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Generalized Simpson-diversity
e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 90–96
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Generalized Simpson-diversity Hans-Rolf Gregorius ∗ , Elizabeth M. Gillet ¨ Forstgenetik und Forstpflanzenzuchtung, ¨ ¨ Gottingen, ¨ ¨ ¨ Institut fur Universitat Busgenweg 2, 37077 Gottingen, Germany
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a b s t r a c t
Article history:
Simpson’s index of diversity equals the probability of drawing without replacement two
Received 16 February 2007
individuals of different type from a given collection. The interpretation of its inverse as the
Received in revised form
effective number of types reflects the biological meaning of diversity as the multiformity
11 July 2007
of the collection. The effective number of types is maximal, equalling the total number of
Accepted 22 August 2007
types, only if all types are uniformly distributed. Simpson’s diversity thus fulfills one of
Published on line 29 October 2007
the most basic conceptual criteria for diversity measures. The criterion does not, however, directly carry over to continuously varying differences between types. It is shown that the
Keywords:
common practice of taking averages or special sums of differences cannot serve as the
Simpson
desired generalization. In a new approach, the probabilistic interpretation of Simpson’s
Diversity
index is used to extend the basic criterion of diversity to arbitrary differences between
Continuous-trait
types, thereby retaining the concept of effective number. By considering as the resolution
Effective number
at which differences between individuals distinguish them, we define diversity as the
Resolution
probability D() of sampling without replacement two individuals that differ by more
Multiple-loci
than . It is pointed out that this measure generalizes the classical Simpson-criterion of
Beta-diversity
diversity in that it applies to any decomposition of the collection into groups, such that
Clustering
individuals within a group differ by at most and between groups by at least . Maximum
Wild cherry
diversity is reached if all groups are of equal size. This maximum property of the new measure motivates definition of the effective number of types at resolution independently of any particular decomposition into groups and while maintaining the basic diversity criterion. The advantages of and new insights to be derived from this scale-free measure are demonstrated by suggesting an approach to -diversity in ecology that is consistent with the criterion of diversity and by providing a worked example in population genetics using multiple-loci microsatellite data from three wild cherry populations. The example shows that gene associations may strongly affect the ranking of populations for genetic diversity. © 2007 Elsevier B.V. All rights reserved.
1.
Introduction
Among the dictionary definitions of diversity, the variant “multiformity” comes closest to its usage and significance in biology. This is mirrored in the fact that the measurement of diversity is intimately associated with numbers of different types. When types differ in frequency, however, a simple count of types may not be a satisfactory description of diversity. This led to the definition of “effective” numbers (see, e.g.
∗
Crow and Kimura, 1970, p. 324, for the field of population genetics), which are greater than or equal to one and less than or equal to the total number of types (i.e., those represented by at least one individual in the collection under consideration). Total and effective numbers coincide only if all types are equally frequent (i.e., uniformly distributed). This maximization property of effective numbers carries over to general diversity measures. It is one of the basic conceptual and most widely agreed upon criteria for general measures of diversity.
Corresponding author. E-mail addresses: [email protected] (H.-R. Gregorius), [email protected] (E.M. Gillet). 0304-3800/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2007.08.026
e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 90–96
As a consequence of this criterion, even bounded measures of diversity, such as the index of Simpson (1949) which varies between zero and one, can be associated to effective numbers of types via corresponding uniform distributions of types (see, e.g. MacArthur, 1965). Effective numbers thus more explicitly reflect the idea of multiformity that underlies diversity measures. The concept of effective number becomes vague, however, if variable differences between types are to be considered in the measurement of diversity in addition to the number and frequencies of types. Probably the most frequently cited and applied method that considers variable differences in the measurement of diversity is that of Rao (1982), who addressed the average difference between pairs of individuals within a ´ and Papp, 2000, for collection as a “diversity coefficient” (Izsak example, consider this coefficient as linking ecological diversity indices to measures of biodiversity). Although a particular version of Simpson’s index results from Rao’s method as a special case, the general setting of Rao’s method loses the connection to the basic criterion for diversity measures and thus to the concept of effective number. This becomes apparent when recalling that the statistical variance is another special case of Rao’s average difference (with difference defined by half the squared difference between real-valued measurements). For the variance, it is typically possible to obtain any desired value by appropriate choice of an arbitrary number of measurements (types), even just two. Variance is therefore a measure of “spread” but not of multiformity. Average difference and diversity are thus fundamentally different concepts, even though average differences may conform with the concept of diversity in special cases such as Simpson’s index. This discrepancy also shows up in an analysis of Pavoine et al. (2005), who demonstrate that for a given difference matrix and number of types, the average difference may assume its maximum for non-uniform frequency distributions in which not even all of the initially involved types are represented. More specifically, the authors point out that all types are represented (though not necessarily at equal frequencies) at maximum “diversity” if the difference measures are ultrametric. Ultrametricity is, however, very unlikely to be realized in nature, and even if it were, a uniform distribution of types would maximize Rao’s coefficient again only in very special situations. One of the exceptional cases in which ultrametric differences do exist is, of course, the binary difference between types (identical or different), for which Rao’s coefficient reduces to Simpson’s index for infinite population size. In any case, the findings of Pavoine et al. (2005) as well as the above statements on the distinction between average differences and diversity suggest that the classical criterion for measures of diversity requires reconsideration before it can be applied to the situation of variable differences between types. Instead of averaging differences, Weitzman (1992) proposed an iterative (inductive) technique of adding them. He considered it a “basic axiom . . . to impose on any diversity function” that “the addition of any species to a group of species should increase diversity by at least the dissimilarity of that species from its closest relative among the already existing group of species” (Weitzman, 1992, p. 376f). In this approach, frequencies of types (species) are not taken into consideration. Thus, it makes no difference if a newly added species
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is most similar to an existing species that is frequent (dominant) or that is rare in its community. This is at variance with the above-mentioned basic criterion concerning the effect of frequency distributions on diversity measures. In addition, since the same sum or average of differences can be obtained for any number of types, neither Weitzman’s nor Rao’s measures are sufficiently concerned with the concept of multiformity. Another problem that both types of measure share is that their unboundedness only allows for a relative assessment of diversity in the sense that one collection is more “diverse” than another. Absolute assessment would require an intrinsic upper bound for the measure, by which a collection can be said to be “highly diverse” without referring to other collections. An obvious bound is set to “diversity” by the size of the collection, which, on any scale or level of difference between types, allows no more types than the collection has members. In the following we will show that the original probabilistic interpretation of Simpson’s index can be used to consistently extend the classical criterion for diversity to the situation of arbitrarily variable differences between types. As a consequence, the concept of effective number can be retained. This is achieved by recalling that the distinctiveness of types can be viewed at different levels, depending on the desired resolution. At each such level, types are recognized as being “different” if they differ by more than the specified level. Our approach therefore proposes that diversity should be viewed at several levels of difference among individuals. The opportunity to freely choose the level of resolution also lays the basis for objective comparison of diversity between populations. To prevent possible misconception, this paper is concerned with the development of a diversity concept and its measure rather than with the development of estimators of population diversity based on samples. The statistical problem of designing appropriate estimators becomes relevant only after the measure to be estimated is accepted to be meaningful. Thus, whether the collection under consideration is a sample or a fully scored base population does not make a difference for applicability of the concept and its measure. However, if the sample diversity, for example, is used as an estimator of the diversity of the underlying base population, statistical questions as to the appropriateness of this estimator become essential. In any case, confidence intervals for the estimator can be determined via resampling methods.
