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Keywords:

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

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Development of quadratic entropy
  5. H D optimisation
  6. Estimating and comparing levels of diversity
  7. Among-collection diversity
  8. Partitioning diversity
  9. Testing the effects of factors on diversity: directions for future research
  10. Conclusions
  11. Acknowledgements
  12. References
  13. Supporting Information

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.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Development of quadratic entropy
  5. H D optimisation
  6. Estimating and comparing levels of diversity
  7. Among-collection diversity
  8. Partitioning diversity
  9. Testing the effects of factors on diversity: directions for future research
  10. Conclusions
  11. Acknowledgements
  12. References
  13. Supporting Information

Traditionally, biodiversity has been measured by counting categories (e.g., species, alleles) in collections (e.g., communities, populations) of entities (organisms). However, studies now are evolving towards a more synthetic approach in which different scales and explanatory factors are considered. At an ecological level, species have different phylogenies and life histories that make them similar in some aspects and unique in others. At a genetic level, some DNA sequences share more nucleotides than others. According to these considerations, an index of diversity should include: (i) differences among categories (e.g., nucleotide distances among alleles, phylogenetic/functional distances among species); (ii) the proportions of these categories within a collection (e.g., relative abundance of an allele within a population or of a species within a community); and (iii) factors that might impact the level of diversity (e.g., spatio-temporal scales or factor levels in an experimental design).

Quadratic entropy (QE), defined as the expected distance between two entities in a collection, satisfies these requirements (Rao 1982a). In genetics, nucleotide distances among alleles are used to compare genetic diversity among species and to reveal factors that influence populations, including mutation rate and effective population size (Nei & Li 1979). In ecology and conservation, the taxonomic, phylogenetic and functional distances between species are used to prioritise the conservation of species and areas (Faith 1992; Pavoine, Ollier & Dufour 2005a), to evaluate ecosystem services (Lavorel et al. 2010) and to understand the ecological processes that structure community assemblages (Warwick & Clarke 1995; Pavoine & Dolédec 2005; Hardy & Senterre 2007). With this definition, QE is closely related to classical species encounter theory (e.g., Patil & Taillie 1982).

Interestingly, QE can be partitioned across different levels of factors in exactly the same way that variance is partitioned in analysis of variance (ANOVA) (Rao 1982b, 1986). This type of partitioning was thus designated analysis of quadratic entropy, ANOQE (Liu 1991). Through this analysis, it is possible to evaluate the effects of experimental or structural (e.g., spatio-temporal) factors on biodiversity. Despite its development in the 1980s, this approach is still under-exploited, and is still in its infancy in the domain of testing the effects of multiple factors. In addition, the connections between ANOQE and other independent developments are not widely acknowledged in the biological literature (Excoffier, Smouse & Quattro 1992; Gower & Krzanowski 1999; Legendre & Anderson 1999; McArdle & Anderson 2001). Analysis of molecular variance, AMOVA (Excoffier, Smouse & Quattro 1992), which tests for genetic structures in hierarchical subdivisions of a population, is a striking example of such a connected approach (Pavoine 2005), with the original publication cited about 4000 times (ISI Web of Science 2011). The original report (Rao 1982a) on ANOQE publication by contrast has only 232 citations. However, ANOQE has more potential than AMOVA. It can be employed to analyse nested, crossed, fixed, random or mixed factors, and has several ramifications allowing multivariate factorial analysis (Pavoine, Dufour & Chessel 2004; Pavoine & Bailly 2007) as well as decomposition of diversity across a taxonomic (Ricotta 2005) or phylogenetic (Pavoine, Love & Bonsall 2009; Pavoine, Baguette & Bonsall 2010) tree, all of which render the approach practical for use by biologists. Recently, Rao (2010) re-emphasised the importance that ANOQE is likely to have for future research on diversity. In this context, the objectives of this report are to review the developments made thus far with respect to QE and related approaches, to introduce new developments and to propose new directions for future research.

Development of quadratic entropy

  1. Top of page
  2. Summary
  3. Introduction
  4. Development of quadratic entropy
  5. H D optimisation
  6. Estimating and comparing levels of diversity
  7. Among-collection diversity
  8. Partitioning diversity
  9. Testing the effects of factors on diversity: directions for future research
  10. Conclusions
  11. Acknowledgements
  12. References
  13. Supporting Information

H D Function (QE Index)

Measures similar to the QE index have been developed independently in the fields of functional ecology, genetics, taxonomy and economics since the report by Hendrickson & Ehrlich (1971) (see Appendix S1 for details). Rao (1982a) defined the DIVC index (DIVersity Coefficient) as

  • image

where P is the probability distribution function of a variable X and d(X1,X2) is a non-negative symmetric function that measures the difference between two individuals with X = X1 and X = X2. The term ‘generalised quadratic entropy function’ appeared in Rao (1982c), and the simpler version, ‘quadratic entropy’, was retained in subsequent reports (e.g. Rao 1986, 2010; Rao & Nayak 1985).

