Identification of effects and confounding patterns in factorial designs

Abstract

SUMMARY A method is developed for identifying effects and confounding patterns in factorial designs generated by Patterson's (1976) DSIGN algorithm. The method is an extension of that com- monly used for symmetrical designs with prime number of levels. Das (1964) described an equivalent method in which some of the treatment factors are designated as basic factors and the others as added factors. Levels of added factors are derived by combination of the levels of the basic factors over GF(t). White & Hultquist (1965) extended the field method to designs with numbers of levels of treatment factors still prime or prime power but possibly differing from one factor to another. John & Dean (1975) described the construction of a particular class of single replicate block designs, which they call generalized cyclic designs. The levels of treatment factors are now integers 0, 1, ..., t - 1. The essential feature of the method is that the m-tuples giving the treatments of a particular block constitute an Abelian group, the intrablock subgroup. The method gives the same single-replicate designs as the field method when t is prime. Treatment levels are, however, differently represented when t is a power of a prime and so, in general, different designs are obtained. Unlike the field method, the generalized cyclic method is also available when t is neither prime nor prime power. Dean & John (1975) extended their method to asymmetrical designs. Patterson (1976) described a general computer algorithm, called DSIGN, in which levels of treatment factors are derived by linear combinations of levels of plot and block factors. The method provides finite-field, generalized cyclic and other designs. It is not restricted to block designs but is available for any of the simple block structures defined by Nelder (1965) with fractional, single or multiple replication of treatments. In the present paper we are concerned with identification of effects and confounding in the DSIGN method. We tackle the problem by inspecting linear combinations of levels of treatment factors. This approach is, of course, familiar for finite-field designs: Kempthorne (1947) has