Generation of Factorial Designs

JOHN AND DEAN (1975) described a generalized cyclic method for constructing confounded single-replicate factorial arrangements. Unlike many earlier methods, this method is not restricted to treatment factors with prime-power numbers of levels. Nor is it restricted to symmetrical arrangements, i.e. designs in which all factors have the same number of levels. Dean and John (1975) described the construction of asymmetrical arrangements. John and Dean's method appears to be well suited to the production of factorial designs on a computer. Their papers prompt me to compare the generalized cyclic method with a procedure that has been used regularly since 1966 in the routine computer production of designs. This procedure is incorporated in a program called DSIGN, written by Mrs J. Tolmie and used both at Rothamsted and in Edinburgh.t The DSIGN method depends on the specification of required relationships between the levels of different treatment factors. The possibility of constructing designs in this way is, of course, well known. John (1971) gives many examples, particularly of fractional replicates, and numerous references. Possibly the closest antecedent of the DSIGN procedure is the method described by Das (1964) for confounded symmetrical designs in blocks. The DSIGN procedure can be regarded as a generalization of Das's method, giving not only block designs but also designs with more complicated block structure, e.g. rows and columns, split plots. The DSIGN procedure was first developed for construction of symmetrical factorial treatment schemes in rotation experiments (Patterson, 1965). In practice the method has also been found capable of producing many of the most commonly used asymmetrical designs although its mechanism in these applications is less easily understood. The present paper outlines the construction of asymmetrical block designs by the DSIGN method, applies the method to single-replicate designs and considers its relationship with the generalized cyclic method.