In current biodiversity studies, the formula simplifies to

  • image(eqn 1)

where p is a vector of proportions (p1,p2,…,pS) in the set inline image. Contrary to other developments (Izsák & Pavoine 2011), matrix D = (dkl), which contains the differences between categories (e.g., taxonomic, phylogenetic, functional differences among species, nucleotide differences among alleles) is considered here to be independent from p. Based on pair-wise comparisons, HD(p) is thus the mean difference between two random categories and, therefore, it cannot integrate higher levels of inter-relationships among more than two categories.

Lau (1985) and Rao (1986) demonstrated a useful property of HD(p) (see Partitioning Diversity) if D = (dkl) satisfies the following conditions:

  • image(eqn 2)

H D Generalises Well-Known Indices

Let dkl = 1 for all kl (which is equivalent to D = 11tI, with 1 being the S × 1 vector of ones and I the S × S identity matrix); inline image is then equal to the Gini–Simpson index (variation in a qualitative variable, Rao 1982a):

  • image(eqn 3)

In comparison with the Gini–Simpson index, HD can integrate the fact that species are not equivalent but differ in terms of phylogenetic and taxonomic positions, as well as functional traits, or the fact that some alleles might share more nucleotides than others.

In addition, HD also generalises the variance of a quantitative variable (Rao 2010). The variance of the variable Y is usually written as the mean squared deviation between yk values and their mean, y. However, it can be rewritten as the average squared difference among values:

  • image

Using dkl=(ykyl)2/2 leads to Eqn 1. Although the variance only considers a single quantitative trait to characterise the categories, HD can consider multivariate distances among categories, which is necessary, for instance, when measuring taxonomic and phylogenetic distances among species, or functional distances based on multiple traits.

H D Is Generalised

In contrast, two generalisations of HD have been developed in ecology. The objectives of these generalisations are to include a parameter in the index of diversity that modifies the importance given to rare species in comparison with more abundant ones. The structure of a community is then described by a vector where abundant species are progressively given more weight until only the most abundant species dominates. This vector might be used to test whether different processes underlie the presence of rare vs. abundant species.

Ricotta & Szeidl (2006) defined

  • image

With a = 2, Qa = HD; with a = 0, Qa is a generalisation of the richness; and with a tending to 1, Qa is a generalisation of the Shannon (1948) index.

Pavoine et al. (2009) also generalised HD in the context of a phylogenetic tree describing the estimated dates of speciation among species. Each period in a phylogeny defines groups of species that descend from it, just as dividing a genealogical tree at a given time defines sets of related families. The index of biodiversity was defined as

  • image

The values (tKtK−1) are the lengths of periods, and pi,K is the proportion of the ith group defined at period K. With a = 0, Ia generalises the richness and is equal to the famous Faith (1992) index of phylogenetic diversity, apart from the use of an additive constant (the height of the tree). With a = 2, Ia = HD, where D is defined from the tree. When a tends to 1, Ia is a generalisation of the Shannon (1948) index.

Proposed Functions of HD

Recent reports in ecology have proposed using functions of HD instead of HD itself. The objective of this is to obtain intuitive measures that are more easily interpretable by ecologists and conservation biologists. The method consists of obtaining the ‘doubling’ or ‘replication’ property: if N equally diverse, equally large and maximally dissimilar assemblages are pooled, the diversity of the pooled assemblages must be N times the diversity of the individual assemblages. With the condition that the distances (values in D) lie in [0,1], Ricotta, Burrascano & Blasi (2010) suggested using 1/(1−HD). Chao, Chiu & Jost (2010) developed a related formula for the special case of distances among species obtained from rooted phylogenetic trees or functional dendrograms: inline image. Here, the distances among species in D are equal to half the sum of the branch lengths on the shortest path that connect two species in a tree, and inline image is the average distance between a species and the root of the tree.

Given that the function f(x) = 1/(1−x) increases monotonically, these transformations do not change how communities are ranked according to their diversity. Rather, they change the absolute value of the difference between the diversity of two communities, and will thus impact our interpretation of how much more (or less) diversified a community is in comparison to others. These transformations are therefore useful to compare the levels of diversity among distinct communities (Jost 2006). However, considering the developments to date regarding the quadratic entropy index, the function HD, without any transformation, has the advantage of being integrated in a more general, inferential framework: tests have been developed based on HD to evaluate the effects of various factors on diversity.

