Connectedness and optimality in multidimensional designs

Sennetti (1972) showed the existence of minimal multidimensional designs (MMD's) and minimal augmented multidimensional designs (MAMD's) which allow estimation of type I and type II contrasts. For an MMD, only one more design point is required than there are degrees of freedom for the parameter vector. For MAMD's, the number of assemblies added is equal to the difference between the number of degrees of freedom for the parameter vector and the rank of the design matrix.

Using the chain concept of connectedness as defined by Bose (1947), this work suggests a practical procedure to obtain an MMD for estimating type I contrasts and proves the procedure valid. In addition, a procedure is discussed that may be used to obtain an MMD for estimating type II contrasts. After proof of the validity of the procedure, advantages of this procedure over some other possible procedures to obtain an MMD are given. It is shown that only a slight modification of the procedure is necessary to be able to obtain an MAMD for estimating type II contrasts.

If there is a restriction on the number of replicates of factor levels for an experiment, then a different approach is suggested. If m_ij denotes the number of replicates of level j of factor F_i, then it is desired to increase the number of estimators for type I contrasts without altering any of the m_ij. The interchange algorithm used by Eccleston and Hedayat (1974) to accomplish this for a proper, locally connected (l-connected) randomized block design is extended to two-factor, no interaction designs. The design obtained is pseudo globally connected (pg-connected), thus guaranteeing more estimates for main effect contrasts. In addition, the new design will be better than the old with respect to the S-optimality criterion. It is shown that the procedure can also be used in a two or more factor experiment to pg-connect an I-connected design for two factors. The new design obtained will be better than the old with respect to a new criterion, C-optimality. The algorithm described is proved to have no effect on the amount of aliasing (based on a norm suggested by Hedayat, Raktoe, and Federar, (1974)) due to a possibly incorrect assumption of no interaction.

The use of the interchange algorithm to pg-connect a design for level combinations is suggested because of the increased number of estimators for type II contrasts that may be obtained. A theorem is proved which gives the minimum number of estimates that will be available for estimating a type II contrast if a design is pg-connected for level combinations.

The last topic discussed is the use of a criterion for choosing a particular MAMD for estimating type II contrasts. The sequentially S-optimal design is defined. It is shown that the sequentially S-optimal design is easy to obtain and is similar to the S-optimal design.