Course Mixed Models

The course Mixed Models is offered by the Graduate School of Natural Sciences/GSNS and Graduate School of Technical Sciences/GSTS, Aarhus University, August 2025.

Name of course: Mixed Models

ECTS credits: 3.0 ECTS

Course parameters: No. of contact hours/hours in total incl. preparation, assignment(s) or the like: 12h of lectures distributed in two lecture days (9:15-12:00, 13:15-15:00); 39,5h for working in homework, self-study, consultations and interviews with the instructor(s) and oral examination; 42 h for writing a report. The total work load is of 90h . 

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, Generalised Linear Mixed Models and simple Multivariate Generalised Linear Mixed Models.

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

Regarding the statistical knowledge, it is assumed that the PhD student knows:

• Basic statistical inference theory, including basic notions of probability, probability distributons, conditional probability, expectation and variance of random variables;
• Notions of parametric models and statistical inference, including: likelihood based estimation, confidence intervals and hypothesis tests, linear models and basic generalised linear models for non-Gaussian distributed responses (e.g., binomial and Poisson generalised linear models).

The skills mentioned above can be obtained in the PhD courses “Basic Statistical Analysis” (see  https://tildeweb.au.dk/au33031/Courses/BasicStatisticalAnalysis). It is crucial that the students have the pre-requisite at the beginning of the course.

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  https://tildeweb.au.dk/au33031/Courses/Introduction2R ).

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, generalised linear mixed models and simple multivariate generalised 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) formulation of the statistical models used and 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 programme:
Time schedule: 2 lecture days from 9:15am  to 12:00am and from 13:15 to 15:00 supplied by office hours. Additionally, the participants should write a report on the assignments involving two relevant full statistical analyses proposed. The course responsible will offer the participants to give a feedback to a draft of the reports, provided the draft is delivered before a deadline established at the beginning of the course by the course responsible.

The course is divided into the 12 modules listed in the schedule below:

Schedule (tentative):

Week 33, day 1 -14th August 2025:

09:15 - 10:00 - topic 1- Introduction and course (informal) overview

10:15 - 11:00 - topic 2 - A short (directed) review of Gaussian linear models

11:15 - 12:00 - topic 3 - The idea of random components

12:00 - 13:15 - Lunch break

13:15 - 14:00 - topic 5 - Basic inference for simple Gaussian linear mixed models

14:15 - 15:00 - topic 6 - Models for repeated measurements with serial correlation

Week 34,  day 2 - 21st August 2025:

09:15 - 10:00 - topics 7 - A short (directed) review of generalised linear models

                   and  topic 8 - From generalised linear models to generalised linear mixed models

10:15 - 11:00 - topic 9 - Poisson mixed models for counts

11:15 - 12:00 - topic 10 - Binomial and Bernoulli mixed models for proportions

12:00 - 13:15 - Lunch break

13:15 - 14:00 - topic 11 - Gamma and inverse Gaussian models for positive responses

14:15 - 15:00 - topic 12 - Closing section with an informal exposition on multivariate generalised linear mixed models (advanced topic for information only, not included in the evaluation).

The time spent in each topic might vary slightly, according to the pedagogical necessity.

Week 35-38: Elaboration of the written report on two relevant full statistical analyses (choosen from the collection of problems proposed at the beginning of the course). During these weeks there will be set open office hours for the students discuss matters related to the course (in particular the two analyses to be reported).

Oral examination: 25th and 26th September 2025 (45 minutes interview with each participant).

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 studies. Next, the class of generalised linear models is presented, which allow modelling non-Gaussian responses (e.g., binomial, Poisson, Gamma and Inverse Gaussian distributed responses) and non-linear relationships with explanatory variables. Finally, simple multivariate generalised linear mixed models, which are presented and discussed. These last models allow to study several responses (possibly of different nature) simultaneously. In all cases emphasis is put on applications; mathematical and theoretical details will not be emphasised but instead the conscientious use of the models studied will be aimed.

A written report of two full statistical analyses (chosen from a collection of problems presented at the beginning of the course) should be delivered at the deadline established at the beginning of the course and discussed in a final oral examination.

Name of lecturer and course responsible:
Rodrigo Labouriau

Type of course/teaching methods:
Lectures alternated with supervised exercises and self-study including the elaboration of a report on two relevant full statistical analyses proposed at the beginning of the course. The course responsible will offer the participants to give a feedback to a draft of the reports, provided the draft is delivered before a deadline established at the beginning of the course by the course responsible.

Course homepage:
A general description of the course, further details and software can be found at https://tildeweb.au.dk/au33031/Courses/PhDMixedModels

Course assessment:
Attendance to 80% of the lectures is a necessary condition to participation in the oral examination.

In the last part of the course, there will be a final project (two datasets that should be analysed). The final examination constitutes of a written report on the final project and an oral examination based on the report. Using the oral examination and the written report, it will be evaluated whether the PhD student passes the course.

Provider:
Applied Statistics Laboratory (aStatLab) at the Department of Mathematics, Aarhus University in collaboration with the GSNS. Course responsible: Rodrigo Labouriau (aStatLab, MATH AU).

Special comments on this course:
The PhD student must masters some basic statistical techniques and the software R as described in Criteria for participation.

Time:
See the schedule above.

Venue:
Department of Mathematics (at Aarhus University).

No show fee:
Course participants on our transferable skills courses, who do not show up at the course or cancel their course participation after the course cancellation deadline (without providing a doctor’s note), may have to pay a no-show fee, unless someone from the waiting list is able to take part in the course instead.

The no-show fee is DKK 1,200 (the price of one ECTS). The no-show fee has been introduced due to many late cancellations, thus preventing people from the waiting lists to have a seat at the courses.

Registration:

• Participation in the course is without cost for PhD students from Aarhus University

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.

Please be aware that your registration for the course not necessarily equals your admission for the course. You will receive an e-mail after the registration deadline regarding whether you are admitted for the course or if you are registered on the waiting list. Please note that seats are allocated on a first-come-first-served basis.

Course dates:

• 14 August 2025 09:00 - 15:00
• 21 August 2025 09:00 - 15:00
• 25 September 2025
• 26 September 2025