It is very common in statistics that the metric associated with a test is a function of the metric used for an intuitive interpretation. As a result, both HD and its transformation could be used in the future to test and interpret, respectively, the effects of different factors on levels of diversity. Nevertheless, transformations have so far only been proposed for measuring the diversity within and among collections (Ricotta et al. 2010). To my knowledge, transformations associated with the effects of multiple factors affecting biodiversity have not been suggested.

H D optimisation

  1. Top of page
  2. Summary
  3. Introduction
  4. Development of quadratic entropy
  5. H D optimisation
  6. Estimating and comparing levels of diversity
  7. Among-collection diversity
  8. Partitioning diversity
  9. Testing the effects of factors on diversity: directions for future research
  10. Conclusions
  11. Acknowledgements
  12. References
  13. Supporting Information

The previous section introduced QE (function HD) and positioned it among the indices that it generalises and, in contrast, among the indices that generalise it. This section goes one step further in characterising QE through analysis of its maximum. According to HD, what would be the characteristics of a population with maximum possible genetic diversity? What would be the characteristics of a community with maximum ecological diversity? Precise definition of the maximum of a diversity index is important because many conservation strategies are based on preserving maximum possible amount of.

Maximising HD in a Euclidean Space

Maximisation of HD has been studied in the case where matrix D is fixed. All species that could be found in a community, or all alleles that could occur within a population, are therefore known, and the genetic, taxonomic, phylogenetic and functional distances among them are fixed. Thus, HD is optimised over all possible vectors, p. This maximisation approach, in which D is fixed, is coherent with the maximisation of more classical indices of diversity (e.g., Shannon 1948; Simpson 1949), where the number of categories is fixed and the distances among categories are fixed as equal. Regardless, for these more classical indices the maximum is unambiguously obtained for the evenness of the proportions of the categories. When the distances among the categories are not equal, the maximum is quite different. Pavoine, Ollier & Pontier (2005b) have obtained the maximum value and a maximising vector, p, for HD. Pavoine & Bonsall (2009) demonstrated that several vectors p can lead to the maximum. I formulate these findings in a new proposition and provide a demonstration of this in Appendix S2. Figure 1 gives examples of the HD values when three theoretical categories are considered.

image

Figure 1.  Examples of applications of HD with three categories. In all panels HD is standardised to a maximum value to enable analysis of HD independently on the scale at which the distances were measured. In the top left-hand corner of each panel is a Euclidean representation of the distances among the three categories. In the bottom left-hand corner of each panel is given the marginal behaviour of HD (when one of the categories has a null proportion and the relative proportions of the two remaining categories vary). On the right of each panel, a triangle plot is used to highlight the values taken by HD when the proportions of the three categories vary. The vector of proportion is (pA,pB,pC). The maximum value is indicated with an open circle. The maximum is obtained at (0, 0·5, 0·5) for panel a, (inline image) for panel b, approximately (0·20, 0·42, 0·38) for panel c and (0·5, 0, 0·5) for panel d.

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Proposition 1

Let HD be the function defined in Eqn 1, with D being an S × S matrix satisfying condition 2. Given that D satisfies condition 2, there exists a Euclidean space with points Mk, k = 1,…,S, such that ||MkMl||2/2 = dkl for all k and l. Let T be a set of s points on the boundary of the smallest hypersphere (SEH) that encloses the S points Mk, k = 1,…,S. Let Ds = (dkl) be the s×s symmetric square subset of matrix D that corresponds to set T. Any vector that gives the proportions defined as inline image for the s points in T and as zero for the remaining points with the constraint that inline image is the radius of the SEH and that inline image contains non-negative values is a maximiser.

An important consequence of this proposition is that, according to HD, diversity can be maximised by reducing richness (i.e., the number of categories, alleles or species), and this depends on matrix D. The choice of D is therefore critical.

Choice of Matrix D

The fact that HD can be maximised by removing some categories has been considered a poor property for an index of diversity (e.g., Shimatani 2001). This is because the simple counting of categories such as species or alleles is still deeply rooted in traditional biodiversity studies, and strongly influences research associated with new diversity indices. Alternatively, I suggest considering the fact that HD, in its general definition, departs from traditional indices of biodiversity and complements them as a strength. HD is a new, non-redundant synthetic measure of biodiversity.

