Designs for two-way elimination of heterogeneity

TLDR: In this paper, the authors considered balanced incomplete block designs for two-way elimination of heterogeneity and showed that they can be converted to Youden's squares by ordering the varieties in the blocks such that each variety occurs once in each position.

Abstract: Sometimes in a design the position within the block is important as a source of variation, and the experiment gains in efficiency by eliminating the positional effect. The classical example is due to Youden in his studies on the tobacco mosaic virus [1]. He found that the response to treatments also depends on the position of the leaf on the plant. If the number of leaves is sufficient so that every treatment can be applied to one leaf of a tree, then we get an ordinary Latin square, in which the trees are columns and the leaves belonging to the same position constitute the rows. But if the number of treatments is larger than the number of leaf positions available, then we must have incomplete columns. Youden used a design in which the columns constituted a balanced incomplete block design, whereas the rows were complete. These designs are known as Youden's squares, and can be used when two-way elimination of heterogeneity is desired. In Fisher and Yates statistical tables [2] balanced incomplete block designs in which the number of blocks $b$ is equal to the number of treatments $v$ have been used to obtain Youden's squares, and the authors state that "in all cases of practical importance" it has been found possible to convert balanced incomplete blocks of the above kind to a Youden's square by so ordering the varieties in the blocks that each variety occurs once in each position. F. W. Levi noted ([3], p. 6) that this reordering can always be done, in virtue of a theorem given by Konig [4] which states that an even regular graph of degree $m$ is the product of $m$ regular graphs of degree 1. Smith and Hartley [5] give a practical procedure for converting balanced incomplete blocks with $b = v$ into Youden's squares. In this paper I have considered some general classes of designs for two-way elimination of heterogeneity. In Section 3 balanced incomplete block designs for which $b = mv$ have been used to obtain two-way designs in which each treatment occurs in a given position $m$ times. The case $m = 1$ gives Youden's squares. In Section 4 it has been shown that balanced incomplete block designs for which $b$ is not an integral multiple of $v$ can be used to obtain designs for two-way elimination of heterogeneity in which there are two accuracies (i.e., some pairs of treatments are compared with one accuracy, while other pairs are compared with a different accuracy) as in the case of lattice designs for one-way elimination of heterogeneity. In Sections 5 and 6 partially balanced designs have been used to obtain two-way designs with two accuracies. In every case the method of analysis and tables of actual designs have been given.