2.
A new approach
Simpson’s diversity measure can be stated as the probability D of sampling from a given collection two individuals without replacement that differ in type. Under this perspective, individuals are recognized as being either identical or different in type (binary distinction), and the diversity criterion is determined by the number of pairs of individuals that differ in type. The larger the number of such pairs found in a collection of given size, the more diverse is the collection. This intuitively obvious criterion is called the Simpson-criterion in the following. It can be modeled in various ways, one of which rests on a random encounter scenario (see, e.g. Hurlbert, 1971). A more basic argument, however, refers to the fact that any concept of
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diversity invokes the notion of difference. The determination of difference requires consideration of two objects that differ, and, when applied to a collection of objects, it relies on counts of heterotypic pairs of objects. Hence, the Simpson-criterion generalizes the mere count of different types to a count of heterotypic pairs and by this considers the frequency distribution of the types. The relation between counting types and counting heterotypic pairs becomes apparent in that, for given collection size and number of types, the number of heterotypic pairs reaches its maximum only if all types are equally frequent. The Simpson-criterion of diversity becomes blurred, however, for poor resolution among types or, for good resolution, when type differences are highly variable. In these situations, where differences between types are recognized to vary quantitatively, the assessment of diversity depends on the level (or resolution) at which differences are considered to sufficiently distinguish individuals from each other. Once such a level is specified, the binary view of difference can be retained, and Simpson’s diversity criterion can be applied by considering the probability D() of sampling without replacement two individuals that differ by more than an amount . D() can be termed a measure of diversity at resolution . It will be referred to as a measure that reflects the generalized Simpson-criterion of diversity.
Denoting by N the population size, by ni the absolute fre quency of the ith type ( i ni = N), and by dij the difference between the ith and the jth type (dii = 0), then D() can be written as D() =
i,j;dij >
ni · nj N(N − 1)
=
N N−1
pi · pj
(1)
i,j;dij >
where pi :=ni /N is the corresponding relative frequency. This covers the special case of D() = 1 when all individuals differ in type, in which case ni = 1 and the index i runs from 1 to N. It also follows directly from the definition of D() that it is not affected by any rescaling of the differences if is replaced by its rescaled value. For complete resolution, where = 0, this measure equals the classical Simpson-index, i.e., D(0) = D = (N/(N − 1))(1 − 2 p ). Clearly, D() decreases with increasing (decreasing i i resolution), and D() = 0 if is greater than or equal to the maximum difference between two individuals (see Fig. 1). If all individuals differ in type by more than , i.e., if all individuals are distinguishable at resolution , then D() = 1 for all ≤ . In particular, this reflects the fact that diversity is bounded by collection size.
Fig. 1 – Relative (D()) and absolute (ne ()) diversity as a function of two measures of genic difference between genotypes: minimum genic and Jaccard difference. Diversity functions are determined for three stands (Roringen, Settmarshausen, Wibbecke) of wild cherry (Prunus avium L.) scored at six nuclear microsatellite loci. D():= the probability of sampling without replacement two individuals that differ by more than an amount (Simpson-diversity at resolution ); ne ():= effective number of types corresponding to D(); N:= population size. Classical measures of diversity not considering genic differences: Da := relative average single-locus allelic diversity, nea := effective number corresponding to Da , Dg := relative average single-locus genotypic diversity, neg := effective number corresponding to Dg .
e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 90–96
If the members of a collection differ substantially in size or function, the representation of types by numbers of individuals may not be meaningful. It may be replaced, for example, by the total biomass of individuals of a given type or by the area occupied by these individuals. In this case, instead of the relative frequency of individuals of a given type, the proportional representation via biomass or area of that type in the total collection will be used. The probability of drawing individuals of different type then becomes the probability to draw without replacement units of biomass or area that belong to individuals of different type. In this concept, units of biomass or area take the position of individuals, so that D() becomes the probability of drawing without replacement two units belonging to individuals that differ by more than the amount . Since these units can be chosen to be arbitrarily small, effects of finite numbers of individuals will vanish (N/(N − 1) ≈ 1), and, apart from , Eq. (1) is solely determined by the proportions by which the types are represented. As compared to conventional diversity measures, D() provides information that is conditional on freely selectable levels of resolution . By this, the degree of diversity is unambiguously defined by a scale-free quantity which can be compared for different collections on any suitable joint level of resolution. Conversely, two collections can be compared for the level of resolution at which they realize the same diversity. There may, however, be no such level. This would happen if, for two collections with diversity functions D1 () and D2 (), one obtains D1 () > D2 () for all values of with D2 () > 0. In this case, collection 1 is uniformly (unconditionally) more diverse than collection 2. Otherwise, there may exist ranges of resolution between which the diversity ranking of collections changes (see Fig. 1). At this point, it has not yet been clarified how this generalized Simpson-criterion could imply the classical criterion for diversity measures, according to which diversity is maximized if all types are equally frequent. Nevertheless, it is clear that for arbitrarily specified collection size and regime of type differences, maximum diversity is likely to be reached only for more complicated frequency distributions. This problem will be investigated in the following section.