Its connection with traditional indices depends on the choice of matrix D. In particular, D = (dkl) is ultrametric if and only if dkl≥0 ∀k,l; dkl≤max(dkj,djl) ∀j,k,l; and dkk≤minlkdklk (which includes equidistances). Ultrametric distances are important in ecology and conservation as taxonomic distances, and the phylogenetic distances defined as the time from speciation, are ultrametric. With D being ultrametric, HD has several interesting properties: (i) the maximising vector of proportions is unique; (ii) it has no zero values; and (iii) it reflects species’ contributions to diversity (Pavoine et al. 2005a).

Estimating and comparing levels of diversity

  1. Top of page
  2. Summary
  3. Introduction
  4. Development of quadratic entropy
  5. H D optimisation
  6. Estimating and comparing levels of diversity
  7. Among-collection diversity
  8. Partitioning diversity
  9. Testing the effects of factors on diversity: directions for future research
  10. Conclusions
  11. Acknowledgements
  12. References
  13. Supporting Information

Now that HD has been defined, identified as a generalised measure, and described based on an optimisation study as a new integrative measure of diversity, it is important to consider that it is estimated from samples, and that therefore estimators are required. Analysing estimators for HD is a prerequisite for testing the effects of factors on its values.

Nayak Parametric Estimator for HD

Nayak (1983) obtained an estimator for HD with the assumption that the observed proportions are drawn from a multinomial distribution. Let π=(π1,π2,…,πS)t be the unknown vector of proportions. Let Nk be a variable denoting the observed number of entities from category k and n the total number of entities observed; k = 1,…,S. Let inline image and inline image. An estimator for HD(π) is

  • image

Nayak (1983) demonstrated that

  • image

where

  • image

the estimator is asymptotically unbiased, and inline image tends towards 0 when n tends towards infinity. inline image is therefore an appropriate estimator for HD(π) (Nayak 1983).

Generally (see Propositions 4.4.6 and 4.4.7 in Nayak 1983),

  • image

Under this condition, a confidence interval for HD can be specified for large samples (high n values) at level 100(1−α)%, with p as an estimate of π and inline image as an estimate of 4πtDΣDπ obtained by replacing π by p:

  • image

where ɛα/2 is the threshold from the normal distribution N(0,1) associated with α.

From this, Nayak (1983) developed tests for differences between HD estimates assuming that the proportions are drawn from a multinomial distribution, and that the sample size is large. Consider H0:HD(p1)=HD(p2)=…=HD(pr), meaning that the diversities within r samples are equal. Let ni be the number of entities observed in sample i, inline image be the averaged diversity over all samples, and inline image be obtained by replacing π with pi in equation 4πtDΣDπ:

  • image

H0 is rejected with a risk of error α if

  • image

where inline image is the threshold from a inline image distribution associated with α. This test might be used to compare the level of nucleotide diversity among populations or the level of taxonomic, phylogenetic and functional diversity among communities.

Non-parametric Alternatives

Field data will not necessarily satisfy the multinomial assumption. Additionally, in ecological studies, the number of individuals per sample is not always known, with quantities instead being broadly measured as biomass, percentage cover or density. Alternatives to Nayak's HD estimates and tests for differences between HD estimates should therefore be developed.

Other approaches do exist for more traditional indices that do not include a matrix D but that could be adapted for application to HD. For instance, Magurran (2004) suggests that it could be more beneficial to measure the diversity index within a number of samples, instead of within a single large sample. Such samples could be jackknifed to improve diversity estimates, and obtain confidence intervals. Diversity curves analysing HD as a function of the number of samples could be computed. These curves may or may not attain an asymptote. In addition, Ricotta et al. (2010) suggested an estimator for HD based on rarefaction methods, where the relative abundances of species are replaced with species’ contributions to the expected species richness. This estimator uses the presence/absence of species in samples, instead of their abundance. Finally, if a single large sample is available, the applicability of bootstrap approaches will depend on how the data have been collected (e.g., Liu 1991; Liu & Rao 1995).

A further alternative to Nayak (1983) would be to develop a test of the homogeneity of multivariate dispersions (Anderson 2006) accounting for the fact that sampled categories are weighted (e.g., a species is weighted by its biomass). Whether it is possible to compare the level of HD among populations or communities in real datasets is thus still an open question.

Among-collection diversity

  1. Top of page
  2. Summary
  3. Introduction
  4. Development of quadratic entropy
  5. H D optimisation
  6. Estimating and comparing levels of diversity
  7. Among-collection diversity
  8. Partitioning diversity
  9. Testing the effects of factors on diversity: directions for future research
  10. Conclusions
  11. Acknowledgements
  12. References
  13. Supporting Information

Previous sections have focused on measuring diversity within a collection. However, a critical component of diversity is the average distance between collections. In genetics, measuring dissimilarities among populations allows evaluation of the degree of isolation among populations. In ecology, measuring dissimilarities among communities can reveal critical processes, including dispersal and colonisation, environmental filters and competition (Pavoine & Bonsall 2011).

