brinla
INLA analysis of Latin Square design - one fixed and two crossed random effects ================ Julian Faraway 22 September 2020
See the introduction for an overview. Load the libraries:
library(ggplot2)
library(INLA)
Data
data(abrasion, package="faraway")
abrasion
run position material wear
1 1 1 C 235
2 1 2 D 236
3 1 3 B 218
4 1 4 A 268
5 2 1 A 251
6 2 2 B 241
7 2 3 D 227
8 2 4 C 229
9 3 1 D 234
10 3 2 C 273
11 3 3 A 274
12 3 4 B 226
13 4 1 B 195
14 4 2 A 270
15 4 3 C 230
16 4 4 D 225
ggplot(abrasion,aes(x=material, y=wear, shape=run, color=position))+geom_point(position = position_jitter(width=0.1, height=0.0))
Default prior model
formula <- wear ~ material + f(run, model="iid") + f(position, model="iid")
result <- inla(formula, family="gaussian", data=abrasion)
result <- inla.hyperpar(result)
summary(result)
Call:
"inla(formula = formula, family = \"gaussian\", data = abrasion)"
Time used:
Pre = 1.44, Running = 8.07, Post = 0.441, Total = 9.95
Fixed effects:
mean sd 0.025quant 0.5quant 0.975quant mode kld
(Intercept) 260.937 7.149 246.059 261.177 274.424 261.608 0
materialB -38.750 10.110 -57.790 -39.100 -17.689 -39.735 0
materialC -18.178 10.022 -37.230 -18.460 2.523 -18.970 0
materialD -28.819 10.065 -47.861 -29.136 -7.936 -29.710 0
Random effects:
Name Model
run IID model
position IID model
Model hyperparameters:
mean sd 0.025quant 0.5quant 0.975quant mode
Precision for the Gaussian observations 5e-03 2e-03 0.002 5.00e-03 9.0e-03 0.004
Precision for run 2e+04 2e+04 505.736 1.39e+04 7.4e+04 0.028
Precision for position 2e+04 2e+04 505.040 1.39e+04 7.4e+04 0.013
Expected number of effective parameters(stdev): 3.71(0.098)
Number of equivalent replicates : 4.32
Marginal log-Likelihood: -85.52
The run and position precisions look far too high. Need to change the default prior
Informative Gamma priors on the precisions
Now try more informative gamma priors for the random effect 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 apar shape parameter.
apar <- 0.5
lmod <- lm(wear ~ material, abrasion)
bpar <- apar*var(residuals(lmod))
lgprior <- list(prec = list(prior="loggamma", param = c(apar,bpar)))
formula = wear ~ material+f(run, model="iid", hyper = lgprior)+f(position, model="iid", hyper = lgprior)
result <- inla(formula, family="gaussian", data=abrasion)
result <- inla.hyperpar(result)
summary(result)
Call:
"inla(formula = formula, family = \"gaussian\", data = abrasion)"
Time used:
Pre = 1.43, Running = 1.44, Post = 0.149, Total = 3.02
Fixed effects:
mean sd 0.025quant 0.5quant 0.975quant mode kld
(Intercept) 264.406 10.223 243.794 264.424 284.959 264.481 0
materialB -43.814 5.196 -53.596 -44.031 -32.714 -44.368 0
materialC -22.356 5.169 -32.182 -22.539 -11.419 -22.823 0
materialD -33.455 5.182 -43.257 -33.655 -22.434 -33.967 0
Random effects:
Name Model
run IID model
position IID model
Model hyperparameters:
mean sd 0.025quant 0.5quant 0.975quant mode
Precision for the Gaussian observations 0.021 0.011 0.006 0.020 0.047 0.016
Precision for run 0.010 0.008 0.001 0.008 0.031 0.004
Precision for position 0.008 0.006 0.001 0.007 0.025 0.004
Expected number of effective parameters(stdev): 9.27(0.445)
Number of equivalent replicates : 1.73
Marginal log-Likelihood: -81.86
Compute the transforms to an SD scale for the random effect terms. Make a table of summary statistics for the posteriors:
sigmarun <- inla.tmarginal(function(x) 1/sqrt(exp(x)),result$internal.marginals.hyperpar[[2]])
sigmapos <- inla.tmarginal(function(x) 1/sqrt(exp(x)),result$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(sigmarun,silent=TRUE))
restab=cbind(restab, inla.zmarginal(sigmapos,silent=TRUE))
restab=cbind(restab, inla.zmarginal(sigmaepsilon,silent=TRUE))
colnames(restab) = c("mu","B-A","C-A","D-A","run","position","epsilon")
data.frame(restab)
mu B.A C.A D.A run position epsilon
mean 264.41 -43.814 -22.357 -33.455 12.682 14.211 7.5626
sd 10.222 5.1944 5.1675 5.1808 6.502 7.2926 2.1673
quant0.025 243.78 -53.602 -32.197 -43.269 5.7023 6.3527 4.6048
quant0.25 257.95 -47.2 -25.711 -36.827 8.5602 9.588 6.0385
quant0.5 264.39 -44.055 -22.565 -33.682 11.024 12.356 7.1351
quant0.75 270.79 -40.722 -19.262 -30.364 14.745 16.533 8.5983
quant0.975 284.89 -32.754 -11.455 -22.474 29.617 33.205 13.003
Also construct a plot the SD posteriors:
ddf <- data.frame(rbind(sigmarun,sigmapos,sigmaepsilon),errterm=gl(3,nrow(sigmarun),labels = c("run","position","epsilon")))
ggplot(ddf, aes(x,y, linetype=errterm))+geom_line()+xlab("wear")+ylab("density")+xlim(0,35)
Posteriors look OK although no weight given to smaller values.
