Generalized Simpson-diversity

Simpson’s index of diversity equals the probability of drawing without replacement two individuals of different type from a given collection. The interpretation of its inverse as the effective number of types reflects the biological meaning of diversity as the multiformity of the collection. The effective number of types is maximal, equalling the total number of types, only if all types are uniformly distributed. Simpson’s diversity thus fulfills one of the most basic conceptual criteria for diversity measures. The criterion does not, however, directly carry over to continuously varying differences between types. It is shown that the common practice of taking averages or special sums of differences cannot serve as the desired generalization. In a new approach, the probabilistic interpretation of Simpson’s index is used to extend the basic criterion of diversity to arbitrary differences between types, thereby retaining the concept of effective number. By considering ρρ as the resolution at which differences between individuals distinguish them, we define diversity as the probability D(ρ) of sampling without replacement two individuals that differ by more than ρρ. It is pointed out that this measure generalizes the classical Simpson-criterion of diversity in that it applies to any decomposition of the collection into groups, such that individuals within a group differ by at most ρρ and between groups by at least ρρ. Maximum diversity is reached if all groups are of equal size. This maximum property of the new measure motivates definition of the effective number of types at resolution ρρ independently of any particular decomposition into groups and while maintaining the basic diversity criterion. The advantages of and new insights to be derived from this scale-free measure are demonstrated by suggesting an approach to β-diversity in ecology that is consistent with the criterion of diversity and by providing a worked example in population genetics using multiple-loci microsatellite data from three wild cherry populations. The example shows that gene associations may strongly affect the ranking of populations for genetic diversity.