Cubic Lattices

In some experiments, the problem is to compare many unstructured treatments in small blocks, the classical example being the study of new plant varieties on variable land. A common method is to use lattice designs, i.e. block designs based upon rows and columns of a square format, with further replicates being formed, if required, from orthogonal squares applied to the format It has been known for some time that cubes can be used instead; this paper sets out to explore the possibilities. There are two cases. In one case, the blocks are formed from the planes of the cube and, in the other case, from its lines. The cubic lattice basically has three replicates-one from each dimension-but, if two or four replicates are required, a design can be found by omitting or duplicating one of the dimensions. Where standard treatments need to be introduced, a useful device is to reinforce, i.e. supplement each block with additional plots of standards, with each block of a replicate being supplemented in the same way. These possibilities are examined. It emerges that cubic lattices with two or three replicates usefully extend the range of available designs, but that those with four replicates are disappointing. However, there is the alternative of using designs based upon Latin cubes. This matter is not taken far but it is shown that, where the Latin cube exists, it gives a better design. A quick way of calculating an approximate analysis of variance is given, which is applicable in a wide range of cases.