• lme4: Doug Bates, Martin Mächler, Steve Walker
• Data: Josh Banta, Adrian Stier, Sea McKeon, David Julian, Jada-Simone White
• \($\): NSERC (Discovery), SHARCnet

\[ \begin{split} \underbrace{Y_i}_{\text{response}} & \sim \overbrace{\text{Distr}}^{\substack{\text{conditional} \\ \text{distribution}}}(\underbrace{g^{-1}(\eta_i)}_{\substack{\text{inverse} \\ \text{link} \\ \text{function}}},\underbrace{\phi}_{\substack{\text{scale} \\ \text{parameter}}}) \\ \underbrace{{\boldsymbol \eta}}_{\substack{\text{linear} \\ \text{predictor}}} & = \underbrace{{\boldsymbol X}{\boldsymbol \beta}}_{\substack{\text{fixed} \\ \text{effects}}} + \underbrace{{\boldsymbol Z}{\boldsymbol b}}_{\substack{\text{random} \\ \text{effects}}} \\ \underbrace{{\boldsymbol b}}_{\substack{\text{conditional} \\ \text{modes}}} & \sim \text{MVN}({\boldsymbol 0},\underbrace{\Sigma({\boldsymbol \theta})}_{\substack{\text{variance-} \\ \text{covariance} \\ \text{matrix}}}) \end{split} \]

• Best fit is a compromise between two components
  (consistency of data with fixed effects and conditional modes; consistency of random effect with RE distribution)
• Goodness-of-fit integrates over conditional modes

%

• deterministic: various approximate integrals (Breslow 2004)
  • Penalized quasi-likelihood, Laplace, Gauss-Hermite quadrature (Biswas 2015)
    best methods needed for large variance, small clusters
  • flexibility and speed vs. accuracy
• stochastic (Monte Carlo): frequentist and Bayesian (Booth et al. 1999,Sung et al. (2007),Ponciano et al. (2009))
  • usually slower, more finicky, but flexible and accurate

• better than Wald, but still have two problems:
  • "denominator degrees of freedom'' (when estimating scale parameter)
  • for GLMMs, distributions are approximate anyway (Bartlett corrections)
  • Kenward-Roger correction? (Stroup 2014)
• Profile confidence intervals: expensive/fragile

• fit null model to data
• simulate "data" from null model
• fit null and working model, compute likelihood difference
• repeat to estimate null distribution
• should be OK but ??? not well tested
  (assumes estimated parameters are "sufficiently" good)

• If we have a good sample from the posterior distribution (Markov chains have converged etc.) we get most of the inferences we want for free by summarizing the marginal posteriors
• post hoc Bayesian methods: use deterministic/frequentist methods to find the maximum, then sample around it

• fixed: fixed-effect formula
• random: random-effect formula (in lme4, combined with fixed)
  • simplest: 1|g, single intercept term
  • nested: 1|g1/g2
  • random-slopes: x|g
  • independent terms: (1|g)+(x+0|g) or (x||g)
• lme: weights, correlation for heteroscedasticity and residual correlation
• MCMCglmm: options for variance structure

• basic
  • nlme (lme)
  • MCMCglmm
  • inference/tests: lmerTest, afex, pbkrtest
    (car, lsmeans, effects, multcomp)
  • blme (Bayesian regularization)
  • gamm4 (additive models)
  • pretty output: broom, dotwhisker, pixiedust
• advanced/specialized
  • glmmADMB, glmmTMB: zero-inflated and other distributions
  • brms, rstanarm: interfaces to Stan
  • INLA: spatial and temporal correlations

• Small clusters: need AGQ/MCMC
• Small numbers of clusters: need finite-size corrections
  (Kenward-Roger/parametric bootstrap/MCMC)
• Simpler inference: importance sampling, quantiles, MCMC?
• Small data sets: issues with singular fits
  Barr et al. (2013) vs. Bates et al. (2015)
• Big data: speed, storage, parallelization
• Model diagnosis
• Confidence intervals accounting for uncertainty in variances

See also: ecostats chapter example; NCEAS modeling examples; BMB mixed models repo, including GLMM FAQ

• Sometimes blocking takes care of non-independence …
• but sometimes there is temporal or spatial correlation within blocks
• … also phylogenetic … (Ives et al. 2006)
• "G-side" vs. "R-side" effects
• tricky to implement for GLMMs,
  but new possibilities on the horizon (Rue et al. 2009; Rousset et al. 2014) (INLA, spaMM); lme4ord package
• CAR, SAR, Moran eigenvectors … ?
• additive models?

• Complex random effects:
  regularization, model selection, penalized methods (lasso/fence)
• Flexible correlation and variance structures
• Flexible/nonparametric random effects distributions
• hybrid & improved MCMC methods
• Reliable assessment of out-of-sample performance