Testing Three-factor Interaction in Unreplicated Three-way Layouts

For unreplicated three-way layouts, the usual F-test can not be used for testing three-factor interaction and it is common to base the analysis on the model without three-factor interaction. This may lead to misleading results if three-factor interaction is actually present. We propose a simple and flexible procedure for testing three-factor interaction when one of the factors has at least four levels. This procedure is not based on a specific functional form for the interactions, and its usefulness is illustrated with several examples. (We are not aware of any method which is not based on assuming a specific functional form for the three-factor interaction.) We compare the performance of our procedure with several other tests via an extensive simulation study. This simulation study shows that there is no test of interaction which can detect all patterns of interaction. Therefore, in the absence of prior information regarding the structure of three-factor interaction, no test can be preferred to other tests on the basis of power. In practice, unless the researcher is confident about the functional form of the three-factor interaction, we recommend using our procedure on the basis of its simplicity and flexibility.