pngquant)
knitr::knit_hooks$set(setPch = function(before, options, envir) {
  if(before) par(pch = 20)
})
opts_chunk$set(setPch = TRUE)

See the introduction for an overview. Load the libraries:

library(ggplot2)
library(INLA)

Data

Load in and plot the data:

data(irrigation, package="faraway")
summary(irrigation)
ggplot(irrigation, aes(y=yield, x=field, shape=variety, color=irrigation)) + geom_point()

Default INLA fit

formula <- yield ~ irrigation + variety +f(field, model="iid")
result <- inla(formula, family="gaussian", data=irrigation)
result <- inla.hyperpar(result)
summary(result)

Default looks more plausible than one way and RBD examples.

Compute the transforms to an SD scale for the field and error. Make a table of summary statistics for the posteriors:

sigmaalpha <- inla.tmarginal(function(x) 1/sqrt(exp(x)),result$internal.marginals.hyperpar[[2]])
sigmaepsilon <- inla.tmarginal(function(x) 1/sqrt(exp(x)),result$internal.marginals.hyperpar[[1]])
restab=sapply(result$marginals.fixed, function(x) inla.zmarginal(x,silent=TRUE))
restab=cbind(restab, inla.zmarginal(sigmaalpha,silent=TRUE))
restab=cbind(restab, inla.zmarginal(sigmaepsilon,silent=TRUE))
colnames(restab) = c("mu","ir2","ir3","ir4","v2","alpha","epsilon")
data.frame(restab)

Also construct a plot the SD posteriors:

ddf <- data.frame(rbind(sigmaalpha,sigmaepsilon),errterm=gl(2,nrow(sigmaalpha),labels = c("alpha","epsilon")))
ggplot(ddf, aes(x,y, linetype=errterm))+geom_line()+xlab("yield")+ylab("density")+xlim(0,10)

Posteriors look OK.

Informative Gamma priors on the precisions

Now try more informative gamma priors for the precisions. Define it so the mean value of gamma prior is set to the inverse of the variance of the residuals of the fixed-effects only model. We expect the two error variances to be lower than this variance so this is an overestimate. The variance of the gamma prior (for the precision) is controlled by the apar shape parameter.

apar <- 0.5
lmod <- lm(yield ~ irrigation+variety, data=irrigation)
bpar <- apar*var(residuals(lmod))
lgprior <- list(prec = list(prior="loggamma", param = c(apar,bpar)))
formula = yield ~ irrigation+variety+f(field, model="iid", hyper = lgprior)
result <- inla(formula, family="gaussian", data=irrigation)
result <- inla.hyperpar(result)
summary(result)

Compute the summaries as before:

Make the plots:

Posteriors look OK.

Penalized Complexity Prior

In Simpson et al (2015), penalized complexity priors are proposed. This requires that we specify a scaling for the SDs of the random effects. We use the SD of the residuals of the fixed effects only model (what might be called the base model in the paper) to provide this scaling.

lmod <- lm(yield ~ irrigation+variety, irrigation)
sdres <- sd(residuals(lmod))
pcprior <- list(prec = list(prior="pc.prec", param = c(3*sdres,0.01)))
formula <- yield ~ irrigation+variety+f(field, model="iid", hyper = pcprior)
result <- inla(formula, family="gaussian", data=irrigation)
result <- inla.hyperpar(result)
summary(result)