Mixed models

Introduction

Mixed models are models which combine “random” and “fixed” effects.

• Fixed effects are effects whose parameters we wish to estimate
  • The slope of a phosphorus response
  • The difference in shield thickness between males and females
• Random effects are effects that we wish to account for without estimating individually
  • Neuron identity
  • Village of residence

Modern and classic approaches

The “modern” approach to mixed models involves explicitly estimating “random” parameters and their variances (see below).

Compared to the classic ANOVA approach to mixed models, the modern approach is:

• conceptually clear
• flexible
• powerful
• computationally difficult

Example

Testing the effects of acid rain on spruce tree growth.

We have a complex manipulation (what kind of air the trees are exposed to). As a result:

• We have a small number of trees
• We take a lot of samples from each tree

Model

“Treatment” is a fixed effect

• We want to estimate the difference in growth with clean air and dirty air

“Tree” is a random effect

• We want to know what about the distribution of tree effects (so we can control for it); we are not specifically interested in the difference in growth between tree 3 and tree 5

Random effects

Random effects are based on unordered factors

• the levels of the factor are conceptualized as random samples from a larger population
• the effect of each level is therefore a random variable
• the essential parameters we estimate are not the effect of each level, but the mean and variance of the distribution.

Is this a random effect?

Treating something as a random effect means treating the levels as interchangeable from the point of view of your scientific hypothesis.

There is sometimes controversy about when it is appropriate to model a predictor using random effects. Here are some criteria …

philosophical questions

• are the levels chosen from a larger population?
• are the levels chosen randomly?
• are the levels a non-exhaustive sample of possible levels?
• do you want to be able to make predictions about new (unobserved) levels?
  • or inferences that include them?
• are you interested in the distribution of levels/variability among levels?
• are you uninterested in testing hypotheses about specific levels?

Types of analysis

• multilevel or hierarchical models
  • each identifiable level could have its own random effect: e.g., country, village, household
• repeated measures
  • individual identity can be associated with a random effect
  • this can also be done with residual structures

inferential questions

The choice to make something a random effect will affect your inferences about other variables than on your inferences about the focal variable

• Inferring using a fixed effect means we are inferring across the group of levels we have measured (only)
• Inferring using a random effect means we are inferring across a population represented by the group of levels we have measured

Examples

Influenza vaccination experiment

• Year as a fixed effect: did vaccination “help” on average over these years (with a particular set of flu strains, etc.)
• Year as random effect: will vaccination “help” on average over a wider set of years (assuming observed years are representative)

Spruce trees

• May not have enough data to do a random effects analysis
• Our statistical inference is then limited to the areas we studied
• Whether we can broaden our scientific inference is a scientific question (not statistical)

Practical questions

• Modern methods
  • have you measured a sufficient number of levels to estimate a variance (>5, preferably >10)?
  • May need more for more complicated models
• Classic methods
  • Do you have a balanced design?
• Otherwise
  • Limit your inference
  • Try an even more advanced approach (usually Bayesian)

Random slopes

• Are only intercepts affected by random factors, or also responses?
  • e.g., spruce trees
• Random slopes are often conceptually appealing, but computationally challenging
  • There is a (questionable) culture of not worrying about them
• Conceptually we are thinking about interactions between a fixed effect (e.g., level of acidity) and a grouping variable (spruce trees)

Fitting

Modern mixed-model packages (see below) can fit a wide variety of models. You just need to specify which effects are random.

• Works with unbalanced designs
• Works with crossed random effects
• May require a lot of data (and sometimes a lot of levels) for a reliable fit
  • Depends on model complexity

Too few levels

If you have something that should be a random-effect predictor, but you don’t have enough levels, you can’t fit a modern mixed model

It’s OK to treat your random effect as a fixed effect, as long as this is properly reflected in your scientific conclusions.

• The scope of your analysis covers only the sampled levels, not the population they were sampled from

Residual structure and group structure

Standard random effects: we attach a random effect to the group

• Sometimes called G-side modeling
• Group membership is binary (same or different)
• Hierarchy is possible
  • You and I are in the same country, but not the same village
  • But the model doesn’t understand if our villages are close together

R-side modeling: we impose a covariance structure on the residuals

• Arbitrary variance-covariance structure (time, space, etc.)
  • I am very close to Ben, and sort of close to you
• Can also allow for heteroscedasticity
• Disadvantages
  • Harder to combine effects
  • Easier to mis-specify the model
  • Difficult to implement (especially for generalized models)

How modern methods work

It’s complicated!

How modern methods work

Typically based on marginal likelihood: probability of observing outcomes integrated over different possible values of the random effects.

Balance (dispersion of RE around 0) with (dispersion of data conditional on RE)

Shrinkage: estimated values get “shrunk” toward the overall mean, especially in small-sample/extreme units

How do we do it?

Different for linear mixed models (LMMs: normally distributed response) and generalized linear mixed models (GLMMs: binomial, Poisson, etc.)

• LMMs: REML vs ML
  • analogy: \(n-1\) in estimation of variance
  • analogy: paired \(t\)-test
  • what do I do when I have more than 2 treatment levels per block?
• GLMMs: PQL, Laplace, Gauss-Hermite

Or:

• Bayesian approach: put a combined prior on the parameters

Practical details in R

• Classical designs
  • aov + Error() term
• Modern methods
  • lme (nlme package): older, better documented, more stable, does R-side models, complex variance structures, gives denominator df/\(p\) values
  • (g)lmer (lme4 package): newer, faster, does crossed random effects, GLMMs
  • many others (see task view)