A Calculus for Factorial Arrangements

1. Introduction and summary. This paper introduces a special calculus for the analysis of factorial experiments. The calculus applies to the general case of asymmetric factorial experiments and is not restricted to symmetric factorials as is the current theory which relies on the theory of finite projective geometry. The concise notation and operations of this calculus point up the relationship of treatment combinations to interactions and the effect of patterns of arrangements on the distribution of relevant quantities. One aim is to carry out complex manipulations and operations with relative ease. The calculus enables many large order arithmetic operations, necessary for analyzing factorial designs, to be partly carried out by logical operations. This should be of importance in programming the analysis of factorial designs on high speed computers. The principal new results of this paper, aside from the new notation and operations, are (i) the further development of a theory of confounding for asymmetrical factorials (Section 4) and (ii) a new approach to the calculation of polynomial regression (Section 5). In particular, the use of the calculus enables one to write the inverse matrix of the normal equations for a polynomial model as a partitioned matrix. As a result it only requires inverting matrices of smaller order. 2. Elements and operations of the calculus. Consider an asymmetric factorial experiment with n factors A1, A2, * **, An such that the number of levels of factor Ai is mi. A particular selection of levels i = (i, i2 , * **, in) is termed a treatment combination, where i8 denotes the i8th level of A8. The total number of treatment combinations is v = IfJ=' mM. Let Yi(i = 1, 2, * * *, v) denote the observation on the ith treatment combination. Then the effect3 of the ith treatment combination is defined to be ti = E( Yi) - Ei=, E( Yi) Iv. Due to the factorial structure of the experiment, the model for the treatment effects can be further expressed in terms of the usual main effect and interaction parameters. We shall denote the main effect, twofactor interaction, *.., n-factor interaction parameters by