Other Uses for Symmetrical Incomplete Blocks

CONFOUNDING THE PROBLEM OF RESTRICTED BLOCK SIZE IN FACTORIAL EXPERIMENTS (M 

In Chapter I we mentioned that for factorial experiments of the type 2 (n factors all at 2 levels) the number of runs required, being 2 , rapidly increased and for 4, 5 and 6 factors is 16, 32 and 64 respectively. These numbers may be too large to get into a single block, and the problem then arises of how to split the experiment up into sinaller units. 

Let us introduce an appropriate symbolism. Let the small letters p, q, r, etc., stand for the condition that the factor referred to is at its upper level. For example rs would mean that p and q are at their lower levels and r and s at their upper levels. Let the capital letter P denote the difference between the sum of all the upper levels of P and the sum of all the lower levels of P, and similarly for the other factors. Then for a three factor experiment, 

The first order interaction PQ is given by the difference between P at the lower level of Q, namely, pr 4- p — r — 1 

It will be noted that we get exactly the same result if we regard PQ as the difference between Q at the lower level of P, namely, qr 4- q — r — 1 

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