Balanced Factorial Experiments

Abstract

Usually, in a factorial experiment, the block size of the experiments is not large enough to permit all possible treatment combinations to be included in a block. Hence we resort 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 [2] and Fisher [4], [5]. An excellent summary of the results of this research appears in Kempthorne [6]. Some examples of asymmetric factorial designs can be found in Yates [14], Cochran and Cox [3], Li [8], Kempthorne [6] and Nair and Rao [9], [10]. Nair and Rao [11] have given the statistical analysis of a class of asymmetrical two-factor designs in considerable detail. The author [13] has considered the problem of achieving "complete balance" over various interactions in factorial experiments. In the present paper a class of factorial experiments, balanced factorial experiments (BFE) (Definition 4.2) is considered. The theorems proved in Section 4 outline a detailed analysis of BFE's, including estimates of various interactions at different levels. Finally, a method of constructing BFE's is given in Section 6. It should be noted that Theorems 5.2 to 5.5 are generalisations of the corresponding theorems by Zelen [15], and the method of construction in Section 6 is a general form of the one indicated by Yates [14], Nair and Rao [9], [10] and Kempthorne [6] (Section 18.7).