• AMOVA;
  • analysis of variance;
  • Euclidean geometry;
  • functional diversity;
  • Monte Carlo tests;
  • nucleotide diversity;
  • optimization;
  • parametric tests;
  • phylogenetic diversity;
  • quadratic entropy


1. Quadratic entropy (QE) was developed as a fundamental function for measuring the diversity within a collection, such as a community, or population, from indices of abundance and distance among categories, such as species or alleles. Based on a literature review in the fields of genetics, ecology and statistics and new developments, I analyse the potential of this function for biodiversity studies.

2. Quadratic entropy was established as a generalisation of well-known diversity indices, and has been widely used in molecular ecology and genetics research. It is now integrated within more general frameworks for analysing functional and phylogenetic diversity in ecology.

3. Quadratic entropy can be maximised by removing categories, and several collections can share the maximum diversity, even with highly distinct compositions. Clarifying these statements, I identify all potential indices of the abundance of the categories that maximise QE.

4. By quantifying changes in diversity when mixing collections together, QE can measure differences among collections. Here, I provide a geometric interpretation of these differences that demonstrates their relevance as classical geometric distances.

5. A critical aspect of these distances is obtained if QE is strictly concave; that is, diversity always strictly increases by mixing distinct collections together. More generally, QE can be used to evaluate the effects of various factors on diversity in a framework designated ANOQE (analysis of QE). Generalising ANOVA (analysis of variance), ANOQE uses QE to measure distances between centroids.

6. Importantly, QE is estimated from sampled data and thus requires estimators. Based on these estimators, tests have been developed to compare levels of diversity. Tests of factor effects are evaluated by parametric, jackknife, bootstrap and permutational approaches. However, the procedures associated with these tests that have been suggested thus far only treat a few factors.

7. There is an urgent need for the development of such approaches in biology to deal with experimental factors, observed population and community structure, and different spatial and temporal scales. Together, QE and the ANOQE procedure are likely to have a critical impact on all scientific disciplines interested in any form of diversity.