3.
Grouping by quantitative differences
The idea to distinguish types only if they differ on a specified level of resolution tacitly assumes that types can be classified into groups, such that types are “indistinguishable” when they belong to the same group and distinguishable only when they belong to different groups. In some sense, these groups correspond to a new typification that reflects the resolution level. More explicitly, this requires that for a given level , members of the same group never differ by more than , while members of different groups always differ by more than . If it were possible to decompose the collection into such groups, D() would exactly equal the probability of sampling without replacement two individuals of types that belong to different groups. Groups with this property result naturally from ultrametric differences, since these metrics establish equivalence relations. Such an equivalence relation is defined for any by
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considering two individuals to be related if they differ by at most the amount . Within each of the resulting equivalence classes, individuals differ by at most , and they differ by more than if they belong to different equivalence classes. Hence, these equivalence classes establish a decomposition of the collection into groups of types with the desired property. This decomposition at level can be directly utilized to calculate D() from Eq. (1). To see this, denote the k th group by Ck () and the proportion of individuals with types in Ck () by p˜ k (). Consider the set {(i, j)|dij > } of pairs of types over which the summation in Eq. (1) extends. Then this set can be identically obtained by collecting all pairs of types from different Ck ()’s. 2 Hence, p · pj = p˜ () · p˜ l () = 1 − p˜ () . It foli,j;d > i k=l k k k ij
lows immediately that maximum diversity is attained for given differences and level if each group Ck () is represented by the same number (proportion) of individuals. If there are n groups, this implies p˜ k () = 1/n for k = 1, . . . , n and thus D() = (1 − 1/n) · N/(N − 1). This consistently extends the well known property of the classical Simpson-index. However, each group may consist of several types, so that the sum of their frequencies is determined only by the group frequency. Thus, on the level of type frequencies, a continuum of distributions maximizes diversity. Unfortunately, real measures of difference rarely fulfill the requirements of an ultrametric. This need not prevent us from applying any reasonable method of clustering to the pertinent difference matrix and selecting from the resulting hierarchy of groups a decomposition of the collection by “cutting stems” in the associated dendrogram at level . Without proper reasoning, however, this is an arbitrary way of proceeding, since it may lead to different numbers of groups depending on the applied method of clustering. On the other hand, the number of groups of types that are “effectively” involved in the determination of the value of D() is implicit in the expression (1 − 1/n) · N/(N − 1), which represents the diversity for the ideal situation, where each of n types is equally frequent in a collection of size N. This line of thought will be detailed in the next section. Irrespective of whether or not D() is conceived of as resulting from a decomposition of the collection, a particular level of resolution may turn out to be more distinctive than other levels, in that D() is less sensitive to changes in . In particular, if D() is constant over an extended interval of -values, this is tantamount to the absence of pairs of individuals with differences in that interval. The interval thus marks a region of distinct difference among individuals, and the length of the interval extending around a particular level of resolution can be viewed as its degree of distinctiveness. Biological interpretations of distinctive levels of resolution for the measurement of diversity are manifold, ranging from numbers of well-separated subpopulations or species to levels of biological organization. Levels of organization correspond to the levels of resolution over which diversity values remain constant for awhile, following a strong decline and preceding the next strong decline. This behavior appears as a distinct and broad step in the D() function. Several such steps indicate the existence of a distinctly hierarchical organization, the diversity of which of course decreases as one moves up in the hierarchy.
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4.
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5. Diversity versus differentiation among collections: -diversity
The effective number of types
According to MacArthur (1965), an effective number of types can be extracted from measures of diversity by equating the observed value of the measure to the value that would result for a uniform (even) distribution and then solving the equation for the number of types involved in the distribution. This accords with the general concept of effective number, in which “ideal populations” are compared with “real populations” for a characteristic variable (Gregorius, 1991). In the present context of generalized Simpson-diversity, the characteristic variable for which real and ideal populations are to be compared is D(). To maintain the connection to the original definition of an effective number of types, the “ideal population” must refer to a situation where all types are equally frequent, and where two individuals either have the same type or always differ by the same amount ˆ (binary differences), where the value of ˆ is irrelevant. In such an ideal population of size N and with n equally frequent types, D() = (1 − 1/n) · N/(N − 1) for 0 ≤ < , ˆ as mentioned above. Setting this expression equal to some D() measured in a real population and solving for n yields the effective number of types at resolution as ne () =
1 . 1 − D() · ((N − 1)/N)
(2)
The major difference between the diversity measures D() and the corresponding effective numbers consists in the fact that diversity is measured by the former in relative terms and by the latter in absolute terms. Collections of different size may have the same effective number of types, so that in small collections this number may almost equal the collection size and will thus show up as a high measure of relative diversity. For constant effective number, relative diversity drops with increasing collection size, though this tendency quickly levels out on the order of N/(N − 1). On the other hand, if for conservation purposes it is required that each type be repeated in a sufficient number of copies in order to prevent loss by drift, the quantity of interest is N/ne (). By Eq. (2) this equals N(1 − D()) + D(). In this situation, an appropriate balance between collection size N and relative diversity D() is essential. As was done in the previous section with D(), ne () can again be considered as a function of the resolution . The one-to-one correspondence between the two functions, as seen in Eq. (2), allows us to adopt all of the earlier statements on maximum diversity, distinctiveness of resolution levels, and associated biological interpretations for ne (). In fact, some of these statements gain intuitive power when expressed in terms of absolute rather than relative diversity. Another example is provided in a quantitative genetic context, where differences between genotypes can be associated with differences in effects on selection. In order to be discernible by selective forces, two genotypes might be required to differ by more than a specified proportion of their genes. The effective genetic potential for selection would then be specified by the effective number ne () of genotypes discernible (resolvable) at level .
