Linear mixed models

An introduction with applications in veterinary research

Table of Contents

Luc Duchateau

Biometrics Unit
International Livestock Research Institute (ILRI)
Nairobi, Kenya

Paul Janssen

Center for Statistics
Limburgs University Center (LUC)
Diepenbeek, Belgium

John Rowlands

Biometrics Unit
International Livestock Research Institute (ILRI)
Nairobi, Kenya

International Livestock Research Institute
P.O. Box 30709, Nairobi, Kenya

Center for Statistics, Limburgs University Center
Universitaire Campus, B-3590 Diepenbeek, Belgium

To Pascale: who fills my life with joy
Luc Duchateau

The International Livestock Research Institute (ILRI) began operations on 1 January 1995. ILRI is one of 16 centres in a worldwide agricultural research networks sponsored by the Consultative Group on Agricultural Research (CGIAR). ILRI conducts strategic research in the biological, animal and social sciences to improve livestock productivity in sustainable agricultural systems throughout the developing world. The objectives of the institute's research programmes are to improve animal health, nutrition and productivity (milk, meat, traction) in ways that are sustainable over the long term, to characterise and conserve the genetic diversity of indigenous tropical forage species and livestock breeds, to promote sound and equitable national policies for animal agricultural and natural resource management, and to strengthen the animal husbandry research programmes of developing countries.

The Limburgs University Center (LUC, Belgium), founded in 1973, offers education in medicine, economy and science. The Center for Statistics operates within the Faculty of Sciences. The three main activities of the centre are teaching, research and consultancy. One of the important activities is the organization of the international course programme 'Master of Science in Biostatistics'. This advanced study programme offers a solid education in statistics in general and biostatistics in particular. Completion of the full programme provides qualifications for positions in research centres, governmental services and industry. This programme is also ideally suited for students who want to pursue research towards a doctoral degree in statistics or in biostatistics. The Belgian Administration for Developmental Cooperation (BADC, Brussels) provides a number of scholarships to students from developing countries who want to take part in this biostatistics programme (a preparatory programme of three months, followed by the regular programme of twelve months).

This publication was printed by ILRI, P.O. Box 5689, Addis Ababa, Ethiopia

ISBN 92–9146–038–9

Correct citation: Duchateau L., Janssen P. and Rowlands G.J. 1998. Linear mixed models. An introduction with applications in veterinary research. ILRI (International Livestock Research Institute), Nairobi, Kenya. 159 pp.

Table of Contents

Preface

1 Introduction

1.1 The linear mixed model framework

1.2 Examples

1.2.1 Comparing three doses of a drug: a completely randomised design 
1.2.2 Comparing two doses of a drug: a randomised block design 
1.2.3 Comparing two drugs: a completely randomised design with repeated observations 
1.2.4 Comparing two drugs at two different doses: two sizes of experimental units 
1.2.5 Comparing two drugs at two different doses: a splitplot design 
1.2.6 Variation in ELISA data: a random effects model 
1.2.7 Helminth-resistance in sheep: another random effects model 
1.2.8 Breed differences in susceptibility to trypanosomosis: a repeated measures design

1.3 Advantages of the mixed model

1.3.1 A general framework for data description 
1.3.2 Correct data analysis from the mixed model description 
1.3.3 Ability to use the wanted inference space 
1.3.4 Predicting specific random effects

1.4 Notation

2 The mixed model methodology

2.1 Introduction

2.2 The fixed effects model

2.2.1 The fixed effects model with one independent variable: one-way ANOVA
2.2.2 The fixed effects model with two independent variables: two-way ANOVA

2.3 A general description of the mixed model

2.3.1 The mixed model description based on individual observations
2.3.2 Mixed model description by example
2.3.3 The mixed model description in matrix form

2.4 Variance component estimation and blup

2.4.1 Defining variance components
2.4.2 Estimating variance components

2.4.3 Estimating variance components

2.5 Statistical inference for fixed effects

2.5.1 Estimation of fixed effects and their variance
2.5.2 Estimation of linear combinations of fixed effects and their variance
2.5.3 Confidence intervals
2.5.4 The appropriate inference space

2.6 Hypothesis testing

3  Into the realm of mixed models

3.1 Introduction
3.2 Designed experiments. Example 3.1
3.3 Mixed models with emphasis on random effects. Example 3.2
3.4 Covariance structures in the mixed model. Example 3.3

Appendix A Data sets

Appendix B SAS programs

Appendix C Linear algebra useful for linear models

Appendix D The normal distribution

Appendix E Variances of linear combinations

Appendix F Denominator degree of freedom

Appendix G Expected mean squares

References