This book presents a basic introduction to the theory of mixed models and relies heavily on examples from veterinary research to explain the important statistical concepts involved in mixed models. To a large extent, the mixed model has been developed in animal breeding and selection programs (Henderson, 1990). Lately, the mixed model methodology has been improved and it is now being used in several other disciplines. In this book, we show how it can be advantageous to use the mixed model in veterinary medicine. The idea to write a practical book on mixed models, with applications in veterinary research, originates from a course on mixed models, organised at the International Livestock Research Institute (ILRI) from 15 to 17 September 1997, and which was attended by researchers both from international (ICIPE, ILRI) and national research institutes in Kenya (KARI, KEMRI, KETRI, University of Nairobi and Ministry of Health). All data sets used as examples were kindly provided by investigators working at ILRI. We are grateful to Leyden Baker, Mark Eisler, Sonal Nagda and Kathy Taylor, who contributed data sets for this course.
Part of the theoretical material in this book is based on material that is also discussed (in greater depth) in Duchateau and Janssen (1997), a chapter that we contributed to the lecture notes on mixed models edited by Verbeke and Molenberghs (1997). We also benefited from Bryk and Raudenbush (1992), Goldstein (1995) and Searle, Casella and McCulloch (1992).
The book consists of three chapters. In Chapter 1 we define random and fixed effects factors, and based on these definitions we explain the difference between fixed effects models, random effects models and mixed models. Eight different data sets are described and classified according to one of the three models. We then give an heuristic discussion on the advantages of the mixed model methodology. Finally we introduce notation used in the remainder of the book. Chapter 2 deals with the mixed model methodology. The fixed effects model is explained first so that the mixed model can be constructed from it and both models can be compared. Next, it is shown how data structures can be expressed as mixed models. The data structure of the eight data sets is given in mixed model notation. For two of these we write the mixed model in matrix notation. The final two sections deal with variance component estimation and fixed effects estimation and hypothesis testing in the mixed model framework.
Chapter 3 contains three examples of complex data sets which are fully analysed using the mixed model methodology. All data sets are given in Appendix A. For the statistical analysis, we rely on PROC MIXED, a procedure of the Statistical Analysis System (SAS 6.11, 1996). The SAS programs are listed in Appendix B. Useful publications related to the use of the mixed procedure of SAS are Littell et al. (1996) and Roger (1993). Basic concepts of linear algebra useful for linear mixed models are collected in Appendix C. The multivariate normal distribution, which is the distributional assumption of all the data used in this book, is explained in Appendix D. The derivation of the variance of an estimable function in the mixed model framework is explained in Appendix E. The Satterhwaite procedure for obtaining denominator degrees of freedom for the F statistic of a hypothesis test in the mixed model is shown in Appendix F. Finally, Appendix G contains some technical derivations of expected mean squares.
The authors are grateful to the Limburgs Universitair Centrum, Diepenbeek, Belgium for providing financial support to publish this book.
Nairobi, Kenya
April 1998
Luc Duchateau
Paul Janssen
John Rowlands
