Design Tableau

Alison Smith and Brian Cullis 

One of our main research interests over the past 30 or more years has been the use of linear mixed models (LMM) for the analysis of data from crop improvement programmes. These data arise from comparative experiments in which the aim, typically, is to select the “best” varieties. In order to maximise the accuracy of selection we have developed analytic procedures that involve LMM with complex variance and correlation structures. For example, we use separable auto-regressive models to mitigate the impact of spatial trend within experiments conducted in the field, and factor analytic models to extract key information about variety by environment interaction in the analysis of multi-environment trial data.

We were fortunate enough to have trained and worked as young biometricians when analysis of variance (ANOVA) techniques were the primary method of analysis for comparative experiments.

Our tool of trade was the GENSTAT package, so that the elegant notation of Wilkinson and Rogers and the framework of Block and Treatment structures became ingrained in our statistical thinking. So, despite the complexity of the LMM we now use, we appreciate the importance of maintaining these fundamental concepts, in particular the link between the analysis and the experimental design. We are concerned that this view is shared by only a few, as is evidenced by what we regard as a widespread mis-use of LMM for comparative experiments. This may either be due to an unintentional lapse in transitioning from ANOVA to LMM or a complete lack of exposure to traditional methods of analysis for comparative experiments.  

Over recent years, we have made it a priority to fill in this gap for our young statistical colleagues at the University of Wollongong. In particular we have attempted to provide a link between ANOVA and LMM and to explain how to derive LMM that reflect the randomisation employed in the design of the experiment, no matter how complex. We found this to be a non-trivial task and tried numerous educational tools but without great success. A turning point was Brian’s introduction of an Honours Statistics course on experimental design at the University of Wollongong. He based this course on Rosemary Bailey’s book and found words of wisdom that have inspired us to develop an approach that we have termed “Design Tableau” (DT). The main aim of DT is to provide a simple, but general series of steps for specifying the LMM for a comparative experiment. It is founded on the seminal work of Sir Ronald Fisher, John Nelder, Rosemary Bailey and Robin Thompson. The motivation and concepts underlying Design Tableau will form the basis of our presentation. We will discuss the formal link between ANOVA and LMM, describe the steps that constitute DT, illustrate DT for simple cases in which the LMM may be used to re-produce an ANOVA and finally demonstrate how DT can be applied in a wide range of complex comparative experiments.    

 

Brian Cullis is Senior Professor of Biometry at the University of Wollongong having been appointed to this position in July 2011 after working as a biometrician for more than 30 years with NSW Department of Primary Industries. His interests relate to the design and analysis of agricultural and biological data. He has co-authored more than 180 journal articles. Much of his work has involved the analysis of experimental data using linear mixed models, and he is a co-author of the ASReml software package and the R package ASReml-R. He has served as a Co-Editor of Biometrics and is currently an Associate Editor of the Australian and New Zealand Journal of Statistics and the Journal of Agricultural Science Cambridge.