By G. Dunn

ISBN-10: 0486477533

ISBN-13: 9780486477534

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First similarity and distance measures for qualitative characters will be introduced, and then those for quantitative characters. 1 Qualitative characters A distance measure that geneticists have used when describing popUlations in terms of gene frequencies has the following form. 17) where i Cos 0 is a measure of genetic similarity between the two populations, A and B, PiA and PiB being the gene frequencies for the ith allele at a given locus in the two populations. The angular transformation for the proportions has a variance-stabilizing role, and the distance, dAB' in geometrical terms is the chord subtended by the angle 0 on a hypersphere of unit radius (see Edwards & Cavalli-Sforza, 1964; and Cavalli-Sforza & Bodmer, 1971, Chapter 11).

When used as a measure of similarity for two OTUs its calculation involves averaging over the states of different quantitative characters to produce an 'average character state' for each OTU. Jardine & Sibson (1971) remark that such a procedure is 'absurd'. It has often been suggested that the correlation coefficient will be a useful measure of similarity in those situations where absolute size alone is regarded as less important than shape. Thus, in classifying plants and animals, the absolute sizes of the organisms or their parts are often of less importance than their shapes, and in such cases the investigator requires a similarity coefficient which will take the value unity whenever the set of character-state values describing two OTUs are parallel, irrespective of how far apart they are; for example, the following sets of scores are parallel in this sense: OTUI OTU2 OTU3 10 15 30 5 10 25 15 20 35 3 8 23 20 25 40 The correlation coefficient meets this requirement (it takes the value unity for each pair of OTUs in the above example); unfortunately, the converse is not true since the correlation may take the value one even when the two sets of values are not parallel.

The reader interested in a more detailed discussion of this problem is referred to Fitch (1970). 6 Summary . The important and difficult issues of the choice of characters, the weighting of characters, and homology have been described only 24 Taxonomic characters relatively briefly in this chapter. Much fuller discussions are ayailable in Sneath & Sokal (1973) and Jardine & Sibson (1971). However, we will now assume that the organisms to be classified and the characters to describe them have been selected, and that numerical methods are required first to quantify their similarity, then to construct the classification, and finally to produce a diagnostic key; techniques for each of these procedures will be covered in the remaining chapters of this text.

### An Introduction to Mathematical Taxonomy by G. Dunn

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