A Unified Approach for Diversity and Dissimilarities

Diversity and dissimilarities are two related concepts, as the diversity of a collection is zero if all of its entities are the same. Increasing numbers of studies therefore incorporate nucleotide, taxonomic, phylogenetic and functional dissimilarities among organisms. These dissimilarities should also be considered when measuring the dissimilarities among whole collections.

According to Rao & Nayak (1985), any non-negative and concave index of diversity, H, might be partitioned as

  • image(eqn 4)

where C(pi,p) is a measure of differences between a collection i with the vector of proportions pi, and a theoretical average collection with the vector of proportions p=∑iλipi, λi is a weight given to collection i with ∑iλi=1. If C is symmetric (C(p,q)=C(q,p)), then Eqn 4 can be rewritten as

  • image(eqn 5)

where

  • image(eqn 6)

D(pi,pj) is a measure of dissimilarity, induced by the function H, between a collection with the vector of proportions pi and a collection with the vector of proportions pj. Rao & Nayak (1985) demonstrate that C is a symmetric function if, and only if, H can be written as a form of HD. HD therefore unifies the diversity and dissimilarity concepts because the diversity of a collection is measured based on dissimilarities among categories, and dissimilarities among collections are calculated from diversity. Equation 5 corresponds to a first, simple version of diversity partitioning, where the diversity in the collections pooled together (SST=HD(p)) is equal to the sum of the averaged diversity within collections (inline image), and the diversity among collections (SSB = SST−SSW). If D satisfies condition 2, HD is concave (Rao 1986), which ensures that D(pi,pj)≥0 and SSB ≥ 0.

A Geometric Interpretation

A measure of the distance between populations or communities should indicate the degree of differences among these populations or communities in terms of the organisms they contain, and in terms of differences among these organisms. From a functional point of view, two communities that share no species might be evaluated as similar if the species they contain have similar life histories. The biological meaning of D(pi,pj), as a measure of the distance among populations or communities, has been questioned. Pavoine & Bonsall (2009) demonstrate that the dissimilarity among two collections with maximum diversity is zero, even if the two collections have different compositions and thus fall into different categories. There is therefore a need to clarify how D(pi,pj) measures the distance among two collections.

According to Eqn 6, these distances among collections are defined as the degree to which diversity increases by mixing two collections together. As Ricotta (2005) highlights, using this excess of diversity as a distance among collections is not necessarily straightforward. However, the geometric interpretation of D(pi,pj) provided below demonstrates that D is definitely a meaningful distance metric.

If D satisfies condition 2, then it is possible to define a Euclidean space in which each category k will be positioned at a point Mk, such that ||MkMl||2/2=dkl for all k and l. Let M be the S×n matrix of coordinates, with points (categories) as rows and axes as columns, n is the dimension of the Euclidean space. Then, each collection might be positioned at a centroid of the category points: a collection i with the vector of proportions pi is positioned at point Gi with the vector of coordinates inline image, such that ||GiGj||2/2=D(pi,pj) (Champely & Chessel 2002; Pavoine et al. 2004). The distance between two collections is therefore equal to half the squared distance between their centroids. In this Euclidean space, the decomposition SST = SSW+SSB given above is a decomposition of geometric variability (Pavoine et al. 2004; Cuadras 2008).

A consequence of this geometric interpretation is that D(pi,pj) = 0 if, and only if, the collections i and j have identical centroids. Examples are given in Fig. 2. In the extreme case where the distances among categories can be plotted in only one dimension, that is where ANOQE = ANOVA, then D(pi,pj) compares the means. Thus, the geometric interpretation here given provides justification for the use of D(pi,pj): the ANOVA framework defines the distances between two collections as differences between means, whereas the ANOQE framework more generally defines the distances between two collections as differences between centroids. The fact that distances are measured from means (or centroids) must be kept in mind when interpreting values from the D(pi,pj) index. For instance, in Fig. 2a, if points are species and if the Euclidean space is defined by the body size of the species, then a community with species A, D and E and a community with species B, C and F might be considered as equivalent because the species within them have equal mean body sizes.

image

Figure 2.  Examples of zero distance between distinct collections. (a) Six theoretical categories are placed on an axis with the following coordinates: 0, 1, 2, 5, 8 and 10. Categories have been named in this order from A to F. A collection with categories A, D and E with even proportions will be positioned at the same value (at the mean = 13/3, closed dot) on this axis as a collection with categories B, C and F with even proportions. According to Eqn 6, the distance between these two collections is zero, even if their composition is different. (b) Four theoretical categories are represented in a Euclidean space as the vertices of a square. A collection including categories A and C with even proportions will be positioned at the centre of the square. A collection with categories B and D with even proportions will also be positioned at the centre of the square. According to Eqn 6, the distance between these two collections will therefore be zero, even if their composition is different. More concretely, the categories in these two panels might be species. The Euclidean space could be defined by a single quantitative functional trait in panel a and by two traits in panel b.

