Measuring diversity from dissimilarities with Rao's quadratic entropy: Are any dissimilarities suitable?

Rao has developed quadratic entropy to measure diversity in a set of entities divided up among a fixed set of categories. This index depends on a chosen matrix of dissimilarities among categories and a frequency distribution of these categories. With certain choices of dissimilarities, this index could be maximized over all frequency distributions by eliminating several categories. This unexpected result is radically opposite to those obtained with usual diversity indices. We demonstrate that the elimination of categories to maximize the quadratic entropy depends on mathematical properties of the chosen dissimilarities. In particular, when quadratic entropy is applied to ultrametric dissimilarities, all categories are retained in order to reach its maximal value. Three examples, varying from simple one-dimensional to ultrametric dissimilarity matrices, are provided. We conclude that, as far as diversity measurement is concerned, quadratic entropy is most relevant when applied to ultrametric dissimilarities.