STRUCTURAL VALUE: A CONCEPT USED IN THE CONSTRUCTION OF TAXONOMIC CLASSIFICATIONS

Summary The number of hierarchical clustering methods now available in numerical taxonomy is such as to be capable of yielding a bewilderingly large number of dendrograms. Moreover, mathematical criteria for choice of method have met with limited acceptance. The construction of dendrograms, whether on phenetic, eclectic, or cladistic principles, is not, however, the entire process of classification. Although both dendrograms and hierarchical classifications are ultrametric representations of relationship, the latter are always rank‐defined while the former may be defined on continuous distances. Even rank‐defined dendrograms are often so complex in terms of ranks and branching points (nodes) as to be unacceptable in practice as taxonomic classifications. In such cases a process of simplification is necessary beyond that involved in the ultrametric transformation. Although methods exist for assessing best ultrametric transformation, little consideration has been given to the subsequent simplification process. In this paper the criteria used by taxonomists in determining whether a classification shows an acceptable degree of simplification (e.g., by the compression or splitting of a dendrogram) are analysed. The extent to which a classification meets these criteria is termed its structural value. Psychological research has shown that the hierarchical organisation of stimuli aids memory and that the number of stimuli that can readily be considered at one time shows an upper limit of about five to seven. Hierarchical classification subdivides large groups into mentally more manageable ones (‘leaflets’ of a hierarchical tree) and it is suggested that the fewer additional taxa (nodes) and additional ranks required to do this, the higher the structural value. A structural value statistic (V) is developed which is the one‐complement of a loss function with terms representing the extent to which the leaflet sizes ( n ij ) nodes ( T ) and ranks ( r ) are in excess. In the simplest case, where a is the desired leaflet size and T ∗ and r ∗ are the consequent ‘best’ number of nodes and ranks (not necessarily integral), the structural value of a classification of N basic taxa can be expressed as V = 1 ‐ ∑ i ∑ j ( n i j ‐ a ) / ( N + T ) ( T ‐ T * ) / ( N ‐ 1 ) ‐ ( r ‐ r * ) / 1 n N ,calculated only over n i j > a, r > r ∗, T > T ∗. The structural value statistic is applied to a number of hypothetical situations and to a range of published classifications dating from 1753 to 1976. It is suggested that the statistic, which is essentially independent of the data used in assessing similarity, supplements the existing data‐dependent criteria for determining the quality of a classification and serves to specify more precisely one aspect of the hierarchical classificatory process.