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Consequence of the Strict Concavity of HD

It is clear that if two sets of points in a multidimensional space are identical and associated with equal weights, their centroids must be identical. However, the converse is not true, except under particular conditions. More formally, D(pi,pj) = 0 if pi = pj, but the converse is true only if HD is strictly concave; diversity always strictly increases by mixing distinct collections together. Here, I provide further conditions on D such that HD is strictly concave (see Appendix S3). In this case D(pi,pj) = 0 if and only if pi = pj, meaning that the distance between two collections is zero if and only if their compositions are identical. If HD is not strictly concave, the distance might be zero even if the compositions are not identical, and thus the strict concavity for HD could be a prerequisite for a more intuitive measure of among-collection diversity.

Proposition 2

Let p ∈ AS and D be an S×S matrix satisfying condition 2, and let inline image, where 1 is the S×1 vector of ones and I the S×S identity matrix. HD(p) = ptDp is strictly concave if and only if S points with coordinates X, defined as −QDQ = XXt, are embedded in exactly S−1 dimensions; that is to say, if and only if rank(−QDQ) = S−1.

Note that ultrametric distances, including equidistances and, thus, the particular case of the Gini–Simpson index, satisfy this property. If HD is strictly concave, then there is a single collection with a maximum value of HD.

Partitioning diversity

  1. Top of page
  2. Summary
  3. Introduction
  4. Development of quadratic entropy
  5. H D optimisation
  6. Estimating and comparing levels of diversity
  7. Among-collection diversity
  8. Partitioning diversity
  9. Testing the effects of factors on diversity: directions for future research
  10. Conclusions
  11. Acknowledgements
  12. References
  13. Supporting Information

Eqn 5 is a first diversity partitioning that analyses the effect of one factor on HD, but QE can be more generally applied to estimate the effects of multiple factors.

Evaluating the Contributions of Factors on Values of HDANOVA is generalised

If and only if matrix D satisfies condition 2, the quadratic entropy is completely concave (Lau 1985; Rao 1986). This property means that an ANOVA-like analysis (ANOQE) can be performed using HD instead of variance (Rao 1986, 2010), which implies that it is possible to test the effects of experimental factors or observational factors, such as spatial and temporal scales, or subdivisions of populations or communities into demes or patches on diversity. HD is decomposed into a number of non-negative components, assigned to specified nested and/or crossed factors and interactions. For instance, consider two crossed factors, X1 with r levels and X2 with s levels. Let pijk be the proportion of the kth category for the ith level of X1 and the jth level of X2. Let pij=(pij1,…,pijk,…,pijS) be the corresponding vector of proportions. Let λij be the weight attributed to this sample; for instance, λij=∑kpijk/∑i,j,kpijk. Let λi=∑jλij; λj=∑iλij; pi=∑jλijpij/λi; pj=∑iλijpij/λj; and inline image. Then

  • image(eqn 7)

SSB(X1) and SSB(X2) are the main effects of X1 and X2, respectively.

  • image
  • image

SSB(X1×X2) is the component of diversity stemming from the interaction between factors X1 and X2. It can be simply deduced from Eqn 7 and has a simple form only if λij=λiλj (independence of prior probabilities):

  • image

where pc=pijpipj+p•• (Nayak 1983). It is also possible to evaluate the conditional effect of X2 given X1 as follows (Nayak 1986a):

  • image(eqn 8)

Similar approaches have been developed by Gower & Krzanowski (analysis of distance, 1999), Legendre & Anderson [distance-based redundancy analysis (db-RDA) 1999], Anderson (nonparametric multivariate analysis of variance 2001) and McArdle & Anderson (extended development of db-RDA for applications to dissimilarity matrices that do not satisfy condition 2, 2001). In addition to these closely related approaches, ANOQE generalises the well-known ANOVA method (Fisher 1925) and its matching categorical analysis of variance (Light & Margolin 1971). Despite the extensive use of ANOVA and this generalised character of ANOQE, use of ANOQE in ecology and genetics is still scarce.