Even though -diversity seems to be an established concept in ecology, it repeatedly enjoys lively discussion (for more recent examples see the paper of Vellend (2001) or Ricotta (2006), and the papers cited therein in response to earlier comments). In essence, the concept is conceived of as the gain in diversity that results by adding collections (principle of complementarity). The gain is measured in various ways by considering the difference between the average diversity within collections (␣-diversity) and the diversity of the totality of collections (␥diversity). Thus, -diversity is not an independent measure that quantifies the diversity of any specific type of collection. Strictly speaking, usage of the term “diversity” is not even appropriate in this context. This becomes particularly evident from the ambivalent usage of related terminology in population genetics. There, various normalizations of the difference between the average variance of genetic traits within populations and the total variance of such traits (usually indexed by ST such as in FST , GST , RST , ˚ST , QST , etc.) are interchangeably addressed as diversity or differentiation among populations. Obviously, the general approach is the same as with the measurement of -diversity in ecology. According to the intention to quantify genetic differences among populations, however, the correct term should be “differentiation”. Yet, as was pointed out repeatedly (for a more recent reference see Gregorius et al., 2003), the ST-type measures do not satisfy basic requirements for differentiation measures. The term differentiation is in fact well-defined, and appropriate measures are available for discrete and continuous traits (see Gregorius et al. (2003)). Though introduced in a chiefly population genetic context, these measures are equally applicable to the measurement of differences (differentiation) between arbitrary collections of objects. Among these are differences in species composition among sites or communities, when considering both frequencies and differences in species-specific traits. The principle of complementarity and thus “-diversity” could be captured more consistently by such measures of differentiation. It remains to give meaning to the notion of -diversity as a proper index of diversity. Since diversity addresses the multiformity of a collection of objects, this requires us to consider objects that by themselves are specified as collections (or populations) as well as measures of difference between such objects. Such measures belong to the above-mentioned category of measures of differentiation between collections. An appropriate index of diversity must therefore quantify secondorder diversity in the sense that it quantifies the diversity of a collection of collections. This kind of second-order diversity is distinguished by the fact that in most cases the individual objects (collections) are not classified into types. Each object may therefore represent a unique “type”, so that second-order diversity is solely determined by differences between types but not by type frequencies. Consequently, the measurement of second-order diversity can again be based on the generalized Simpson-criterion with its implied specification of effective numbers. The dij ’s in Eq.
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(1) now refer to the difference between the ith and jth collection, and D() denotes the probability of drawing without replacement two collections that differ by at least an amount . Analogously, ne (), as given by Eq. (2), specifies the effective number of collections at resolution (depending on context, the wording “number of collections effectively differing at resolution ” might be preferable). Since the level of resolution refers to differences between collections, one might expect that second-order diversities are closely associated with the differentiation among collections. That this is generally not true follows from the simple fact that any specific degree of differentiation among populations can just as well be realized by two collections as by any higher number. Hence, secondorder diversity and differentiation are largely independent notions.
6.
A population genetic application
In almost all population genetic studies, genetic diversity is measured as the allelic diversity at individual gene loci. Multiple loci are considered by taking averages of allelic diversities over the loci studied. In terms of Simpson’s index D (= D(0)), allelic diversity at diploid loci refers to 2N individual genes, so that the population size N must be replaced by 2N in D. Analogously, genotypic diversity is obtained by averaging single-locus diversities over loci (where the population size is N). The notations Da and Dg will be used to distinguish average (relative) allelic from genotypic diversity. The corresponding effective numbers nea and neg are obtained by replacing D() in Eq. (2) by Da and Dg , respectively. The practice of averaging over loci of course ignores the associations of alleles both within and between loci in the individual genotypes and can therefore be expected to give only a limited impression of the genic diversity residing in the members of the population. Hence, to conclude the present paper, a brief numerical demonstration will be given of the effects of including multiple loci in the measurement of genic diversity. This requires specification of genic differences between population members for the gene loci under consideration. Genic differences between two individuals can be determined in various ways, most of which consider differences in either number of allele copies or number of allelic states. Irrespective of the degree of ploidy, the difference between genotypes in the copy number of each allele results from subtraction of its number in one genotype from that number in the other genotype, and from summation of the absolute values of the differences for all alleles over all loci. Since the difference in copies per locus equals at most the degree of ploidy, this sum is divided by the product of the degree of ploidy times the number of loci in order to obtain a measure that is normalized to 1. If, for example, haplotypes are studied (such as in gametes, extranuclear cell organelles, or endosperms in conifers), the measure reduces to the proportion of loci for which two haplotypes differ in allelic state. This measure will be called the minimum genic difference in the following. Analogously, the difference between two genotypes in number of allelic states is determined for each gene locus by
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the number of alleles present in one but not in the other genotype plus that number present in the other but not in the first genotype. These numbers are summed over all loci, and the sum is divided by the total number of allelic types present at all loci in both genotypes together. The result is akin to what is commonly known as Jaccard’s index. Note that this definition again applies equally to all degrees of ploidy. Both measures address numbers or proportions of genes by which two individuals or genotypes differ, but they do so to different extents. This is demonstrated by the representations in Fig. 1, which are based on data for six nuclear microsatellite loci obtained in three stands of wild cherry (data from the dis¨ sertation of Holtken, 2005). Generally, Jaccard differences lead to larger relative and absolute diversities than minimum genic differences (of course with the exception of = 0). This goes along with distinctly more steps for the Jaccard differences, which is a consequence of the fact that the normalization of genic differences varies strongly with genotypes. Although for high resolution (small ) relative diversities (upper graphs in Fig. 1) are large and very similar for all three populations, they differ distinctly