On the mutual relatedness of evenness measures

Several evenness measures have been proposed for quantifying the distribution of abundance among community species, but none seems to be generally preferred. Since these measures have the common objective of summarizing community structure, they may be expected to be intercorrelated. In this paper, seven standard measures of evenness were calculated for 65 sample plots of ruderal vegetation within the archaeological sites of Paestum and Venosa (southern Italy). Principal component analysis was used to identify the primary aspects of community structure being characterized by these seven indices. The first two principal components explained 96% of total variance. A comparison of the first two principal components with the analyzed measures of evenness provides insight into what aspects of community structure they are expressing. While the first principal component is most sensitive to the relative abundances of rare species, the second principal component is clearly associated to changes in the abundance of the dominant community species. Abbreviation: PCA Principal Component Analysis. (Smith and Wilson 1996). This requirement is based on the assumption that community diversity can be partitioned into two components, species richness and evenness. If the separation is incomplete, so that evenness is affected by the number of species, then differences in evenness values could result from differences in the species count rather than any fundamental difference in community organization (Sheldon 1969). As a precise formulation for this notion of independence of species richness, Hill (1973) proposed that replication should not change the value of community evenness. Consider an N-species community characterized by the relative abundance vector p = (p1, p2,..., pN) such that 0 ≤ pi ≤ 1 and Σpi = 1. It seems reasonable that replicating the N-species sequence n-times (and renormalizing) should multiply richness by n but leave evenness unchanged. Notice that this replication property is part of Taillie’s (1979) more general requirement that an evenness index maintains the natural ordering introduced by the Lorenz curves used by economists to compare wealth distributions. The Lorenz curve is obtained by plotting the cumulative species relative abundances as abscissa against corresponding cumulative proportions of species as ordinates. Arrange the components of the species relative abundance vector p of a given community in descending order so that the ranked abundance vector p# = is obtained, where The Lorenz curve is then defined as the polygonal path joining the successive points: π0 = (0, 0), π1 = (p1, 1/N), π2 = (p1+p2, 2/N),..., πN = (p1+p2+... + pN , N/N ) ≡ (1, 1) (Figure 1). The resulting diagram is similar to the intrinsic diversity profile proposed by Patil and Taillie (1979, 1982) for defining the concept of intrinsic diversity order: both use as abscissa the cumulative species relative abundances. However, the intrinsic diversity profile uses as ordinate the cumulative number of species, whereas the Lorenz curve uses as ordinate the cumulative proportion of species. Patil and Taillie (1979, 1982) defined community A to be intrinsically more diverse than community B without reference to indices, provided B leads to A by a finite sequence of forward transfers of abundance (for mathematical details, see Patil and Taillie 1979, 1982). Following this definition, the hypothetical community A is intrinsically more diverse than community B if and only if community A has its intrinsic diversity profile everywhere above that of community B. Notice that the ordering is only partial in that two communities need not be intrinsically comparable. In this latter case, the intrinsic diversity profiles of both communities cross one another. Similarly, community A is intrinsically more even than community B if and only if community A has its Lorenz curve everywhere above that of community B. Consequently, a measure of evenness E that is invariant under species replication maintains the Lorenz ordering provided that E is consistent with the intrinsic diversity ordering when restricted to communities with the same number of species (Taillie 1979). For instance, when diversity comparisons are restricted to communities with the same number of species, since there is no fundamental difference between diversity and evenness when species richness is held constant, the intrinsic diversity ordering is identical to the corresponding Lorenz ordering.