Mixed Models, Autumn 2018

ECTS credits: 3 ECTS


Course parameters:

Language: English

Level of course: PhD

Time of year: September 2018

No. of contact hours/hours in total incl. preparation, assignment(s) or the like: 18h of lectures distributed in three lecture days (9:00-12:00, 13:00-16:00); 36h  for working in homework, self-study and consultations and interviews with the instructor(s);  26h for writing a report and preparing an internal seminar. The total work load is of 80h (18h+36h+26h).

Capacity limits: Minimum 10 and maximum 15.


Objectives of the course:

The course aims to provide the basic tools to use Mixed Models (including Gaussian Linear Mixed Models, Models for Repeated Measures, Generalized Linear Mixed Models and simple Multivariate Generalized Linear Mixed Models).


Learning outcomes and competences:

At the end of the course, the student should be able to:

1) Describe and discuss the use and applicability of classic statistical models for dependent responses based on random components, including: Gaussian linear mixed models, generalized linear mixed models and simple multivariate generalized linear mixed models.

2) Conduct (under supervision) statistical analysis of data with dependence structure using the models abovementioned, including:

a) the identification of pertinent models for answering the biologic/scientific question of interest,

b) identification of the key assumptions related to those statistical models,

c) conduction of the analysis using modern software (R),

d) model control and verification of the key assumptions, and e) draw reasonable conclusions from those analyses and report written and orally the results obtained.


Compulsory program:

A written report of the final assessment related to a concrete mini-project should be delivered at the end of the course and discussed in an internal seminar.


Course contents:

The course starts by revising the basic theory of Gaussian Linear Models (i.e. linear models based on the normal distribution); these models are extended to the class Gaussian Linear Mixed Models that incorporate random components representing structures of dependency commonly found in dependent experimental and observational data. Next, the class of generalized linear models are presented, which allow to model non-Gaussian responses (e.g. binomial, Poisson, Gamma and Inverse Gaussian distributed responses) and non-linear relationships with explanatory variables. Finally, simple multivariate generalized linear mixed models, which are presented and discussed. These last models are models allow to model several responses (possibly of different nature) simultaneously. In all cases emphasis is put in application; mathematical and theoretical details will not be emphasised but instead the conscientious use of the models studied will be aimed.



Tentative time schedule: 3 lecture days from 9:00 am (sharp) to 12:00 am and 3 practical lectures from 13:00 to 16:00 disposed according to the following time-schedule:

Tuesday 4 September 2018:

Lecture 1: Introduction, Practical Matters and Gaussian Linear Mixed Models

a) Introduction and overview of the course the course (carrousel of examples)

b) Short review of Gaussian Linear Models (GLM): model definition, fitting GLM, inference and model control of GLM

c) Gaussian linear models with random components: estimation and inference of random effects, prediction of random effects, techniques for investigating the covariance structure, repeated measures and longitudinal data

Practical Lecture 1: Simulation of linear and generalized linear mixed models

a) Simulation of simple linear and generalized linear mixed models

b) Monte Carlo power calculations for tests involving linear and generalized linear mixed models

c) Case studies

Homework 1: Tutorials on simulations, case studies, literature reading (including strategic chosen articles and chapters of classic and modern books).


Tuesday 11 September 2018:

Lecture 2: Generalized Linear Mixed Models

a) Short review of Generalized Linear Models (GLIM): Binomial, Poisson, Gamma and Inverse Gaussian models, basic theory of GLIM, inference and model control of GLM

b) Basic theory of Generalized Linear Mixed Models (GLIMM, i.e. GLM with random components: Definition of GLIMMs, estimation and hypothesis tests for GLIMM (via parametric bootstrap)

Practical Lecture:

a) Tutorials and discussion on the construction of tests for GLMM via parametric bootstrap

b) Tutorials and discussion on model control techniques for GLMM

c) Case studies.

Homework 2: Case studies, literature reading (including strategic chosen articles and chapters of classic and modern books). Elaboration of a report on a mini project based on a concrete case.


Tuesday 25 September 2018:

Lecture 3: Multivariate Generalized Linear Mixed Models and Closing

a) Simple Multivariate Generalized Linear Mixed Models: Initial examples, definition of the models, basic inference theory and case-studies.

b) Internal seminar presentation (mini-projects), general discussion on potential use of the techniques introduced and discussed during the course.

c) Course evaluation



The PhD student must masters some basic statistical techniques and the software R as described below.

Regarding the statistical knowledge, it is assumed that the PhD student knows basic inference theory for parametric models, including: estimation, confidence interval and hypothesis testing, linear models and basic generalized linear models. These skills can be obtained in the courses “Basic Statistical Analysis in Life and Environmental Sciences” (see home.math.au.dk/rodrigo/Courses/BasicStatisticalAnalysis ).

The course will use the software R as a tool, but it is NOT a course on R. It will be assumed that the PhD students have the software R installed on their computers, and that they know the basic notions of R programming. This includes knowing to: read and write data in R, perform basic operations with variables and vectors, make simple tabulations, use simple functions, use repeated and conditional calculations and draw simple graphs. These skills can be obtained in the course “Introduction to R” (see home.math.au.dk/rodrigo/Courses/Introduction2R ).

Name of lecturer:

Rodrigo Labouriau.



GSST / Department of Mathematics (Applied Statistics Laboratory)


Type of course/teaching methods:

Lectures alternated with supervised exercise, homework and self-study including the elaboration of an internal seminar based on a simple concrete mini-project.


Place: Aarhus Universitet, room to be announced.



Deadline for registration is 24 August 2018.


No show fee:

Please note the following: As from 2017, a no-show fee is introduced at GSST’s transferable skills courses for course participants who do not show up at the course or cancel their course participation after the course registration deadline – unless they can provide a Doctor’s note. The no-show fee will be DKK 1,200 (the price of one ECTS). The no-show fee is introduced because GSST has experienced many late cancellations, thus preventing people from the waiting lists to have a seat at the courses.

Due to an Agreement between Danish Universities coming into force as of 1 January 2011, participants from other universities than Aarhus University will have to pay DKK 1,200 per ECTS. In principle this also applies to external parties, but exemption can be granted under specific circumstances.


Registration for participants from Aarhus University:



Registration for participants from other universities:



If you have any questions, please contact PhD Partner Karen Konradi, GSST (konradi@au.dk).