Other Specific HD Partitions in Ecology

Other kinds of HD partitioning complementing the ANOQE framework have already been developed in the field of ecology. First, taxonomic diversity can be measured by HDTax, where DTax represents taxonomic distances among species that depend on the hierarchy in the taxonomy. The distance between two species from the same genus is 1, while that between two species from different genera but the same family is 2, and so on. Consider a taxonomy including species, genera, families and orders. Let p, g, f and o be the vectors of the relative abundances of the species, the genera, the families and the orders, respectively. According to Shimatani (2001),

  • image

where G is the Gini–Simpson index (see HDgeneralises well-known indices). This stems from an additive property of HD (if D=D1+D2, HD=HD1+HD2) and from the fact that inline image (Eqn 3). This taxonomic decomposition (or any other additive partitioning of matrix D) can be applied to any component of ANOQE, such that these two complementary approaches allow studies of the effects of factors on the diversity of different taxonomic levels (Ricotta 2005). As Pavoine et al. (2009) highlighted, a similar approach can be applied to phylogenies divided into evolutionary periods, and a more complex decomposition of HD on a phylogenetic tree can be found in Pavoine et al. (2010).

For understanding the respective roles of the proportions and the distances between categories in the measurement of diversity, another decomposition for HD is as follows (Shimatani 2001),

  • image

where G is the Gini–Simpson index, A is the unweighted average distance, and B evaluates covariations between proportions and distances. This equation is therefore related to the well-known decomposition of the expected product of two variables: E(XY)=E(X)E(Y)+Cov(X,Y). In an ecological setting, analysis of component B in the context of understanding the determinants of community diversity and structure may be very interesting. Indeed, processes that limit similarity among species (including some cases of competition, or mutualism) are expected to lead to positive covariance between abundance and distance, whereas processes that increase similarity among species, including environmental pressures, are expected to lead to negative covariance between abundance and distance.

These particular decompositions of HD complement the ANOQE framework, providing additional details on the relative impacts of D and p on each individual component of ANOQE.

Testing the effects of factors on diversity: directions for future research

  1. Top of page
  2. Summary
  3. Introduction
  4. Development of quadratic entropy
  5. H D optimisation
  6. Estimating and comparing levels of diversity
  7. Among-collection diversity
  8. Partitioning diversity
  9. Testing the effects of factors on diversity: directions for future research
  10. Conclusions
  11. Acknowledgements
  12. References
  13. Supporting Information

A crucial step to allow for concrete applications of ANOQE to real data sets will be the development of the tests associated with ANOQE and although tests have been suggested, there is a critical need for further development.

One Factor

The decomposition for a single factor is given in A unified approach for diversity and dissimilarities. The effect of one factor on diversity has been evaluated with parametric tests by assuming multinomial distributions and large samples (Nayak 1986b), with bootstrap (Liu 1991; Liu & Rao 1995) and with permutational schemes (Anderson 2001; Pavoine & Dolédec 2005; Hardy & Senterre 2007; Hardy 2008). Anderson (2001) emphasised that one of the assumptions of these tests might be that observations have similar distributions, which means that a test for the equality of HD measured per level of a factor (see Estimating and comparing levels of diversity) could be necessary before evaluating the effect of the factor. Nevertheless, a critical issue was observed by Nayak (1986b), who demonstrated that the components SST (total diversity) and SSB (diversity between levels of factors; i.e., the effect of a factor) are asymptotically independently distributed. Contrary to ANOVA, SSB/SST rather than SSB/SSW should therefore be used in statistical inference associated with the ANOQE. In addition, this ratio has a direct interpretation as the proportion of explained diversity. Another important definition of these tests is that of the null hypothesis, H0. For instance, the Nayak (1986b) test is based on H0 = the true proportion vectors for each level of the factor being equal; that is H0 = π1 = … = πr. H0 implies that the dissimilarities among collections are zero (SSB = 0), but the converse is true only if HD is a strictly concave function (see Consequence of the strict concavity ofHD).

Two Factors

Few solutions have been proposed with more than one factor. For nested factors, solutions have been developed with permutational schemes (Excoffier, Smouse & Quattro 1992; Pavoine & Dolédec 2005).

