Introduction to Generalized linear mixed models?

Harnessing non linearity random effects

Julien Martin

BIO 8940 - Lecture 8

2024-09-19

Questions after reading Bolker et al 2009

• Difference between fixed and random effects

• When to transform data?

  • if you have a funky looking distribution of continuous data, is it always ok to transform to achieve normality if you don’t violate any test assumptions?

• Walkthrough Figure 1 ?

• Get rid of non-significant fixed effects?

  • If important for my hypothesis, should I always keep them?
  • What if I have a fairly small dataset?

• How to choose a link function? Why not using the default?

• Can we go through example in Box 1?

GLMM: What are they?

GaCha Life Minie Movie

Video game allowing you to dress-up anime style characters

Generalized linear mixed model

An extension to Generalized linear model and an extension to linear mixed model

GLMM expresses the transformed conditional expectation of the dependent variable y as a linear combination of the regression variables X

Model has 3 components

• a structural component or additive expression \(\beta_0 + \beta_1 X_1 + ... + \beta_k X_k\)
• a link function: \(g(\mu)\)
• a response distribution: Gaussian, Binomial, Bernouilli, Poisson, negative binomial, zero-inflated …, zero-truncated …, …

\[ g(\mu_i) = \beta_0 + \beta_1 X_1 + ... + \beta_k X_k \]

and

\[ \mu_i = E(y_i | x_i) = g(\mu_i)^{-1} \]

How do you fit them?

In R:

• glmer() from lme4 📦 same as lmer() but with a family argument
• glmmPQL() from MASS 📦 (based on lme())
• glmmADMB() from - glmmADMB 📦 works well and flexible be beware
• glmmTMB() from glmmTMB 📦 works well and flexible be beware
• asreml() from glmmTMB 📦 great but not-free
• MCMCglmm() from MCMCglmm 📦 great but Bayesian
• Choose you bayesian flavor 📦:
  • stan: brms, rethinking, rstan, …
  • BUGS: runjags, rjags, …

Model assumptions

• Easy answer none or really few

• More advanced answer I am not sure, it is complicated

• Just check residuals I as usual

• Technically only 3 assumption:
  • Variance is a function of the mean specific to the distribution used
  • observations are independent
  • linear relation on the latent scale

Warning

Generalized Linear Models do not care if the residual errors are normally distributed as long as the specified mean-variance relationship is satisfied by the data

Choosing a link function

A link function should map the stuctural component from \((-\infty,\infty)\) to the distribution interval (e.g. (0,1) for binomial)

So number of link function possible is extremley large.

Choice of link function heavily influenced by field tradiditon

For binomial models

• logit assume modelling probability of an observation to be one
• probit assume binary outcome from a hidden gaussian variable (i.e. threshold model)
• logit & probit are really similar, both are symmetric but probit tapers faster. logit coefficient easier to interpret directly
• cologlog not-symmetrical

Estimating repeatability ?

Latent scale

Business as usual ?

Observed scale ??????

• Using rptR 📦 is the easiest or QGGlmm 📦 (see associated citation for reference and explanations)

Marginalized vs Conditioned estimates

Difference between marginalized and conditioned coefficients?

GLMMadaptive 📦 is the only way I know to do easily get marginalized coefficients

Practical

Walkthrough Example box 1

Happy modelling