A- and Mv-efficient Block Designs for Comparing a Set of Controls to a Set of Test Treatments

SUMMARY. In this paper, we consider the problem of comparing s controls to t test treatments using experimental units which are arranged in b blocks of size k. Methods are given for constructing highly A-and MV-efficient designs for such situations. The methods of construction given combine optimality results obtained for one way classification models with the optimal augmentation of blocks with controls of block designs which are known to be efficient for making comparisons among the test treatments. Examples are given to show how the results obtained can be applied. 1. Introduction In this paper we consider experimental situations in which t ≥ 2 test treatments are to be compared to some set of s ≥ 2 standard or control treatments, s ≤ t, using a block design consisting of b blocks of size k. We shall use 1, · · · , v to label the s+t = v treatments being studied with 1, · · · , s denoting the controls and s + 1, · · · , v denoting the test treatments. We shall use d to denote an arbitrary block design which can be used in such a setting. The combinatorial structure of d is given by its v × b incidence matrix N d whose entries n dij give the number of times treatment i(i = 1, · · · , v) occurs in block j(j = 1, · · · , b). We say d is binary if n dij = 0 or 1 for all i, j. The ith row sum of N d is denoted by r di and represents the number of times treatment i is replicated in d. Using A to denote the transpose of a matrix A, N d N d is called the concurrence matrix of d and its entries are denoted by λ dij. Observations Y ijp obtained after applying the ith treatment to experimental units occurring in the jth block are assumed to follow the usual additive two-way model, i.e.