Collections of simple effects and their relationship to main effects and interactions in factorials

When analyzing data from a cross-classified experiment, one of the primary interests lies in estimating and testing main effects and interactions. Fre quently, one is also interested in comparing the levels of one factor at a given level of another factor, particularly if interactions are present. Su ch collections of simple main effects are termed slices herein. Based on un folded designs in which effects are represented by generic sets of orthogon al contrasts among cell means, the relationships between the contrast sets defining main effects,, interactions, and slices in terms of Kronecker repr esentations and projection properties are examined. The material is appropr iate in a first course on linear models and/or a course in experimental des ign with linear algebra prerequisite to demonstrate the relationship and in terpretation of various effects in a factorial setting. An example from pro duction quality control is presented.