Evenness indices once again: critical analysis of properties

Various properties have been advocated for biological evenness indices, with some properties being clearly desirable while others appear questionable. With a focus on such properties, this paper makes a distinction between properties that are clearly necessary and those that appear to be unnecessary or even inappropriate. Based on Euclidean distances as a criterion, conditions are introduced in order for an index to provide valid, true, and realistic representations of the evenness characteristic (attribute) from species abundance distributions. Without such value-validity property, it is argued that a measure or index provides only limited information about the evenness and results in misleading interpretations and evenness comparisons and incorrect results and conclusions. Among the overabundant variety of evenness indices, each of which is typically derived by rescaling a diversity measure to the interval from 0 to 1 and thereby controlling or adjusting for the species richness, most are found to lack the value-validity property and some lack the property of strict Schur-concavity. The most popular entropy-based index reveals an especially poor performance with a substantial overstatement of the evenness characteristic or a large positive value bias. One evenness index emerges as the preferred one, satisfying all properties and conditions. This index is based directly on Euclidean distances between relevant species abundance distributions and has an intuitively meaningful interpretation in terms of relative distances between distributions. The value validity of the indices is assessed by using a recently introduced probability distribution and from the use of computer-generated distributions with randomly varying species richness and probability (proportion) components.