Ann. Inst. Statist. Math. Vol. 43, No. 2, 377-390 (1991) ON THE DETERMINATION AND CONSTRUCTION OF OPTIMAL ROW-COLUMN DESIGNS HAVING UNEQUAL ROW AND COLUMN SIZES MIKE JACROUX AND RITA SAHA RAY* Department of Pure and Applied Mathematics, Washington State University, Pullman, WA 99164-2930, U.S.A. (Received May 19, 1989; revised January 30, 1990) Abstract. In this paper we consider experimental situations requiring usage of a row-column design where v treatments are to be applied to experimental units arranged in bl rows and b2 columns where row i has size kl~, i -- 1,..., bl and column j has size k2j, j = 1,..., b2. Conditions analogous to those given in Kunert (1983, Ann. Statist., 11, 247-257) and Cheng (1978, Ann. Statist., 6, 1262-1272) are given which can often be used to establish the optimality of a given row-column design from the optimality of an associated block design. In addition, sufficient conditions are derived which guarantee the existence of an optimal row-column design which can be constructed by appropriately arranging treatments within blocks of an optimal block design. Key words and phrases: Row-column design, block design, incidence matrix, balanced unequal block design. 1. Introduction In this paper we consider experimental settings in which v treatments are to be tested using a row-column design where n experimental units are arranged in bl rows and b2 columns where row i has size kli, i = 1,..., bl and column j has size k2j, j = 1,..., b2. If we let d denote some design which can be used in such an experimental setting, then we shall let Nd 1 : [ndij )'~ (1)~ Nd2 : ~(n(2)~dij] a n d Nd3 ~n (3) ~ : ~ dij] denote, respectively, the v × bl treatment-row incidence matrix, the v × b2 treatment-column incidence matrix, and the bl x b2 row-column incidence m a t r i x , i.e., °(1)'~dij = the number of times treatment i occurs in row j, ~'°dij(2) = the number of times treatment i occurs in column j, and n(~) = the number of a~3 observations obtained in row i and column j. The model assumed here for the data obtained from some design d specifies that an observation Yijkz obtained after * Visiting from the Indian Statistical Institute. 377