Augmenting Factorial Plans to Accommodate Additional Two-Level Factors

Abstract

In the application of factorial patterns there sometimes arise situations where the experimenter feels that one or more additional factors should have been included in an experiment which has just been performed. He may then (i) disregard the original experimental data and perform an entirely new experiment using a factorial plan that also permits the estimation of the additional factors or (ii) augment the data so that the desired information on the additional factors can be obtained without losing the information obtained with the original design. To illustrate the problem, consider an experiment performed on four 2-level factors. If the plan consists of the 16 treatment combinations of the complete 24 factorial pattern, and all 3 and 4-factor interactions are absent, then the mean, the 4 main-effects, the six 2factor interactions, and the experimental error can be estimated by standard techniques. If a fifth 2-level factor is to be introduced into the plan, how can it be accomplished? The original 24 factorial plan can be considered to be a ' replicate of a 25 experiment where the fifth factor is held constant at its 0 level. In other words, if the additional factor is denoted by E, the half replicate is defined by the identity relationship I = Eo , where the subscript indicates the level at which the factor E occurs in each treatment combination. An obvious procedure for augmenting the 2 replicate to permit estimation of the effect of E and the interaction effects involving E is to add the treatment combinations defined by I = E1 , which is the remaining 2 replicate of the original plan. The addition of this 2 replicate involves the addition of 16 treatment combinations, so that a complete replicate with 32 treatment combinations is used in the augmented experiment. The problem which will be considered in this paper is, in the frame-