On Balancing in Factorial Experiments

Abstract

R. C. Bose [1] has considered the problem of balancing in symmetrical factorial experiments. In all the designs considered in that paper, the block size is a power of $S$, the number of levels of a factor. The purpose of the present paper is to consider a general class of designs, where a `complete balance' is achieved over different effects and interactions. It is proved in this paper (Theorems 4.1 and 4.2) that if a `complete balance' is achieved over each order of interaction, the design must be a partially balanced incomplete block design. Its parameters are found. The usual method of analysis (of a PBIB design [2]) which is not so simple, can be simplified a little for these designs (section 5), on account of the balancing of the interactions of various orders. The simplified method of analysis is illustrated by a worked out example 5.1. Finally, the problem of balancing is dealt with for asymmetrical factorial experiments also. Incidentally, it may be observed that the generalised quasifactorial designs discussed by C. R. Rao [4] are the same as found by the author, from considerations of balancing.