for their corresponding absolute diversities, i.e., effective numbers (lower graphs). Whereas at intermediate resolutions (0.40 < < 0.75 for minimum genic differences, 0.55 < < 0.85 for Jaccard differences) the three populations are most strongly distinguished for their relative diversities, their absolute diversities already converge after a strong decline. This observation confirms the above remarks that relative and absolute measures of diversity lead to different statements on diversity. In particular it demonstrates that the three populations display their differences in genetic diversity most clearly for their effective numbers of genotypes at low resolution. For relative diversities their differences become more evident for intermediate resolutions. Concerning the diversity ranking of the three populations, it turns out that Settmarshausen is uniformly more diverse than Wibbecke for both the minimum genic and the Jaccard differences. In contrast, Roringen changes diversity ranking with respect to both of the other two populations, starting with the largest diversity at high resolution and falling below Settmarshausen and Wibbecke in succession with decreasing resolution. Hence, in comparison to the other populations, the assessment of the genetic diversity of Roringen depends on the chosen resolution. In fact, genotypes that are distinguished from other genotypes by many genes can be expected to normally occur at low frequencies. To increase the frequency of such genotypes and to thus increase D() as well as ne () for larger values of , systems of reproduction are required that increase genic associations. Among such systems is clonal propagation, which is known to occur regularly in wild cherry by root suckering. Interestingly, it was shown ¨ in the work of Holtken (2005) that the degree of cloning was lowest in Roringen, distinctly larger in Settmarshausen, and largest in Wibbecke. Thus, the presumably smaller genic associations in Roringen would explain its high genotypic diversity values for high resolution and its comparatively strong decline with decreasing resolution. These findings call for comparison with the results of a classical assessment of genetic diversity of the three populations. As is seen in table of Fig. 1, even for complete
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resolution ( = 0) the ranking of the populations differs from that obtained for both relative average allelic (Da ) and average genotypic (Dg ) diversity (the same holds for the corresponding effective numbers nea and neg ). While the ranking of Settmarshausen and Wibbecke is maintained, Roringen shows the lowest average allelic diversity. However, it ranks well between Settmarshausen and Wibbecke for its average genotypic diversity. Consequently, there is little correspondence between the diversity rankings obtained for multiple-locus diversity functions and average single-locus diversities. A straightforward explanation is suggested by the negligence of gene associations in classical diversity analyses. Finally, there is no indication of distinct steps in the diversity functions such as would be expected for hierarchical organization of genetic variation. As was explained above (last paragraph of the section on grouping for quantitative differences), the existence of genetically differentiated subpopulations (or demes) should be recognizable in this way at lower levels of resolution, since genetic differences within subpopulations should be distinctly smaller than between subpopulations. The population substructure due to clonal reproduction does not show up in the diversity functions, since it is limited to = 0. In fact, the small population sizes as well as the supposed selective neutrality of the microsatellite gene markers give little reason to expect hierarchical structure. Software for computing and drawing diversity functions D() and choropleths for illustration of difference matrices is available from the co-author.
Acknowledgements The authors are thankful for the suggestions of anonymous reviewers that helped to improve several of our formulations.
references
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Generalized Simpson-diversity Hans-Rolf Gregorius ∗ , Elizabeth M. Gillet ¨ Forstgenetik und Forstpflanzenzuchtung, ¨ ¨ Gottingen, ¨ ¨ ¨ Institut fur Universitat Busgenweg 2, 37077 Gottingen, Germany
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a b s t r a c t
Article history:
Simpson’s index of diversity equals the probability of drawing without replacement two
Received 16 February 2007
individuals of different type from a given collection. The interpretation of its inverse as the
Received in revised form
effective number of types reflects the biological meaning of diversity as the multiformity
11 July 2007
of the collection. The effective number of types is maximal, equalling the total number of
Accepted 22 August 2007
types, only if all types are uniformly distributed. Simpson’s diversity thus fulfills one of
Published on line 29 October 2007
the most basic conceptual criteria for diversity measures. The criterion does not, however, directly carry over to continuously varying differences between types. It is shown that the
Keywords:
common practice of taking averages or special sums of differences cannot serve as the
Simpson
desired generalization. In a new approach, the probabilistic interpretation of Simpson’s
Diversity
index is used to extend the basic criterion of diversity to arbitrary differences between
Continuous-trait
types, thereby retaining the concept of effective number. By considering as the resolution
Effective number
at which differences between individuals distinguish them, we define diversity as the
Resolution
probability D() of sampling without replacement two individuals that differ by more
Multiple-loci
than . It is pointed out that this measure generalizes the classical Simpson-criterion of
Beta-diversity
diversity in that it applies to any decomposition of the collection into groups, such that
Clustering
individuals within a group differ by at most and between groups by at least . Maximum
Wild cherry
diversity is reached if all groups are of equal size. This maximum property of the new measure motivates definition of the effective number of types at resolution independently of any particular decomposition into groups and while maintaining the basic diversity criterion. The advantages of and new insights to be derived from this scale-free measure are demonstrated by suggesting an approach to -diversity in ecology that is consistent with the criterion of diversity and by providing a worked example in population genetics using multiple-loci microsatellite data from three wild cherry populations. The example shows that gene associations may strongly affect the ranking of populations for genetic diversity. © 2007 Elsevier B.V. All rights reserved.
1.
Introduction
Among the dictionary definitions of diversity, the variant “multiformity” comes closest to its usage and significance in biology. This is mirrored in the fact that the measurement of diversity is intimately associated with numbers of different types. When types differ in frequency, however, a simple count of types may not be a satisfactory description of diversity. This led to the definition of “effective” numbers (see, e.g.
∗
Crow and Kimura, 1970, p. 324, for the field of population genetics), which are greater than or equal to one and less than or equal to the total number of types (i.e., those represented by at least one individual in the collection under consideration). Total and effective numbers coincide only if all types are equally frequent (i.e., uniformly distributed). This maximization property of effective numbers carries over to general diversity measures. It is one of the basic conceptual and most widely agreed upon criteria for general measures of diversity.
