In glmmTMB: Generalized Linear Mixed Models using Template Model Builder

One commonly requested feature is to be able to run a post hoc Markov chain Monte Carlo analysis based on the results of a frequentist fit. This is often a reasonable shortcut for computing confidence intervals and p-values that allow for finite-sized samples rather than relying on asymptotic sampling distributions. This vignette shows an example of such an analysis. Some caveats:

• when using such a "pseudo-Bayesian" approach, be aware that using a scaled likelihood (implicit, improper priors) can often cause problems, especially when the model is poorly constrained by the data
• in particular, models with poorly constrained random effects (singular or nearly singular) are likely to give bad results
• as shown below, even models that are well-behaved for frequentist fitting may need stronger priors to give well-behaved MCMC results
• as with all MCMC analysis, it is the user's responsibility to check for proper mixing and convergence of the chains before drawing conclusions
• the first MCMC sampler illustrated below is simple (Metropolis with a multivariate normal candidate distribution). Users who want to do MCMC sampling on a regular basis should consider the tmbstan package, which does much more efficient hybrid/Hamiltonian Monte Carlo sampling.

library(knitr)
opts_chunk$set(echo = TRUE,
               eval = identical(Sys.getenv("NOT_CRAN"), "true"))
rc <- knitr::read_chunk
rc(system.file("vignette_data","mcmc.R",package="glmmTMB"))

Load packages:

library(glmmTMB)
library(coda)     ## MCMC utilities
library(reshape2) ## for melt()
## graphics
library(lattice)
library(ggplot2); theme_set(theme_bw())

Fit basic model:

Set up for MCMC: define scaled log-posterior function (in this case the log-likelihood function); extract coefficients and variance-covariance matrices as starting points.

This is a basic block Metropolis sampler, based on Florian Hartig's code here.

Run the chain:

L <- load(system.file("vignette_data", "mcmc.rda", package="glmmTMB"))

(running this chain takes r round(t1["elapsed"],1) seconds)

Add more informative names and transform correlation parameter (see vignette on covariance structures and parameters):

colnames(m1) <- c(names(fixef(fm1)[[1]]),
                  "log(sigma)",
                  c("log(sd_Intercept)","log(sd_Days)","cor"))
m1[,"cor"] <- sapply(m1[,"cor"], get_cor)

xyplot(m1,layout=c(2,3),asp="fill")

The trace plots are poor, especially for the correlation; the effective sample size backs this up, as would any other diagnostics we did.

print(effectiveSize(m1),digits=3)

In a real analysis we would stop and fix the mixing/convergence problems before proceeding; for this simple sampler, some of our choices would be (1) simply run the chain for longer; (2) tune the candidate distribution (e.g. by using tune to scale some parameters, or perhaps by switching to a multivariate Student t distribution [see the mvtnorm package]); (3) add regularizing priors.

Ignoring the problems and proceeding, we can compute column-wise quantiles or highest posterior density intervals (coda::HPDinterval) to get confidence intervals. Plotting posterior distributions, omitting the intercept because it's on a very different scale.

ggplot(reshape2::melt(as.matrix(m1[,-1])),aes(x=Var2,y=value))+
         geom_violin(fill="gray")+coord_flip()+labs(x="")

tmbstan

The tmbstan package allows direct, simple access to a hybrid/Hamiltonian Monte Carlo algorithm for sampling from a TMB object; the $obj component of a glmmTMB fit is such an object. (To run this example you'll need to install the tmbstan package and its dependencies.)

(running this command, which creates 4 chains, takes r round(t2["elapsed"],1) seconds)

However, there are many indications (warning messages; trace plots) that the correlation parameter needs to a more informative prior. (In the plot below, the correlation parameter is shown on its unconstrained scale; the actual correlation would be $\theta_3/\sqrt{1+\theta_3^2}$.)

library(png)
library(grid)
img <- readPNG(system.file("vignette_data","tmbstan_traceplot.png",package="glmmTMB"))
grid.raster(img)

To do

• solve mixing for cor parameter
• more complex example - e.g. Owls

glmmTMB documentation built on Oct. 7, 2023, 5:07 p.m.