Block Designs

Abstract

In the previous chapter, we were concerned with the study of complete randomized block designs. In biological studies involving animals, however, it is not always possible to compare several treatments within litters since the size of the litter will be a function of the particular species used. In such cases, it is then necessary to consider various types of incomplete experimental designs. The methodology presented here rests on the concept of compatibility and the extended notion of distance between rankings. This approach provides a natural extension of the well-known Friedman and Durbin statistics to some partially balanced incomplete designs. The tests developed are also applicable to general block designs with ties and multiple observations per cell.

Chapter Notes

Alvo and Cabilio (1996) consider non-doubly balanced incomplete block designs (DBIBD), whereby each triplet of objects is not necessarily presented to the same number of judges and they obtain the asymptotic distribution for the Kendall statistic in this case. Although we have concentrated on the measures of Spearman and Kendall, the methods presented here for the analysis of block designs are quite general and applicable to other measures of similarity between rankings, particularly to those that can be written as inner products of score vectors. These methods have been applied in Alvo and Cabilio (1998) to the case of Hamming measure. On the whole, unlike various other approaches to such problems, the resulting tests have forms which are easily calculated immediate extensions of their versions in the complete block situation.

The asymptotic distributions are linear combinations of independent chi squares, with coefficients that are given analytically for many designs based on the Spearman or Kendall measures and which can in any case be calculated quite simply. Once the coefficients are determined, the critical values can be approximated using a procedure such as that in Jensen and Solomon (1972).

The statistics may be modified in order to simplify their asymptotic distributions to chi square, but this comes at the cost of making the statistics more complex. The example given previously for the Kendall case further shows that such statistics may have exact distributions whose support is less dense than that of the forms derived here.

Caution should be exercised in the use of the large sample critical values in conducting a small sample test. Various studies indicate that at least for complete block and BIBDs, other approximations to the small sample critical values may be a great deal more accurate (see for example Alvo and Cabilio 1995b). One approach which may have some value in dealing with small samples and with unbalanced designs is to generate the p-values of the test by simulation methods.

Ties for Hamming distance are also discussed in Alvo and Cabilio (1998). The discussion on the choice of scores follows closely the development in Alvo and Cabilio (2005) for the incomplete case where some simulation results are reported. This in turn was motivated by the work of Sen (1968)

A companion result to Lemma 5.4 showing a further use of Chebyshev polynomials appears in Alvo and Cabilio (2000). In particular, it is shown that one can compute values of the hypergeometric distributions recursively.