Random effects models for complex designs

Plaid designs are characterised by having one set of treatments applied to rows and another set of treatments applied to columns. In a 2003 publication, Farewell and Herzberg presented an analysis of variance structure for such designs. They presented an example of a study in which medical practitioners, trained in different ways, evaluated a series of videos of patients obtained under a variety of conditions. However, their analysis did not take full account of all error terms. In this paper, a more comprehensive analysis of this study is presented, informed by the recognition that the study can also be regarded as a two-phase design. The development of random effects models is outlined and the potential importance of block-treatment interactions is highlighted. The use of a variety of techniques is shown to lead to a better understanding of the study. Examination of the variance components involved in the expected mean squares is demonstrated to have particular value in identifying appropriate error terms for F-tests derived from an analysis of variance table. A package such as ASReml can also be used provided an appropriate error structure is specified. The methods presented can be applied to the design and analysis of other complex studies in which participants supply multiple measurements under a variety of conditions.