THE THEORY OF CONFOUNDING IN FACTORIAL EXPERIMENTS IN RELATION TO THE THEORY OF GROUPS

Abstract

When the treatments in an experiment introduce all combinations of n factors, each at two levels, so that they are 2^n in number, it has long been known that it is possible in many cases to divide each replication onto two, four, or more blocks in such a way that the contrasts between blocks constitute only such interactions between the primary factors as are of minor importance to the experimenter. The great advantage of limiting the size of the block lies in the fact that in this way the contents of each block may be made much more homogenous than if it had been of larger size. In a very large class of experiments of this type we are concerned to evaluate the effcts of each individual factor with high precision, and to discover whether any pair of the factors tested show an appreciable interaction. If in these respects the precision of the experiment can be increased, it is usually advantageous to do so at the expense of foregoing information as to the reality of one or more of the interactions involving three of more factors. Various systems of confounding, using factors up to six in number, have been discussed by Barnard (1936) and Yates (1937). In the present paper I propose to develop the connexion of the subject with that of Abelian groups, to prove a general proposition connecting the minimal size of block required with the number of factors involved, and to supply a catalogue of systems of confounding available up to fifteen factors.