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**The design of optimum multifactorial experiments.**
*(English)*
Zbl 0063.06274

From the introduction: A problem which often occurs in the design of an experiment in physical or industrial
research is that of determining suitable tolerances for the components of a certain assembly;more generally of ascertaining the effect of quantitative or qualitative alterations in the various components upon some measured characteristic of the complete assembly. It is sometimes possible to calculate what this effect should be; but it is to the more general case when this is not so that the methods given below apply. In such a case it might appear to be best to vay the components independently and study separately the effect of each in turn. Such a procedure, however, is wasteful either of labour or accuracy, while to carry out a complete factorial experiment (i.e. to make up assemblies of all possible combinations of the \(n\) components) would require \(L^n\) assemblies, where \(L\) is the number of values (assumed constant) at which each component can appear. For \(L\) equal to 2 this number is large for moderate \(n\) and quite impractible for \(n\) greater than, say, 10. For larger \(L\) the situation is even worse. What is required is a selection of \(N\) assemblies from the complete factorial design which will enable the component effects to be estimated with the same accuracy as if attention had been concentrated on varying a single component throughout the \(N\) assemblies. Designs are given below for \(L=2\) and all possible \(N\leq 100\) except \(N=92\) (as yet not known), and for \(L=3,4,5,7\) when \(N=L^r\) (for all \(r\)).

The following results have been obtained:

(a) When each component appears at \(L\) values, all main effects may be determined with the maximum precision possible using \(N\) assemblies, if and only if \(L^2\) divides \(N\), and certain further conditions are satisfied.

(b) For \(L=2\), the solution of the problem is for practical purposes complete. In designs of the form \(N=L^r\), the effects of certain interactions between the components may also be estimated with maximum precision.

The precision naturally increases with the number of assemblies measured, and to this extent depends on the judgement of the experimenter.

The following results have been obtained:

(a) When each component appears at \(L\) values, all main effects may be determined with the maximum precision possible using \(N\) assemblies, if and only if \(L^2\) divides \(N\), and certain further conditions are satisfied.

(b) For \(L=2\), the solution of the problem is for practical purposes complete. In designs of the form \(N=L^r\), the effects of certain interactions between the components may also be estimated with maximum precision.

The precision naturally increases with the number of assemblies measured, and to this extent depends on the judgement of the experimenter.