Corresponding author. E-mail addresses: [email protected] (H.-R. Gregorius), [email protected] (E.M. Gillet). 0304-3800/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2007.08.026
e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 90–96
As a consequence of this criterion, even bounded measures of diversity, such as the index of Simpson (1949) which varies between zero and one, can be associated to effective numbers of types via corresponding uniform distributions of types (see, e.g. MacArthur, 1965). Effective numbers thus more explicitly reflect the idea of multiformity that underlies diversity measures. The concept of effective number becomes vague, however, if variable differences between types are to be considered in the measurement of diversity in addition to the number and frequencies of types. Probably the most frequently cited and applied method that considers variable differences in the measurement of diversity is that of Rao (1982), who addressed the average difference between pairs of individuals within a ´ and Papp, 2000, for collection as a “diversity coefficient” (Izsak example, consider this coefficient as linking ecological diversity indices to measures of biodiversity). Although a particular version of Simpson’s index results from Rao’s method as a special case, the general setting of Rao’s method loses the connection to the basic criterion for diversity measures and thus to the concept of effective number. This becomes apparent when recalling that the statistical variance is another special case of Rao’s average difference (with difference defined by half the squared difference between real-valued measurements). For the variance, it is typically possible to obtain any desired value by appropriate choice of an arbitrary number of measurements (types), even just two. Variance is therefore a measure of “spread” but not of multiformity. Average difference and diversity are thus fundamentally different concepts, even though average differences may conform with the concept of diversity in special cases such as Simpson’s index. This discrepancy also shows up in an analysis of Pavoine et al. (2005), who demonstrate that for a given difference matrix and number of types, the average difference may assume its maximum for non-uniform frequency distributions in which not even all of the initially involved types are represented. More specifically, the authors point out that all types are represented (though not necessarily at equal frequencies) at maximum “diversity” if the difference measures are ultrametric. Ultrametricity is, however, very unlikely to be realized in nature, and even if it were, a uniform distribution of types would maximize Rao’s coefficient again only in very special situations. One of the exceptional cases in which ultrametric differences do exist is, of course, the binary difference between types (identical or different), for which Rao’s coefficient reduces to Simpson’s index for infinite population size. In any case, the findings of Pavoine et al. (2005) as well as the above statements on the distinction between average differences and diversity suggest that the classical criterion for measures of diversity requires reconsideration before it can be applied to the situation of variable differences between types. Instead of averaging differences, Weitzman (1992) proposed an iterative (inductive) technique of adding them. He considered it a “basic axiom . . . to impose on any diversity function” that “the addition of any species to a group of species should increase diversity by at least the dissimilarity of that species from its closest relative among the already existing group of species” (Weitzman, 1992, p. 376f). In this approach, frequencies of types (species) are not taken into consideration. Thus, it makes no difference if a newly added species
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is most similar to an existing species that is frequent (dominant) or that is rare in its community. This is at variance with the above-mentioned basic criterion concerning the effect of frequency distributions on diversity measures. In addition, since the same sum or average of differences can be obtained for any number of types, neither Weitzman’s nor Rao’s measures are sufficiently concerned with the concept of multiformity. Another problem that both types of measure share is that their unboundedness only allows for a relative assessment of diversity in the sense that one collection is more “diverse” than another. Absolute assessment would require an intrinsic upper bound for the measure, by which a collection can be said to be “highly diverse” without referring to other collections. An obvious bound is set to “diversity” by the size of the collection, which, on any scale or level of difference between types, allows no more types than the collection has members. In the following we will show that the original probabilistic interpretation of Simpson’s index can be used to consistently extend the classical criterion for diversity to the situation of arbitrarily variable differences between types. As a consequence, the concept of effective number can be retained. This is achieved by recalling that the distinctiveness of types can be viewed at different levels, depending on the desired resolution. At each such level, types are recognized as being “different” if they differ by more than the specified level. Our approach therefore proposes that diversity should be viewed at several levels of difference among individuals. The opportunity to freely choose the level of resolution also lays the basis for objective comparison of diversity between populations. To prevent possible misconception, this paper is concerned with the development of a diversity concept and its measure rather than with the development of estimators of population diversity based on samples. The statistical problem of designing appropriate estimators becomes relevant only after the measure to be estimated is accepted to be meaningful. Thus, whether the collection under consideration is a sample or a fully scored base population does not make a difference for applicability of the concept and its measure. However, if the sample diversity, for example, is used as an estimator of the diversity of the underlying base population, statistical questions as to the appropriateness of this estimator become essential. In any case, confidence intervals for the estimator can be determined via resampling methods.
2.
A new approach
Simpson’s diversity measure can be stated as the probability D of sampling from a given collection two individuals without replacement that differ in type. Under this perspective, individuals are recognized as being either identical or different in type (binary distinction), and the diversity criterion is determined by the number of pairs of individuals that differ in type. The larger the number of such pairs found in a collection of given size, the more diverse is the collection. This intuitively obvious criterion is called the Simpson-criterion in the following. It can be modeled in various ways, one of which rests on a random encounter scenario (see, e.g. Hurlbert, 1971). A more basic argument, however, refers to the fact that any concept of
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diversity invokes the notion of difference. The determination of difference requires consideration of two objects that differ, and, when applied to a collection of objects, it relies on counts of heterotypic pairs of objects. Hence, the Simpson-criterion generalizes the mere count of different types to a count of heterotypic pairs and by this considers the frequency distribution of the types. The relation between counting types and counting heterotypic pairs becomes apparent in that, for given collection size and number of types, the number of heterotypic pairs reaches its maximum only if all types are equally frequent. The Simpson-criterion of diversity becomes blurred, however, for poor resolution among types or, for good resolution, when type differences are highly variable. In these situations, where differences between types are recognized to vary quantitatively, the assessment of diversity depends on the level (or resolution) at which differences are considered to sufficiently distinguish individuals from each other. Once such a level is specified, the binary view of difference can be retained, and Simpson’s diversity criterion can be applied by considering the probability D() of sampling without replacement two individuals that differ by more than an amount . D() can be termed a measure of diversity at resolution . It will be referred to as a measure that reflects the generalized Simpson-criterion of diversity.
Denoting by N the population size, by ni the absolute fre quency of the ith type ( i ni = N), and by dij the difference between the ith and the jth type (dii = 0), then D() can be written as D() =
i,j;dij >
ni · nj N(N − 1)
=
N N−1
pi · pj
(1)
i,j;dij >
where pi :=ni /N is the corresponding relative frequency. This covers the special case of D() = 1 when all individuals differ in type, in which case ni = 1 and the index i runs from 1 to N. It also follows directly from the definition of D() that it is not affected by any rescaling of the differences if is replaced by its rescaled value. For complete resolution, where = 0, this measure equals the classical Simpson-index, i.e., D(0) = D = (N/(N − 1))(1 − 2 p ). Clearly, D() decreases with increasing (decreasing i i resolution), and D() = 0 if is greater than or equal to the maximum difference between two individuals (see Fig. 1). If all individuals differ in type by more than , i.e., if all individuals are distinguishable at resolution , then D() = 1 for all ≤ . In particular, this reflects the fact that diversity is bounded by collection size.
