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, presenting solutions of exercises, and handing in miniproject reports.
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 two
mini-projects:
First miniproject: Disease for cucumbers -
ANOVA with random effects with data.
Second miniproject: Linear mixed
model with AR(1) errors. Along with the project description
comes some code
and more code.
List of teaching sessions: (remember to refresh to get latest version of handouts).
- Handouts (with exercises, updated 02.02.23). Read M & T page 157-168.
- Handouts (with exercises, updated 08.02.23). Read M & T page 169-171 and 175-182.
- Handouts (with exercises) (updated 08.02.2023). 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.
- 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 27.02.23).
- We consider inference for anova-models Handouts (with exercises) (under construction) (section 5 in the ANOVA notes). Some R-code and data for the gene-expression example.
- We consider asymptotic inference for general linear mixed
models. Handouts (updated 31.01.24). Some R-code
for a simulation study and parametric bootstrap.
- Self-study. Groups work on and complete mini-projects.
- Self-study. Groups work on and complete mini-projects.
- Self-study. Groups work on and complete mini-projects.
- Self-study. Groups work on and complete mini-projects.
-
Prediction. Handouts (updated
11.04.23). Read M & T page 171-174 and 182-186.
- Conditional
independence, unnormalized densities (updated 17.04.2022) and Bayes
statistics (updated 13.04.2023).
- Bayesian statistics
continued. We conclude course by discussing a
few case studies regarding
mixed model analyses of randomized studies.
- Lectures on assorted topics for mixed
models. Bonus slides
and slides on autocorrelated noise
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