Generalized linear mixed models in R: nitty-gritty

Licensed under the Creative Commons attribution-noncommercial license. Please share & remix noncommercially, mentioning its origin.

library(lme4)
library(broom)
library(broom.mixed)
library(dotwhisker)

Overview

Most aspects of GLMMs are carried over from LMMs (random effects) and GLMs (families and links).

integration methods

You do have to decide on an approximation method.

• penalized quasi-likelihood
  • fastest/least accurate
  • uses quasi-likelihood (no AIC etc.?)
• Laplace approximation
  • fast, flexible compromise
• adaptive Gauss-Hermite quadrature
  • slowest
  • most accurate
  • most limited

In lme4, use the nAGQ= argument; nAGQ=1 (default) corresponds to Laplace approximation

integration rules

• it only really matters in relatively extreme cases (small numbers of binary obs per group)
• use the most accurate method that’s available and feasible

Laplace-approximation diagnostics

library(lattice)
aspect <- 0.6
xlab <- "z"; ylab <- "density"; type <- c("g","l"); scaled <- FALSE
mm <- readRDS("../data/toenail_lapldiag.rds")
print(xyplot(y ~ zvals|id, data=mm,
             type=type, aspect=aspect,
             xlab=xlab,ylab=ylab,
             as.table=TRUE,
             panel=function(x,y,...){
    if (!scaled) {
        panel.lines(x, dnorm(x), lty=2)
    } else {
        panel.abline(h=1, lty=2)
    }
    panel.xyplot(x,y,...)
}))

comparing integration methods

g1 <- glmer(incidence/size ~ period + (1|herd),
            family=binomial,
            data=cbpp,
            weights=size)
g2 <- update(g1,nAGQ=5)
g3 <- update(g1,nAGQ=10)
g4 <- MASS:::glmmPQL(incidence/size ~ period,
                     random = ~1|herd,
                     data=cbpp,
                     family=binomial,
                     weights=size)

## iteration 1

## iteration 2

## iteration 3

## iteration 4

dwplot(list(Laplace=g1,AGQ5=g2,AGQ10=g3,glmmPQL=g4))

overdispersion

• reminder: “too much” variance
• only applies to families with estimated variance:
  e.g. Poisson, binomial (\(N>1\))
• sum^2 of Pearson residuals / residual degrees of freedom
• e.g. aods3::gof()
• observation-level random effects

dealing with overdispersion

• quasi-likelihood
• observation-level random effects (Elston et al. 2001)
• overdispersed distributions
  (negative binomial, beta-binomial, etc.)
  glmmTMB/brms (Brooks et al. 2017)

other diagnostics

• assumption of Normal residuals may not be very good
• simulated residuals (DHARMa)

zero-inflation

• glmmTMB etc.
• Owls example

Complete separation

• penalized regression
• add priors!

Brooks, Mollie E., Kasper Kristensen, Koen J. van Benthem, Arni Magnusson, Casper W. Berg, Anders Nielsen, Hans J. Skaug, Martin Maechler, and Benjamin M. Bolker. 2017. “Modeling Zero-Inflated Count Data With glmmTMB.” BioRxiv, May, 132753. doi:10.1101/132753.

Elston, D. A., R. Moss, T. Boulinier, C. Arrowsmith, and X. Lambin. 2001. “Analysis of Aggregation, a Worked Example: Numbers of Ticks on Red Grouse Chicks.” Parasitology 122 (5): 563–69.