© 1966 by Biometrika Trust
Balanced confounding of factorial experiments
Universities of Aberdeen and Natal
The paper discusses the use of pseudo-factors for the construction of balanced factorial arrangements. The method is developed for s×q designs and then more generally for s×q1×q2×...,×qn designs, where s is necessarily a prime power. Usually blocks of st plots are wanted. The procedure is to replace each factor after the first by one or more pseudo-factors at s levels, and to form a parent design that confounds sT in blocks of st for a suitably large T. An arbitrary correspondence is then established between the combinations of levels of pseudo-fators and the qu, levels of factor Qu. This gives a set of initial blocks for the required design, and a series of permutations then introduces balance. Chief emphasis is placed on designs for two factors. The special case of s×q designs, with s>q blocks of s plots, is related to the use of balanced incomplete block designs in constructing asymmetrical factorial designs. The relation with partially balanced arrays of strength two is also discussed. Methods for reducing the number of blocks first produced are examined, as are the effects of choice of correspondence of factor levels and choice of permutations on the number of replicates, the loss of information on main effects, and resolvability. Blocks of fewer than st plots m considered.