TOC.md

Mixed Models := ((--| generalized | non-) Linear | Generalized Additive) Mixed-Effects Models

i.e., LMM, GLMM, NLMM. GAMM

Table of Contents -- Lectures

Part I: Linear MM

1. Introduction - 1- and 2-way ANOVA and beyond

'The' nlme book: Pinheiro, J.C., and Bates, D.M. (2000) Mixed-Effects Models in S and S-PLUS, Springer.

available at https://link.springer.com/book/10.1007%2Fb98882 (download PDF: inside ETH VPN)

a. nlme (book) chapter 1: Rail, ... .. "anova" notation: 1-way anova, 2-way R(markdown) script: R/ch01.Rmd -- and nicely rmarkdown::render()ed html.

b. lMMwR (not-yet-book by Doug Bates) chapter 1: sleepstudy --

c. Comparison aov(), lme(), lmer().

d. Classical solution: Assume given Sigma (called G). Wikipedia Mixed_model

e. Classical solution does not scale to large q (q = ncol(Z))

f. --> New approach to ML and REML

2. General Notation: No "grouping" assumed --> general Z matrix

3. Inference: "No P values" vs lmerTest et al.

a. nlme (book) chapter 2, notably 2.4 R(markdown) script: R/ch02.Rmd, rmarkdown::render()ed html.

b0. Browse the ?pvalues help page (in the lme4 package).

b. The lmerTest package has an accompanying publication in JSS (Dec, 2017): https://www.jstatsoft.org/article/view/v082i13

from which we used the (slightly extended/modified) R code, lmerTest_v82i13.R in 2018, and an extended help page example on ham tasting data, lmerTest-ham-ex.R in 2019.

An extended version of JSS paper's Appendix A, is now available at Rune Christensen's Satterthwaite_for_LMMs -- rendering correctly at MM's MEMo-pages -- providing nice derivations on how the degrees of freedom are approximated.

• When can we trust the confidence intervals / P values ??
• Profiled Likelihood - based intervals; profile pairs
• Bootstrap -- non-iid?

c. Random slopes : non-scalar random effects etc: Ch. 3 of the lMMwR lecture notes, lMMwR.pdf; R code: ChLongitudinal.R.

Part II: Generalized LMM, e.g. for Count Data (Ecology)

• Example (contraceptive) of logistic GLMM: Ch. 6 of the lMMwR lecture notes, lMMwR.pdf; R code: ChGLMMBinomial.R.

• Likelihood for logistic GLMM

• From binary Bernoulli to Binomial (number of 1's) via aggregation (assuming groups of identical x_i)

• 2020: COVID-19 pandemic: News: Air pollution linked with higher COVID-19 death rates study from Harvard School of Public Health; e.g. New York Times, April 7. Harvard publication, updated several times; latest of April 24): https://projects.iq.harvard.edu/covid-pm.

  • mentions that everything is reproducible available on https://github.com/wxwx1993/PM_COVID , and asks to cite

  • Exposure to air pollution and COVID-19 mortality in the United States.
    Xiao Wu, Rachel C. Nethery, Benjamin M. Sabath, Danielle Braun, Francesca Dominici.
    medRxiv 2020.04.05.20054502; https://doi.org/10.1101/2020.04.05.20054502

  • I have forked the R code for the Generalized Linear Mixed Model (GLMM) which is their model to https://github.com/mmaechler/PM_COVID in order to add more.

• Venables and Ripley (2002) Modern Applied Statistics with S (= MASS), 4th ed.; chapter 7. Generalized Linear Models, intro pages 183-187.

• Somewhat gentle introduction to GLMs -- Springer Text Peter K. Dunn and Gordon K. Smyth (2018) Generalized Linear Models With Examples in R, from Ch. 4, downloadable from inside ETHZ.

• Modeling Count Data (by GLMs), including Hurdle and Zero-Inflation Models: Regression Models for Count Data in R vignette countreg from CRAN package pscl, slightly commented by MM, as Rnw, and pdf.

• Model formulation; likelihood approximations:

  • Laplace Approximation; generalized to Adaptive Gauss-Hermite Quadrature (=: AGQ) (in R, glmer(*, nAGQ = k)).

  • CRAN package GLMMadaptive promising special emphasis on AGQ. We use its GLMMadaptive Basics vignette, extended by MM. Get the Rmarkdown source Rmd, or R script, R, rendered html or pdf.

• (Outlook only:) CRAN package glmmTMB: Zero-inflation, hurdle models, etc; fast algorithms via automatic differentiation numerics (TMB = Template Model Builder).
  Two examples, comparison with other R pkgs in R journal paper

Part III: Nonlinear MM (NLMM / NLME)

a. Motivation and Examples: nlme (book) chapter 6 R(markdown) script: R/ch06.Rmd, rmarkdown::render()ed html.

b. Glimpses into theory: nlme book ch.7, notably p.306--309.

c. Outlook into PK/PD modeling (pharmacology): New R package nlmixr, incl. non-linear functions defined via differential equations (ODE).

d. Non-linear Mixed-Effects (NLME): nlme (book) chapter 8, adapted R(markdown) script: R/ch08.Rmd, rmarkdown::render()ed html.

Not treated in lecture:

• GAMM (Generalized Additive Mixed Models), e.g. mgcv::gamm() in R
• Robust Linear MM, e.g., R package robustlmm