Use of Resolvable Designs in Field and on Farm Experiments

1. Introduction Resolvable designs, though introduced in the combinatorial context, have found their utility in field experiments. The concept of resolvability is due to Bose (1942) who defined this for a balanced incomplete block (BIB) design. A BIB design is an arrangement of v treatments into b blocks such that each block contains k (< v) distinct treatments, each treatment appears in r blocks and each pair of treatments appear together in λ blocks. A BIB design is called resolvable if the blocks of the BIB design can be divided into r groups such that each group contains each of the v treatments exactly once. Clearly, b/r = x, where x is the number of blocks in each resolvable group. Example 1: The following BIB design with the parameters v = 4, b = 6, r = 3, k = 2, λ = 1, is resolvable. The columns represent blocks and the symbols within blocks are treatments in the block 1 3 1 2 1 2 2 4 3 4 4 3 The concept of resolvability was extended to other block designs like partially balanced and cyclic designs. In general a proper block design, is said to be resolvable if its blocks can be grouped into r sets of blocks, each set containing b/r = x blocks, such that every treatment appears in each set precisely once. It was also extended to orthogonal arrays. The concept of resolvability of orthogonal arrays was subsequently exploited to construct main-effect orthogonal plans for asymmetrical factorials by Gupta, Nigam, Dey (1982) which are quite useful for variance estimation of non-linear statistics in complex surveys. Shrikhande and Raghavrao (1963) further extended the concept of resolvability to α-resolvability where in each of the resolvable groups, each treatment appears exactly α times. Example 2: The following group divisible design with parameters v = b = 4, r = k = 2, λ 1 = 0, λ 2 = 1 is also resolvable. The columns represent blocks and the symbols within blocks are treatments in the block.