Optimal design of three way layouts without interactions

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Abstract
In a goneral three way layout without interactions: a particular choice of the system {n ijk } of numbers of observations at the levels i, j, k of the three factors represent a specific (nomandomized) design of the three way layout. Classes of uniformly optimal designs under different conditions on the availability or observations, i.e. in different sets of admitted designs, are characterized. In cases where uniformly optimal designs do not exist, D and A optimal designs are given. A particular resut is the following: If the numbers of observations are restricted by the systems {n i }, {n j }, {n j } of the total numbers of observations at each level of every factor, then a design dealing with inference with respect to the first factor is uniformly if and only if it satisfics the conditions {n ij = n i n j /n} and {n ik =n k /n} for all i and j. The results obtained establish, in particular optimality statements concerning incomplete designs such as those presented by J. KIEFER in [Ann. Math. Statist. 29 (1958), 675-699] and extension thereof which are given by V. KUROTSOHKA [Symp. on Symmetric Functions in Statisties, Windsor, Ont., 1971].

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