11 May 2016

## Acknowledgments

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

## (Generalized) linear mixed models

(G)LMMs: a statistical modeling framework incorporating:

• combinations of categorical and continuous predictors,
and interactions
• (some) non-Normal responses
(e.g. binomial, Poisson, and extensions)
• (some) nonlinearity
(e.g. logistic, exponential, hyperbolic)
• non-independent (grouped) data

## Technical definition

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

## What are random effects?

A method for …

• accounting for among-individual, within-block correlation
• compromising between
• complete pooling (no among-block variance)
• fixed effects (large among-block variance)
• handling levels selected at random from a larger population
• sharing information among levels (shrinkage estimation)
• estimating variability among levels
• allowing predictions for unmeasured levels

## Random-effect myths

• levels of random effects must always be sampled at random
• a complete sample cannot be treated as a random effect
• random effects are always a nuisance variable
• you can't say anything about the predictions of a random effect
• you should always use a random effect no matter how few levels you have

## Maximum likelihood estimation

• 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

%

## Estimation methods

• 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

## Wald tests

• typical results of summary()
• exact for ANOVA, regression:
approximation for GLM(M)s
• fast
• approximation is sometimes awful (Hauck-Donner effect)

## Likelihood ratio tests

• 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

## Parametric bootstrapping

• 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)

## Bayesian inference

• 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

## formula formats

• 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

## On beyond lme4

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

## On beyond R

• Julia: MixedModels package
• SAS: PROC MIXED, NLMIXED
• AS-REML
• Stata (GLLAMM, xtmelogit)
• AD Model Builder; Template Model Builder
• HLM, MLWiN
• MCMC: JAGS, Stan, rethinking package

## Challenges

• 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

## Spatial and temporal correlations

• 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 … ?

## Next steps

• 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

## Sales pitch

• http://ms.mcmaster.ca/~bolker/misc/private/14-Fox-Chap13.pdf
• supplementary material
• B. M. Bolker (2015)