Fig. 1 – Relative (D()) and absolute (ne ()) diversity as a function of two measures of genic difference between genotypes: minimum genic and Jaccard difference. Diversity functions are determined for three stands (Roringen, Settmarshausen, Wibbecke) of wild cherry (Prunus avium L.) scored at six nuclear microsatellite loci. D():= the probability of sampling without replacement two individuals that differ by more than an amount (Simpson-diversity at resolution ); ne ():= effective number of types corresponding to D(); N:= population size. Classical measures of diversity not considering genic differences: Da := relative average single-locus allelic diversity, nea := effective number corresponding to Da , Dg := relative average single-locus genotypic diversity, neg := effective number corresponding to Dg .
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If the members of a collection differ substantially in size or function, the representation of types by numbers of individuals may not be meaningful. It may be replaced, for example, by the total biomass of individuals of a given type or by the area occupied by these individuals. In this case, instead of the relative frequency of individuals of a given type, the proportional representation via biomass or area of that type in the total collection will be used. The probability of drawing individuals of different type then becomes the probability to draw without replacement units of biomass or area that belong to individuals of different type. In this concept, units of biomass or area take the position of individuals, so that D() becomes the probability of drawing without replacement two units belonging to individuals that differ by more than the amount . Since these units can be chosen to be arbitrarily small, effects of finite numbers of individuals will vanish (N/(N − 1) ≈ 1), and, apart from , Eq. (1) is solely determined by the proportions by which the types are represented. As compared to conventional diversity measures, D() provides information that is conditional on freely selectable levels of resolution . By this, the degree of diversity is unambiguously defined by a scale-free quantity which can be compared for different collections on any suitable joint level of resolution. Conversely, two collections can be compared for the level of resolution at which they realize the same diversity. There may, however, be no such level. This would happen if, for two collections with diversity functions D1 () and D2 (), one obtains D1 () > D2 () for all values of with D2 () > 0. In this case, collection 1 is uniformly (unconditionally) more diverse than collection 2. Otherwise, there may exist ranges of resolution between which the diversity ranking of collections changes (see Fig. 1). At this point, it has not yet been clarified how this generalized Simpson-criterion could imply the classical criterion for diversity measures, according to which diversity is maximized if all types are equally frequent. Nevertheless, it is clear that for arbitrarily specified collection size and regime of type differences, maximum diversity is likely to be reached only for more complicated frequency distributions. This problem will be investigated in the following section.
3.
Grouping by quantitative differences
The idea to distinguish types only if they differ on a specified level of resolution tacitly assumes that types can be classified into groups, such that types are “indistinguishable” when they belong to the same group and distinguishable only when they belong to different groups. In some sense, these groups correspond to a new typification that reflects the resolution level. More explicitly, this requires that for a given level , members of the same group never differ by more than , while members of different groups always differ by more than . If it were possible to decompose the collection into such groups, D() would exactly equal the probability of sampling without replacement two individuals of types that belong to different groups. Groups with this property result naturally from ultrametric differences, since these metrics establish equivalence relations. Such an equivalence relation is defined for any by
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considering two individuals to be related if they differ by at most the amount . Within each of the resulting equivalence classes, individuals differ by at most , and they differ by more than if they belong to different equivalence classes. Hence, these equivalence classes establish a decomposition of the collection into groups of types with the desired property. This decomposition at level can be directly utilized to calculate D() from Eq. (1). To see this, denote the k th group by Ck () and the proportion of individuals with types in Ck () by p˜ k (). Consider the set {(i, j)|dij > } of pairs of types over which the summation in Eq. (1) extends. Then this set can be identically obtained by collecting all pairs of types from different Ck ()’s. 2 Hence, p · pj = p˜ () · p˜ l () = 1 − p˜ () . It foli,j;d > i k=l k k k ij
lows immediately that maximum diversity is attained for given differences and level if each group Ck () is represented by the same number (proportion) of individuals. If there are n groups, this implies p˜ k () = 1/n for k = 1, . . . , n and thus D() = (1 − 1/n) · N/(N − 1). This consistently extends the well known property of the classical Simpson-index. However, each group may consist of several types, so that the sum of their frequencies is determined only by the group frequency. Thus, on the level of type frequencies, a continuum of distributions maximizes diversity. Unfortunately, real measures of difference rarely fulfill the requirements of an ultrametric. This need not prevent us from applying any reasonable method of clustering to the pertinent difference matrix and selecting from the resulting hierarchy of groups a decomposition of the collection by “cutting stems” in the associated dendrogram at level . Without proper reasoning, however, this is an arbitrary way of proceeding, since it may lead to different numbers of groups depending on the applied method of clustering. On the other hand, the number of groups of types that are “effectively” involved in the determination of the value of D() is implicit in the expression (1 − 1/n) · N/(N − 1), which represents the diversity for the ideal situation, where each of n types is equally frequent in a collection of size N. This line of thought will be detailed in the next section. Irrespective of whether or not D() is conceived of as resulting from a decomposition of the collection, a particular level of resolution may turn out to be more distinctive than other levels, in that D() is less sensitive to changes in . In particular, if D() is constant over an extended interval of -values, this is tantamount to the absence of pairs of individuals with differences in that interval. The interval thus marks a region of distinct difference among individuals, and the length of the interval extending around a particular level of resolution can be viewed as its degree of distinctiveness. Biological interpretations of distinctive levels of resolution for the measurement of diversity are manifold, ranging from numbers of well-separated subpopulations or species to levels of biological organization. Levels of organization correspond to the levels of resolution over which diversity values remain constant for awhile, following a strong decline and preceding the next strong decline. This behavior appears as a distinct and broad step in the D() function. Several such steps indicate the existence of a distinctly hierarchical organization, the diversity of which of course decreases as one moves up in the hierarchy.
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4.
