Experimenting with Blocks of Natural Size

Abstract

SUMMARY Most experimental designs have been evolved to meet the requirements of agricultural field trials, the branch of biology where modern statistical methods began. Hence, it is usually assumed that blocks should all be made the same size and as small as possible. For other biological problems requirements may be different. Thus, the need may be for blocks of a given size, whether large or small. This happens when the plots are individual organisms that exist in groups, like animals in a litter or trees planted in rows and columns before the experiment was conceived. Although such groups can be divided if need arises, this often leads to loss of efficiency. Also, many useful designs exist in extended blocks, i.e., in which the number of plots in a block exceeds the number of treatments. Such natural blocks are not necessarily equal in size. Thus, litters may have differing numbers of animals and groups of trees may develop gaps. Unwanted plots can always be discarded but this can be very wasteful. However, it is not difficult to design experiments with varying block sizes, because the properties of a nonorthogonal design can be found from a matrix, which can be built up as a sum of components, each element having a component from each block. If, therefore, for each available block size a list is made of all possible combinations of treatments and if the components are worked out for each of the possibilities, it is sometimes easy to produce a design with the properties desired, thus extending the usual range of experimental designs.

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