Three-Dimensional Incomplete Block Designs for Interaction Models

Abstract

Suppose we are running an experiment with m varieties of corn, n types of fertilizer, and p insecticides (and with land assumed homogeneous), and suppose we wish to run some number of observations h which is different from the full mnp. A very simple design, the Latin square, is available to us if we are assuming an additive (no-interaction) model and if m = n = p. If we are assuming an additive model but m, n, and p are not all equal, then we must consider designs more general than the Latin square; another paper (Potthoff [1962]) lays the groundwork for a variety of such 3DIB (three-dimensional incomplete block) designs and thereby generalizes the previous work in this area (Youden [1937], Shrikhande [1951]). In many experimental situations, however, the experimenter will not feel that he can safely assume an additive model as required for the Latin square and the designs of Youden [1937], Shrikhande [1951], and Potthoff [1962], and he will want to use some 3DIB design (i.e., a design whose number of observations h is not necessarily equal to mnp) based on some sort of interaction model. There would normally be four possible types of interaction models which he might consider: (i) a model with one two-factor interaction; (ii) a model with two twofactor interactions; (iii) a model with all three two-factor interactions; and (iv) a model with three-factor interaction. For example, if the experimenter admits the possibility of corn-fertilizer interaction and of corn-insecticide interaction but is confident that there is no fertilizerinsecticide interaction, then model (ii) would be the one to use. This paper deals with the analysis of certain types of 3DIB designs for the four models (i-iv); 2DIB designs for interaction models are also covered by way of introduction. Section 1 is concerned mainly with explaining our notation. Then Section 2 presents a summary of all the basic results and formulas (without proofs) pertaining to estimation