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. Frequently, one is also interested in comparing the levels of one factor at a given level of another factor, particularly if interactions are present. Such collections of simple main effects are termed slices herein. Based on unfolded designs in which effects are represented by generic sets of orthogonal contrasts among cell means, the relationships between the contrast sets defining main effects, interactions, and slices in terms of Kronecker representations and projection properties are examined. The material is appropriate in a first course on linear models and/or a course in experimental design with linear algebra prerequisite to demonstrate the relationship and interpretation of various effects in a factorial setting. An example from production quality control is presented.