In julianfaraway/brinla: Bayesian Regression with INLA

library(knitr)
opts_chunk$set(cache=FALSE,comment=NA, fig.path="figs/", warning=FALSE, message=FALSE, dev ="svglite")
options(digits=5,show.signif.stars=FALSE,width=120)
knitr::knit_hooks$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 summarize the data:

data(vision, package="faraway")
vision$npower <- rep(1:4,14)
vision$eyesub <- paste0(vision$eye,vision$subject)
summary(vision)

Plot the data:

ggplot(vision, aes(y=acuity, x=npower, linetype=eye))+geom_line()+facet_wrap(~ subject)

Default prior model

formula <- acuity ~ power + f(subject, model="iid") + f(eyesub, model="iid")
result <- inla(formula, family="gaussian", data=vision)
result <- inla.hyperpar(result)
summary(result)

The subject precision looks far too high. Need to change the default prior

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 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 cpar parameter in the code.

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

Compute the transforms to an SD scale for the random effect terms. Make a table of summary statistics for the posteriors:

sigmasubject <- inla.tmarginal(function(x) 1/sqrt(exp(x)),result$internal.marginals.hyperpar[[2]])
sigmaeye <- inla.tmarginal(function(x) 1/sqrt(exp(x)),result$internal.marginals.hyperpar[[3]])
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(sigmasubject,silent=TRUE))
restab=cbind(restab, inla.zmarginal(sigmaeye,silent=TRUE))
restab=cbind(restab, inla.zmarginal(sigmaepsilon,silent=TRUE))
colnames(restab) = c(names(lmod$coef),"subject","eyesub","epsilon")
data.frame(restab)

Also construct a plot the SD posteriors:

ddf <- data.frame(rbind(sigmasubject,sigmaeye,sigmaepsilon),errterm=gl(3,nrow(sigmaepsilon),labels = c("subject","eye","epsilon")))
ggplot(ddf, aes(x,y, linetype=errterm))+geom_line()+xlab("acuity")+ylab("density")+xlim(0,15)

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(acuity ~ power, vision)
sdres <- sd(residuals(lmod))
pcprior <- list(prec = list(prior="pc.prec", param = c(3*sdres,0.01)))
formula = acuity ~ power+f(subject, model="iid", hyper = pcprior)+f(eyesub, model="iid", hyper = pcprior)
result <- inla(formula, family="gaussian", data=vision)
result <- inla.hyperpar(result)
summary(result)