The Use of Group Divisible Designs for Confounded Asymmetrical Factorial Arrangements

A factorial experiment involving $m$ factors such that the $i$th factor has $m_i$ levels is termed an asymmetrical factorial design. If the number of levels is equal to one another the experiment is termed a symmetric factorial experiment. When the block size of the experiment permits only a sub-set of the factorial combinations to be assigned to the experimental units within a block, resort is made to the theory of confounding. With respect to symmetric factorial designs, the theory of confounding has been highly developed by Bose [1], Bose and Kishen [4], and Fisher [11], [12]. An excellent summary of the results of this research appears in Kempthorne [13]. However, these researches are closely related to Galois field theory resulting in (i) only symmetric factorial designs being incorporated into the current theory of confounding; (ii) the common level must be a prime (or power of a prime) number; and (iii) the block size must be a multiple of this prime number. The theory of confounding for asymmetric designs has not been developed to any great degree. Examples of asymmetric designs can be found in Yates [19], Cochran and Cox [9], Li [15], and Kempthorne [13]. Nair and Rao [16] have given the statistical analysis of a class of asymmetrical two-factor designs in considerable detail. Kramer and Bradley [14] discuss the application of group divisible designs to asymmetrical factorial experiments, however their paper is mainly confined to the two-factor case and its intra-block analysis. It is the purpose of this paper, which was done independently of their work, to outline the general theory for using the group divisible incomplete block designs for asymmetrical factorial experiments. The use of incomplete block designs for asymmetric factorial experiments results in (i) no restriction that the levels must be a prime (or power of a prime) number, (ii) no restriction with respect to the dependence of the block size on the type of level, and (iii) unlike the previous referenced works on asymmetric factorial designs, the resulting analysis is simple, does not increase in difficulty with an increasing number of factors, and "automatically adjusts" for the effects of partial confounding. Section 2 states three useful lemmas, Section 3 contains the main results of this paper, and Section 4 outlines the recovery of inter-block information.