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4) That is, dij will take some non-zero value if i and j are not the same OTU. 5) (d) Triangular inequality: Given three OTUs i, j and k, the dissimilarities between them satisfy the inequality ~~~+4 p~ The triangular inequality is also known as the metric inequality, and dissimilarity coefficients satisfying the above properties are known as metrics and generally referred to as distances rather than dissimilarities. The most familiar metric is, of course, Euclidean; it is familiar because we live in a locally Euclidean universe and this tends to give it advantages in numerical taxonomy, where it is very widely used, because our daily experience gives us an intuitive grasp of Euclidean distances and thereby enables us to grasp their properties without difficulty.

Although this idea of weighting characters on the basis of their rareness of occurrence has a certain attraetion, it has been criticised on various grounds by Sneath & Sokal (1973) and they do not recommend Smirnov's coefficient for use in numerical taxonomy. An alternative, and perhaps more straightforward, way of solving this problem is to code the data as several binary characters (see Chapter 2). 4 Similarity measures for quantitative characters Quantitative characters such as length or diameter could be dealt with by simply converting them to binary characters.

Apart from the inevitable procedures of selection or rejection there are two other forms of weighting - a priori and a posteriori. The former means that amongst the selected characters, some axe considered more important than others; for example, it might be suggested that more reliance be placed on characters known to be good diagnostic features in other groups, or those assumed to be good indicators of phylogenetic relationships. This approach was criticized by Adanson in the eighteenth century, and again in the twentieth century by numerical taxonomists such as Sneath and Sokal, because it presupposes a knowledge of the classification one wants to produce before the analysis of the data.

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