Testing non-additivity (interaction) in two-way ANOVA tables with no replication

Testing for any significant interaction between two variables depends on the number of replicates in each cell of the two-way table and structure of the interaction. If there is interaction between two factors model of observations include interaction term and is called 'non-additive model' which makes interaction and non-additivity equivalent in terms of meaning. When there are several observations taken at each level combination of two variables, testing non-additivity can easily be done by usual two-way ANOVA method which cannot be used when there is only one observation per cell. For the cases with only one observation per cell, some methods have been developed starting with Tukey's one-degree-of-freedom test in which interaction is supposed to be the product of two factor's effects. There are other methods which are used for different structures of interaction when there is only one observation. In this paper, we review some of these tests. After presenting general methodology for the two-factor linear model with interaction effect and the general two-way ANOVA method when there are n > 1 observations per cell, we present some methods for testing non-additivity when there is only one observation per cell. Finally, we illustrate these methods on examples.