COMPARISON OF SEARCHING PROBABILITIES FOR SEARCH DESIGNS

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Consider a search design with m factors, and consider two distinct vectors xi_1 and xi_2 of factorial effects. It is assumed that the effects of xi_1 are completely unknown. It is known that at most k effects of xi_2 are nonzero and the remaining effects are zero (k is a given small number). However it is not known which are the nonzero effects. The problem is to search the k nonzero effects and estimate them along with xi_1 by using the observation vector y based on a search design. Suppose zeta(kx1) is a nonzero vector of xi_2. Then two procedures for searching them are considered : (1) Compute S(zeta)^2 the sum of squares due to error (SSE) based, on an estimate y'"*"(zeta) obtained from estimates "xi"_2 and "zeta". Find a vector zeta_1 for which S(zeta_1)^2 turns out to be a minimum for all possible zeta of 3xi_2. Then take zeta_1 as the possibly nonzero vector of xi_2. (2) Compute the estimate "gamma"(zeta)^2 of "*"zeta"*"^2. Find a vector zeta_2 for which "gamma"(zeta_2)^2 turns … More out to be a maximum. Then take zeta_2 as the possibly nonzero vector of xi_2. In this study, as a generalization of (2) we proposed a new procedure : (3) Compute the estimate tau"("zeta)^2 of tau="*"Qzeta"*"^2, where Q is a certain positive definite matrix. Find a vector zeta_3 for which tau"("zeta_3)^2 turns out to be a maximum. Then take zeta_3 as the possibly nonzero vector of xi_2. We gave a geometrical property for the (3) and a relation to a statistical testing (H_0 : zeta = 0, H_1 : zeta "*" 0). That is, the maximaization of tau"("zeta_3)^2 is equivalent to that of the test F-statistics. As a result, it was shown that this procedure (3) is equivalent to (1), i.e., xi_1 = xi_3. In case of k=l, for search designs which had been obtained up to the present, we calculated searching probabilities for nonzero effects for values (delta^2 =1.0, 1.5 ..., 6.0) of delta^2 ="*"Qzeta_0"*"^2/siguma^2 by using a computer, where zeta_0 is a true nonzero vector of xi_2. As a by-product of a construction of search design, we presented positive useful results on the problems of the classification of weighing matrix, enumeration of certain graphs and optimality of fractional factorial designs in case of correlated errors. Less