# Overview of research into the design and linear model analysis of multitiered experiments

Research into the design of experiments that involve multiple randomization is ongoing, both generally and in specific application areas. In addition mixed model analysis of these experiments, termed multitiered, is being investigated. In general terms it involves incorporating an experiment's randomization into the process of specifying the analysis and mixed model so that one obtains a randomization-based mixed model. The multitiered web site discusses the design and analysis of multitiered experiments. A detailed description of the approach and its theoretical justification is given in Brien (1992), available in a PDF file or a Postscript file. Other references for the design and analysis of multitiered experiments are available in the bibliography.

## The design and structure of multitiered experiments

(joint work involving: Prof. R.A. Bailey, School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS U.K.; Dr. C.J. Brien, School of Mathematics and Statistics, University of South Australia, North Terrace, Adelaide, SA 5000, Australia; Dr. B.D. Harch, CSIRO Mathematical and Information Sciences, 120 Meiers Rd, Indooroopilly, Qld 4068, Australia; and Dr. R.L. Correll, Rho Environmentrics, PO Box 366, Highgate, SA 2063 Australia.)

Multitiered experiments are characterized by involving more than one randomization and so are more complicated than standard experimental designs found in textbooks, as the latter involve only a single randomization. A particular application is to two-phase experiments where material from a first experiment is not finished with once it has been produced but must be collected and kept over for further processing or experimentation. One example is a grape growing experiment where the grapes are to be harvested, after which they are taken to a winery to be processed into wine prior to evaluation in a sensory experiment. Here an experimental design is required for the grape-growing experiment and a second design, that accommodates the first design, is needed for the sensory experiment. This is an example of a wider class of experiments that consist of a laboratory/measurement phase subsequent to an initial phase. Other types of experiments that are multitiered include superimposed, some plant and grazing experiments.

The different types of multiple randomization that can occur in multitiered experiments have been identified and described by Brien and Bailey (2006) and they develop randomization diagrams for exhibiting the randomizations. The implications of this for the structure on the factors, and hence for the analysis of variance decomposition, have been investigated and are detailed in Brien and Bailey (2009) and Brien and Bailey (2010). We advocate the use of decomposition tables, similar to analysis of variance tables, in designing experiments. Also, we study the conditions that must be placed on the designs employed to ensure that the analysis remains balanced.

Frequently an initial experiment is followed by analyses conducted in a laboratory and Brien, Harch, Correll and Bailey (2011) discuss the design and analysis of such experiments. The intuitive strategy of processing the produce from the initial experiment in the order that it is obtained from the first phase is discussed. Also considered is the randomization of produce from the first phase to the positions in the laboratory phase using a range of experimental designs. In this, we illustrate situations where one can, and cannot, ignore the treatments that have been randomized to the units in the first phase. An important consideration is the division of the laboratory phase into suitable time periods. We examine the confounding for proposed designs and suggest randomization-based mixed models that could be used to analyse them.

A particular area where multiphase experiments occur is in early generation variety trials in which produce is evalued in the laboratory after the field phase (Cullis et al., 2003). The design of such experiments requires further research.

## Randomization analysis for multitiered experiments

(joint work involving: Prof. R.A. Bailey, School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS U.K.; Dr. C.J. Brien, School of Mathematics and Statistics, University of South Australia, North Terrace, Adelaide, SA 5000, Australia.)

Some suggest that the only basis for statistical inference in experiments is the randomization employed in them. A theory that uses the randomization as a means of testing hypotheses without making the assumption of normality has been developed for experiments in which there is just a single randomization. We are developing a randomization theory for experiments that involve multiple randomizations. The relationship between the analysis of variance and REML-estimates for multitiered experiments will be described.

## Mixed model analysis of multitiered and longitudinal experiments

(joint work involving: Prof. R.A. Bailey, School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS U.K.; Dr. C.J. Brien, School of Mathematics and Statistics, University of South Australia, North Terrace, Adelaide, SA 5000, Australia; Dr. C.G.B. Demétrio, Depatmento de Matematica e Estatistica, ESALQ, Universidade de São Paulo, Caixa Postal 9, 13418-900 Piracicaba, SP, Brasil; Prof. R.W. Payne, Chief Science and Technology Officer, VSN International, 5 The Waterhouse, Waterhouse Street, Hemel Hempstead Herts. HP1 1ES U.K.)

Two methods of analysis, available for the analysis of designed experiments, are analysis of variance and mixed-model estimation. Brien and Payne (1999) show that more than one structure formula is required to derive an analysis of variance table that displays the confounding relationships in a multitiered experiment and, for balanced, multitiered experiments, allows a full analysis. An example, for which three structure formulae are required to achieve this, has been developed and an algorithm formulated for obtaining the sums of squares for an analysis of this and other experiments requiring three or more formulae. The algorithm has been implemented in GenStat in the AMTIER procedure (Brien and Payne, 2006). Brien and Demétrio (1998a) discuss the multitiered analysis of variance, and its implications, for the particular case of grazing trials - an extended version in a PDF file is available.

Brien and Bailey (2006) describe a general method for deriving a randomization-based mixed model that can then be fitted using mixed model software. It is based on the principle that the model used should include, at least, all the terms that are justified by the randomization and takes into account the confounding that occurs in an experiment. This is done by dividing the factors into sets, called tiers, based on the randomization and determining the crossing and nesting relationships between factors. The method is applied to a three-tiered, balanced experiment.

Brien and Demétrio (2009) extend Brien and Bailey (2006) to explicitly identify terms for which autocorrelation and smooth trend arising from longitudinal observations need to be incorporated in the model. The method is applied to formulate mixed models for a wide range of examples of longitudinal experiments. The mixed model analysis of data from a three-phase, longitudinal experiment to investigate the effect of time of refinement on Eucalyptus pulp from four different sources is also described. Cubic smoothing splines are used to describe differences in the trend over time and unstructured covariance matrices between times are found to be necessary.