Abstract
This paper concerns the visualisation of interaction in three-way arrays. It extends some standard ways of visualising biadditive modelling for two-way data to the case of three-way data. Three-way interaction is modelled by the Parafac method as applied to interaction arrays that have main effects and biadditive terms removed. These interactions are visualised in three and two dimensions. We introduce some ideas to reduce visual overload that can occur when the data array has many entries. Details are given on the interpretation of a novel way of representing rank-three interactions accurately in two dimensions. The discussion has implications regarding interpreting the concept of interaction in three-way arrays.
Highlights
“. . .It is important that the final model or models should make sense physically: at a minimum, this usually means that interactions should not be included without main effects nor higher-degree polynomial terms without their lower-degree relatives
If the model is to be used as a summary of the findings of one out of several studies bearing on the same phenomenon, main effects would usually be included whether significant or not
Three-way tables are usually analysed by linear models containing additive terms representing main effects, two-factor interactions, and three-factor interactions
Summary
The error terms εij are assumed to be indepen[110] dently distributed with equal variances Many classical models, such as Tukey’s model 111 for one degree of freedom for non-additivity (Tukey, 1949), can be considered as spe[112] cial cases of a biadditive model. Alternative names under which (4) has appeared, are FANOVA (FActor ANalysis Of VAriance) (Gollob, 1968) and AMMI (Additive Main effects and Multiplicative Interactions) (Gauch, 1992). We prefer the neutral biadditive model terminology which is in line with general statistical usage (Denis and Gower, 1994). These authors were interested in biadditivity because they thought that substantive genetic effects were better modelled in multiplicative rather than additive terms.
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