Mixed models and Bayesian inference

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

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. Mat8/Mat8tek/Mat8øk students must complete an additional miniproject related to Bayesian inference.

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

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

First miniproject: ANOVA for split-plot experiments - theory and practice (updated 06.03.18).

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 05.02.18). Read M & T page 157-168.

2. Handouts (with exercises, updated 05.02.18). Read M & T page 169-171 and 175-182.

3. Handouts (with exercises). 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 01.03.18).

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

6. We consider asymptotic inference for general linear mixed models. Handouts (updated 09.03.18). Some R-code for a simulation study and parametric bootstrap. We also conduct a midway evaluation.

7 - 10. Self-study. Groups work on and complete mini-projects.

11. Groups present mini-projects.

12. Prediction. Handouts (updated 20.04.18). Read M & T page 171-174 and 182-186.

13. Conditional independence, unnormalized densities and the Kalman-filter. Slides (updated 21.04.18) and tutorial.

14. Bayesian inference. Slides (updated 24.04.18). Read sections 6.1, 6.2 and 6.4 in Chapter 6 in M & T.

15. Bayesian inference. Slides.

16. Presentation of exercises and Rasmus presents extra mini-projects for 8. semester students.

Third miniproject: Topics in Bayesian statistics: choose either large sample methods in Bayesian inference (first 5 pages) or choice of priors for variances in hierarchical models. Here is a short introduction to the two options. Some questions for third miniproject.

17. Presentation of miniproject 3 (8. semester students).