Testing for co-directional interactions using union--intersection and intersection--union methods

When interaction terms exist in a two-factor, factorial experiment, the consideration and analysis of main effects are often restricted to those situations where the interaction between factors is not significant. Hinkelman and Kempthorne [4] softened that stance somewhat and advocate testing main effects when the interaction is deemed co-directional but not anti-directional. A test for the main effects in that situation may be pragmatic to the practitioner and appealing to researchers in other disciplines. Intersection--union and union--intersection methods are examined for assessing the directional nature of significant interactions so that the main effects in a two-factor factorial may be evaluated. The tests suggested are conceptually straightforward and practical and maintain the nominal Type-I error rate. Examples are provided to illustrate the methods.