Introduction

F. Guillaume Blanchet – Andrew MacDonald

2024-04-29

The very basic

Internet

Network : jouvence

Password : Orford2021

Dos and don’ts at Jouvence

Kim will give us a quick explanation about this

Course material

Website : https://bios2.github.io/hiermod/

On it, you will find the slides but also practical examples.

You will also find the course syllabus (in French)

Here is a brief overview of the course

Statistiques avancées en sciences de la vie (BIO709/BIO809)

3 credits course at Université de Sherbrooke

It is a pass or fail course…

So, if you get involved, ask questions, try, work… you will pass.

What we want to do during this course

Basic knowldege of statistics

  • General introduction
  • General overview of probability theory… for the life sciences
  • Probability distribution
  • Frequentist and Bayesian modelling
  • Data simulation (how and when)
  • Data simulation using different probability distribution

What we want to do during this course

Statistical computing

  • Introduction to Stan

We will not give an introduction to any programming language (e.g. R, Python, Julia).

We assume you know one and are conformtable using it… We are more comfortable in R, but can work with many other languages (but please no Fortran !)

What we want to do during this course

Simple regression models

  • Refresher in matrix algebra
  • Refresher of linear model
  • Generalized linear model

Simple hierarchical regression models

  • Model with a single hierarchical component on the intercept
  • Model with a single hierarchical component on the slope

What we want to do during this course

Complexe hierarchical regression models

  • Model with multiple hierarchical terms
  • Model with multiple hierarchical levels
  • Hierarchical model with constraints
  • Gaussian process

What we want to do during this course

 

We also want to learn about your research… and this might influence what we see during the course !

Course general structure

General daily schedule

7h00 to 8h30 - Breakfast

8h30 to 10h00 - Lecture/Practice

10h00 to 10h30 - Break

10h30 to 12h30 - Lecture/Practice

12h30 to 13h30 - Lunch

13h30 to 15h00 - Lecture/Practice

15h30 to 16h00 - Break

16h00 to 17h00 - Lecture/Practice

17h00 to 17h30 - Let us know about your research !

17h30 to 18h00 - Break

18h00 to 19h00 - Supper

The course basics

What are hierarchical models ?

For this course, hierarchical models are regression models in which the parameters (the regression coefficients) are not defined by a single value but they are themselves given a probability distribution (Gelman and Hill 2007).

Particularities of hierarchical models

Hierarchical models are

  • A challenging bit of technology (probably more than you might think!)
  • Very flexible models (in many more ways that are usually expected!)
  • Constrained to the same particularities as (generalized) (non) linear models (sometimes to a more severe extent)

A bit of vocabulary

Hierarchical models, as we will see them in this course, are also known under different names

  • Random effect models
  • Mixed models
  • Multilevel models
  • Variance component models
  • Error component models

A bit of vocabulary

We decided to use the term hierachical model to prevent confusions that sometimes arises in the litterature about random and fixed effects, which are terms commonly used when referring to mixed effect models.

Actually, random and fixed effects have multiple definitions, which leads to confusion.

A bit of vocabulary

Multiple Definition of fixed and random effects

(Kreft and De Leeuw 1998) Fixed effects are constant and random effect vary

(Searl et al. 1992) Effects are fixed if they are interesting in themselves or random if there is interest in the underlying population

(Green and Tukey 1960) When a sample exhausts the population, the corresponding variable is fixed; when the sample is a small (i.e., negligible) part of the population the corresponding variable is random

(Roy LaMotte 2014) If an effect is assumed to be a realized value of a random variable, it is called a random effect

(Robinson 1991) Fixed effects are estimated using least squares (or, more generally, maximum likelihood) and random effects are estimated with shrinkage.

A bit of history

Because of the different name used for hierarchial models, the history of this subfield of statistics is a little murky, but we know that a few important figures in statistics dabbed into this field, most notably

Fisher R.A. (1919). The Correlation between Relatives on the Supposition of Mendelian Inheritance. Transactions of the Royal Society of Edinburgh, 52 399–433.

Implementation

  • Hierarchical models have been implemented in many software packages,
  • in R
    • lme4, brms, nlme, glmmTMB, MCMCglmm, …
  • in SAS
    • MIXED, HPMIXED, GLMMIX, …
  • in Julia
    • MixedModels.jl
  • …

Implementation

What we will use

We will not use any of these software packages because under specific circumstances, what may seem like the same implementation of a model may lead to different answers and both can actually be right !

This is because the underlying model implemented in the software package may actually be slightly different.

Instead we will implement our own models from scratch using Stan.

Good reference

A good portion of this course material is based on this book.

Great technical references

Everything is there but it can gets technical !

R and Stan

All of the practical aspect of the course will be done with R and Stan.

RStudio

We strongly (!) encourage you to use RStudio and to start a project for the course.

R package to install

# Package on CRAN
install.packages(c("vegan",
                   "palmerpenguins",
                   "tidybayes",
                   "tidyverse",
                   "posterior"))

# cmdstanr - One last important package
install.packages("cmdstanr",
                 repos = c("https://mc-stan.org/r-packages/",
                           getOption("repos")))
cmdstanr::check_cmdstan_toolchain()
cmdstanr::install_cmdstan(cores = 2) # If necessary