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(wear ~ material, abrasion)
sdres <- sd(residuals(lmod))
pcprior <- list(prec = list(prior="pc.prec", param = c(3*sdres,0.01)))
formula = wear ~ material+f(run, model="iid", hyper = pcprior)+f(position, model="iid", hyper = pcprior)
result <- inla(formula, family="gaussian", data=abrasion)
result <- inla.hyperpar(result)
summary(result)
Call:
"inla(formula = formula, family = \"gaussian\", data = abrasion)"
Time used:
Pre = 1.54, Running = 17.4, Post = 1.3, Total = 20.2
Fixed effects:
mean sd 0.025quant 0.5quant 0.975quant mode kld
(Intercept) 264.435 8.748 246.663 264.475 282.025 264.571 0.000
materialB -43.855 5.179 -53.401 -44.150 -32.279 -44.526 0.001
materialC -22.392 5.142 -32.024 -22.640 -11.051 -22.956 0.001
materialD -33.494 5.160 -43.080 -33.766 -22.031 -34.113 0.001
Random effects:
Name Model
run IID model
position IID model
Model hyperparameters:
mean sd 0.025quant 0.5quant 0.975quant mode
Precision for the Gaussian observations 0.019 0.011 0.004 0.017 0.045 0.012
Precision for run 1.350 55.826 0.002 0.015 0.670 0.006
Precision for position 0.019 0.047 0.002 0.010 0.091 0.005
Expected number of effective parameters(stdev): 8.94(0.75)
Number of equivalent replicates : 1.79
Marginal log-Likelihood: -84.54
Compute the summaries as before:
sigmarun <- inla.tmarginal(function(x) 1/sqrt(exp(x)),result$internal.marginals.hyperpar[[2]])
sigmapos <- inla.tmarginal(function(x) 1/sqrt(exp(x)),result$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(sigmarun,silent=TRUE))
restab=cbind(restab, inla.zmarginal(sigmapos,silent=TRUE))
restab=cbind(restab, inla.zmarginal(sigmaepsilon,silent=TRUE))
colnames(restab) = c("mu","B-A","C-A","D-A","run","position","epsilon")
data.frame(restab)
mu B.A C.A D.A run position epsilon
mean 264.44 -43.856 -22.394 -33.495 8.9199 11.097 8.3418
sd 8.7473 5.1768 5.1396 5.1581 5.0464 5.398 2.7518
quant0.025 246.65 -53.417 -32.028 -43.09 1.227 3.3229 4.7356
quant0.25 259 -47.174 -25.664 -36.789 5.5882 7.4367 6.3684
quant0.5 264.44 -44.172 -22.662 -33.788 8.0419 10.06 7.724
quant0.75 269.83 -40.945 -19.472 -30.579 11.258 13.596 9.6682
quant0.975 281.97 -32.323 -11.092 -22.074 21.508 24.675 15.32
Make the plots:
ddf <- data.frame(rbind(sigmarun,sigmapos,sigmaepsilon),errterm=gl(3,nrow(sigmarun),labels = c("run","position","epsilon")))
ggplot(ddf, aes(x,y, linetype=errterm))+geom_line()+xlab("wear")+ylab("density")+xlim(0,35)
Posteriors put more weight on lower values compared to gamma prior. Some work is necessary to correctly compute the tails of the densities.
Package version info
sessionInfo()
R version 4.0.2 (2020-06-22)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Catalina 10.15.6
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRlapack.dylib
locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8
attached base packages:
[1] parallel stats graphics grDevices utils datasets methods base
other attached packages:
[1] INLA_20.03.17 foreach_1.5.0 sp_1.4-2 Matrix_1.2-18 ggplot2_3.3.2 knitr_1.29
loaded via a namespace (and not attached):
[1] pillar_1.4.6 compiler_4.0.2 iterators_1.0.12 tools_4.0.2 digest_0.6.25
[6] evaluate_0.14 lifecycle_0.2.0 tibble_3.0.3 gtable_0.3.0 lattice_0.20-41
[11] pkgconfig_2.0.3 rlang_0.4.7 yaml_2.2.1 xfun_0.16 withr_2.2.0
[16] dplyr_1.0.2 stringr_1.4.0 MatrixModels_0.4-1 generics_0.0.2 vctrs_0.3.4
[21] grid_4.0.2 tidyselect_1.1.0 glue_1.4.2 R6_2.4.1 rmarkdown_2.3
[26] farver_2.0.3 purrr_0.3.4 magrittr_1.5 splines_4.0.2 scales_1.1.1
[31] codetools_0.2-16 ellipsis_0.3.1 htmltools_0.5.0.9000 colorspace_1.4-1 Deriv_4.0.1
[36] labeling_0.3 stringi_1.4.6 munsell_0.5.0 crayon_1.3.4