# Mixed models

This course is concerned with statistical inference for linear mixed models. We will mainly consider likelihood based inference from a frequentist point of view. We also consider briefly Bayesian statistics.

The textbook for the course is

• Madsen, H. and Thyregod, P. (2011) Introduction to General and Generalized Linear Models, Chapman and Hall/CRC, parts of chapter 5 and 6.
I will to a large extent cover material from M & T but I will not follow the book page by page. I will also discuss some material not covered in M & T. I for instance also use the notes The curriculum for the course consists of the pages you are asked to read in M & T, the ANOVA notes, and the handouts provided (note: the exercises and their solutions are also part of the curriculum).

You may find it useful to also consult Faraway (2006) Extending the linear model with R, Chapman and Hall/CRC, chapter 2 and 8, which is a useful reference regarding more practical aspects of fitting mixed models. Pages 41-60 in Demidenko (2013) Mixed Models: Theory and Applications with R, Second Edition, give technical background on MLE and RMLE. Pages 9-49 in West et al (2014) Linear Mixed Models: A Practical Guide Using Statistical Software, Second Edition, also give background on linear mixed models and MLE.

The original reference regarding analysis of variance for orthogonal designs is Tjur, Tue (1984) Analysis of variance models in orthogonal designs, International Statistical Review, 52, 33-65.

The course will be evaluated by active participation in solving exercises and presenting solutions of exercises.

Some data sets used in the course. The orthodontic data set "Orthodont" is a part of the nlme R package.

During the course you will work on and present several mini-projects:

First miniproject: ANOVA for split-plot experiments - theory and practice.

Second miniproject: Linear mixed model with AR(1) errors - parts 1-3. Along with the project description comes some code and more code.

The following will be updated during the course (remember to refresh to get latest version of handouts).

1. Handouts (with exercises, updated 02.02.21). Read M & T page 157-168.

2. Handouts (with exercises, updated 08.02.21). Read M & T page 169-171 and 175-182.

3. Handouts (with exercises) (updated 19.02.2021). We first consider practical implementation of ML and REML for linear mixed models. Secondly we consider a geometric approach to one-way anova (sections 1-2 in the ANOVA notes). We also discuss the so-called hierarchical principle - try this R-code to get a better understanding of the importance of the hierarchical principle and the implications of different parametrizations.

4. We continue with two-way ANOVA for balanced designs and some examples of data analyses (sections 3-4 in the ANOVA notes). Handouts (with exercises)(updated 03.03.21).

5. Students present solutions of exercises.

6. We consider inference for anova-models Handouts (with exercises) (updated 03.03.21) (section 5 in the ANOVA notes). Some R-code and data for the gene-expression example.

7. We consider asymptotic inference for general linear mixed models. Handouts (updated 09.03.21). Some R-code for a simulation study and parametric bootstrap.

8. Presentation of exercises.

9 - 12. Self-study. Groups work on and complete mini-projects.

13 Prediction. Handouts (updated 25.03.21). Read M & T page 171-174 and 182-186.

14 Students present miniprojects.

16 Bayesian statistics continued.