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(penicillin, package="faraway")
summary(penicillin)
library(ggplot2, quietly=TRUE)
ggplot(penicillin,aes(x=blend,y=yield,group=treat,linetype=treat))+geom_line()
ggplot(penicillin,aes(x=treat,y=yield,group=blend,linetype=blend))+geom_line()

Default prior model

Fit the default INLA model:

formula = yield ~ treat+f(blend, model="iid")
result = inla(formula, family="gaussian", data=penicillin)
result <- inla.hyperpar(result)
summary(result)

Precision for the blend effect looks implausibly large. Problem with default gamma prior.

Half-normal priors on the SDs

Try a half-normal prior on the blend precision. I have used variance of the response to help with the scaling so these are more informative.

resprec <- 1/var(penicillin$yield)
formula = yield ~ treat+f(blend, model="iid", prior="logtnormal", param=c(0, resprec))
result = inla(formula, family="gaussian", data=penicillin)
result <- inla.hyperpar(result)
summary(result)

Looks more plausible. Compute the transforms to an SD scale for the blend 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$marginals.fixed, function(x) inla.zmarginal(x,silent=TRUE))
restab=cbind(restab, inla.zmarginal(sigmaalpha,silent=TRUE))
restab=cbind(restab, inla.zmarginal(sigmaepsilon,silent=TRUE))
restab=cbind(restab, sapply(result$marginals.random$blend,function(x) inla.zmarginal(x, silent=TRUE)))
colnames(restab) = c("mu","B-A","C-A","D-A","alpha","epsilon",levels(penicillin$blend))
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(0,15)

Posterior for the blend SD is more diffuse than the error SD. Posterior for the blend SD has non-zero density at zero.

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 fixed-effects model residuals. 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 - smaller values are less informative.

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

Compute the summaries as before:

Make the plots:

Posterior for blend SD has no weight near zero.

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