Nayak (1986a) tackled the question of two crossed factors. Let πij, πi, πj and π•• be the unknown vectors of proportions associated with level i of factor X1 and level j of factor X2, level i of factor X1, level j of factor X2, and the whole studied collection (e.g., a population, a community, a region), respectively (see Evaluating the contributions of factors on values of HDANOVA is generalised). Considering multinomial distributions and the hypotheses H0:πi=π••i, H0:πj=π••j, H0:πij=πi+πjπ••i,j, Nayak (1983) found that the asymptotic distributions of SS(X1), SS(X2) and SS(X1×X2) under these respective null hypotheses depend on unknown parameters. However, he subsequently provided tests for the conditional effect of one factor given the other (ρ2(2|1), see Eqn 8) (Nayak 1986a). For example, the test for a conditional effect of X2 given X1 corresponds to the null hypothesis H0=πi1=…=πiS, for all i = 1,…,r. Under H0, the asymptotic distribution of r(S−1)(n•••r)ρ2(2|1) can be approximated by inline image (Nayak 1986a). This test might help to distinguish the roles of, for example, space vs. time or of two experimental treatments on the level of diversity.

Alternative solutions based on permutational testing were discussed by Legendre & Anderson (1999) in the context of the related db-RDA approach and the issues raised should be considered in developing permutational tests for crossed factors with ANOQE. A critical step is to decide whether the factors considered are fixed, random or mixed. For instance, AMOVA was developed only for nested, random factors.

Restrictions

A key challenge in the analysis of real data is that data need to satisfy all of the hypotheses on which tests are based. Most tests developed in the last two sections are associated with restrictive hypotheses that the vectors of proportions should satisfy: for instance that there should be independence among individual samplings, multinomial distributions and large samples Nayak (1986a,b). In the context of nested ANOQE, Pavoine & Dolédec (2005) tackled the question of non-independence of sampled individuals. As organisms may be distributed patchily, sampling within a patch may lead to a high number of individuals from a single species being drawn simultaneously. Considering a single factor, Hardy (2008) introduced various tests that provide inferential solutions in the case of correlations between the vectors of proportions p and the matrix D, as well as in the case of spatial autocorrelation. It is therefore crucial that constraints related to field or experimental data in biology be acknowledged to allow for further development of the ANOQE framework.

Conclusions

  1. Top of page
  2. Summary
  3. Introduction
  4. Development of quadratic entropy
  5. H D optimisation
  6. Estimating and comparing levels of diversity
  7. Among-collection diversity
  8. Partitioning diversity
  9. Testing the effects of factors on diversity: directions for future research
  10. Conclusions
  11. Acknowledgements
  12. References
  13. Supporting Information

The quadratic entropy index (QE) was used here based on vectors of proportions (index HD), and I have highlighted how this function is maximised and how it can be estimated from sampled data. QE provides a novel view of diversity and a powerful, rigorous statistical framework for analysing biodiversity. Within the ANOQE framework, QE has the potential to identify and test nested and crossed factors underlying biodiversity, while placing ANOQE in a geometric view reinforces its connections with ANOVA. The title ‘analysis of variance’, abbreviated ANOVA by Tukey, stems from the fact that variances are used to measure differences among means (Sokal & Rohlf 1995). Here, using the ANOQE approach, we are able to show that quadratic entropies may be used to measure distances between centroids.

QE, ANOQE and related approaches have been developed several times in genetical and ecological research. They may be implemented in several software applications, which are presented in Appendix S4. Despite these developments, further approaches still require advances test procedures, which thus far have been limited to nested factors or two crossed factors. Identification of questions raised by the differences between fixed, random and mixed factors is also required. Finally, for concrete application to experimental or observational data, these developments should be made in association with research from different biological fields, identifying needs for dealing with particular scenarios, such as broad definitions of proportions (e.g., biomass, densities), and spatial and temporal autocorrelations.

I hope to have highlighted that the QE and ANOQE frameworks could have a strong impact and applicability to any scientific field with a focus on the analysis of multivariate data.

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Development of quadratic entropy
  5. H D optimisation
  6. Estimating and comparing levels of diversity
  7. Among-collection diversity
  8. Partitioning diversity
  9. Testing the effects of factors on diversity: directions for future research
  10. Conclusions
  11. Acknowledgements
  12. References
  13. Supporting Information

Supporting Information

  1. Top of page
  2. Summary
  3. Introduction
  4. Development of quadratic entropy
  5. H D optimisation
  6. Estimating and comparing levels of diversity
  7. Among-collection diversity
  8. Partitioning diversity
  9. Testing the effects of factors on diversity: directions for future research
  10. Conclusions
  11. Acknowledgements
  12. References
  13. Supporting Information

Appendix S1. Some details on the history of the development of the function HD.

Appendix S2. Proof for Proposition 1.

Appendix S3. Proof for Proposition 2.

Appendix S4. Implementation.

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