BLOCK DESIGNS: GENERAL OPTIMALITY RESULTS WITH APPLICATIONS TO SITUATIONS WHERE BALANCED DESIGNS DO NOT EXIST (EXPERIMENTAL, INFORMATION MATRIX, SCHUR-CONVEX)

DEXTER C. WHITTINGHILL, Purdue University

Abstract

The problem of optimally allocating v treatments to experimental units in b blocks of size k(,j) (l (LESSTHEQ) j (LESSTHEQ) b) when no balanced block design (BBD) exists is studied. The model is that of one-way elimination of heterogeneity where the expectation of an observation on treatment i in block j is (alpha)(,i) + (beta)(,j) and all observations are uncorrelated with common variance. Optimality is measured with criteria that are functions of the nonzero eigenvalues of the C-matrix derived from the reduced normal equations for the (alpha)(,i) (1 (LESSTHEQ) i (LESSTHEQ) v). The criteria include A-, D- and E-optimality, Cheng's generalized type 1 optimality, and optimality over the class of Schur-convex functions that are nonincreasing in each eigenvalue. A main result is proved which says if a design d* has a C-matrix with eigenvalues of the form O < a < b = ... = b, is of maximum trace and as E-optimal for a class of designs, then d* is optimal over that class of designs for all Schur-convex functions nonincreasing in each eigenvalue. Several other results using similar techniques of weak supermajorization are given. This main result is applied to the case where all k(,j) = v-l and b-l blocks allow a balanced incomplete block design (BIBD) to be constructed, and with partial success where all k(,j) = 2 and b+l blocks allow a BIBD. A result of Cheng (1978) for eigenvalues of the form O < a = ... = a < b is applied to cases where all k(,j) = v-l but b-l blocks allow a BIBD, and all k(,j) = 2 but b+l blocks allows a BIBD. The main result is also applied in some cases of two situations with unequal block sizes. The first has k(,l) = ... = k(,b-l) = pv + q (p (GREATERTHEQ) O, O (LESSTHEQ) q (LESSTHEQ) v-l) and k(,b) = v-l with b-l blocks allowing a BBD to be constructed. The second has k(,1) = ... = k(,b-l) = pv + q and k(,b) = pv + q-l with b blocks of size pv + q allowing a BBD. Next two situations where all k(,j) = pv + q (p (GREATERTHEQ) l) are studied. In these b + m or b-m (l (LESSTHEQ) m (LESSTHEQ) v/q) blocks allow a BBD to be constructed. The main results is applied in some cases, and cases with eigenvalues of the form of Cheng (1978) are also investigated. For some cases of the above situation appropriate A-, D- or E-efficiencies are presented for designs of special interest, or which were not proved optimal. Finally the main result is used to extend an existing theorem for trend-free block designs generated by a BBD with k (LESSTHEQ) v. Important examples limit further extension for k < v.

Degree

Ph.D.

Subject Area

Statistics

COinS