Incomplete Block Designs with Row Balance and Recovery of Inter-Block Information

Abstract

The conditions for Balanced Incomplete Block Designs (Yates [1936]) are well known to statisticians involved in experiment design. They are (a) that no treatment should occur more than once within any block (b) that the number of blocks in which any specified pair of treatments should occur together should be a constant (X) for all possible treatment pairs (c) that all treatments should occur an equal number of times. Such an arrangemelnt permits the elimination of systematic block differences from comparisons of treatment effects and enables all such treatment comparisons to be made with equal precision. A practical advantage of the design is that the number of treatments per block is less than the number of treatments to be tested, thus allowing a large number of treatments to be tested in relatively small blocks of units. In such designs the positions within the block are not considered as being heterogeneous in character and can normally be regarded as unimportant, the allocation of treatments to these positions being carried out by a random procedure. If we regard the design as applied to an agricultural trial with the blocks as strips of land running vertically as columns, the positions within the blocks can be regarded as forming the rows of the design. In many instances these rows may well contribute systematic effects in the observations, and it may be desirable to remove row variability as well as block variability in a manner similar to that employed in the Latin Square design. Youden [1937] was the first to employ such an arrangement in experimental work on tobacco plants where the blocks comprised the separate plants and the tows consisted of leaf positions on the plant, leaves in different positions possessing different susceptibilities. These designs, known as Youden Squares, consisted of Balanced Incomplete Block designs in which the number of blocks equalled the number of treatments, the treatments being arranged in such a way that every treatment occurred once in each position (row). Balanced Incomplete Block designs in general do not possess the above property and the simple arrangement found in the Youden Squares is not always possible. In such cases however