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5. Diversity versus differentiation among collections: -diversity
The effective number of types
According to MacArthur (1965), an effective number of types can be extracted from measures of diversity by equating the observed value of the measure to the value that would result for a uniform (even) distribution and then solving the equation for the number of types involved in the distribution. This accords with the general concept of effective number, in which “ideal populations” are compared with “real populations” for a characteristic variable (Gregorius, 1991). In the present context of generalized Simpson-diversity, the characteristic variable for which real and ideal populations are to be compared is D(). To maintain the connection to the original definition of an effective number of types, the “ideal population” must refer to a situation where all types are equally frequent, and where two individuals either have the same type or always differ by the same amount ˆ (binary differences), where the value of ˆ is irrelevant. In such an ideal population of size N and with n equally frequent types, D() = (1 − 1/n) · N/(N − 1) for 0 ≤ < , ˆ as mentioned above. Setting this expression equal to some D() measured in a real population and solving for n yields the effective number of types at resolution as ne () =
1 . 1 − D() · ((N − 1)/N)
(2)
The major difference between the diversity measures D() and the corresponding effective numbers consists in the fact that diversity is measured by the former in relative terms and by the latter in absolute terms. Collections of different size may have the same effective number of types, so that in small collections this number may almost equal the collection size and will thus show up as a high measure of relative diversity. For constant effective number, relative diversity drops with increasing collection size, though this tendency quickly levels out on the order of N/(N − 1). On the other hand, if for conservation purposes it is required that each type be repeated in a sufficient number of copies in order to prevent loss by drift, the quantity of interest is N/ne (). By Eq. (2) this equals N(1 − D()) + D(). In this situation, an appropriate balance between collection size N and relative diversity D() is essential. As was done in the previous section with D(), ne () can again be considered as a function of the resolution . The one-to-one correspondence between the two functions, as seen in Eq. (2), allows us to adopt all of the earlier statements on maximum diversity, distinctiveness of resolution levels, and associated biological interpretations for ne (). In fact, some of these statements gain intuitive power when expressed in terms of absolute rather than relative diversity. Another example is provided in a quantitative genetic context, where differences between genotypes can be associated with differences in effects on selection. In order to be discernible by selective forces, two genotypes might be required to differ by more than a specified proportion of their genes. The effective genetic potential for selection would then be specified by the effective number ne () of genotypes discernible (resolvable) at level .
Even though -diversity seems to be an established concept in ecology, it repeatedly enjoys lively discussion (for more recent examples see the paper of Vellend (2001) or Ricotta (2006), and the papers cited therein in response to earlier comments). In essence, the concept is conceived of as the gain in diversity that results by adding collections (principle of complementarity). The gain is measured in various ways by considering the difference between the average diversity within collections (␣-diversity) and the diversity of the totality of collections (␥diversity). Thus, -diversity is not an independent measure that quantifies the diversity of any specific type of collection. Strictly speaking, usage of the term “diversity” is not even appropriate in this context. This becomes particularly evident from the ambivalent usage of related terminology in population genetics. There, various normalizations of the difference between the average variance of genetic traits within populations and the total variance of such traits (usually indexed by ST such as in FST , GST , RST , ˚ST , QST , etc.) are interchangeably addressed as diversity or differentiation among populations. Obviously, the general approach is the same as with the measurement of -diversity in ecology. According to the intention to quantify genetic differences among populations, however, the correct term should be “differentiation”. Yet, as was pointed out repeatedly (for a more recent reference see Gregorius et al., 2003), the ST-type measures do not satisfy basic requirements for differentiation measures. The term differentiation is in fact well-defined, and appropriate measures are available for discrete and continuous traits (see Gregorius et al. (2003)). Though introduced in a chiefly population genetic context, these measures are equally applicable to the measurement of differences (differentiation) between arbitrary collections of objects. Among these are differences in species composition among sites or communities, when considering both frequencies and differences in species-specific traits. The principle of complementarity and thus “-diversity” could be captured more consistently by such measures of differentiation. It remains to give meaning to the notion of -diversity as a proper index of diversity. Since diversity addresses the multiformity of a collection of objects, this requires us to consider objects that by themselves are specified as collections (or populations) as well as measures of difference between such objects. Such measures belong to the above-mentioned category of measures of differentiation between collections. An appropriate index of diversity must therefore quantify secondorder diversity in the sense that it quantifies the diversity of a collection of collections. This kind of second-order diversity is distinguished by the fact that in most cases the individual objects (collections) are not classified into types. Each object may therefore represent a unique “type”, so that second-order diversity is solely determined by differences between types but not by type frequencies. Consequently, the measurement of second-order diversity can again be based on the generalized Simpson-criterion with its implied specification of effective numbers. The dij ’s in Eq.
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(1) now refer to the difference between the ith and jth collection, and D() denotes the probability of drawing without replacement two collections that differ by at least an amount . Analogously, ne (), as given by Eq. (2), specifies the effective number of collections at resolution (depending on context, the wording “number of collections effectively differing at resolution ” might be preferable). Since the level of resolution refers to differences between collections, one might expect that second-order diversities are closely associated with the differentiation among collections. That this is generally not true follows from the simple fact that any specific degree of differentiation among populations can just as well be realized by two collections as by any higher number. Hence, secondorder diversity and differentiation are largely independent notions.
6.
A population genetic application
In almost all population genetic studies, genetic diversity is measured as the allelic diversity at individual gene loci. Multiple loci are considered by taking averages of allelic diversities over the loci studied. In terms of Simpson’s index D (= D(0)), allelic diversity at diploid loci refers to 2N individual genes, so that the population size N must be replaced by 2N in D. Analogously, genotypic diversity is obtained by averaging single-locus diversities over loci (where the population size is N). The notations Da and Dg will be used to distinguish average (relative) allelic from genotypic diversity. The corresponding effective numbers nea and neg are obtained by replacing D() in Eq. (2) by Da and Dg , respectively. The practice of averaging over loci of course ignores the associations of alleles both within and between loci in the individual genotypes and can therefore be expected to give only a limited impression of the genic diversity residing in the members of the population. Hence, to conclude the present paper, a brief numerical demonstration will be given of the effects of including multiple loci in the measurement of genic diversity. This requires specification of genic differences between population members for the gene loci under consideration. Genic differences between two individuals can be determined in various ways, most of which consider differences in either number of allele copies or number of allelic states. Irrespective of the degree of ploidy, the difference between genotypes in the copy number of each allele results from subtraction of its number in one genotype from that number in the other genotype, and from summation of the absolute values of the differences for all alleles over all loci. Since the difference in copies per locus equals at most the degree of ploidy, this sum is divided by the product of the degree of ploidy times the number of loci in order to obtain a measure that is normalized to 1. If, for example, haplotypes are studied (such as in gametes, extranuclear cell organelles, or endosperms in conifers), the measure reduces to the proportion of loci for which two haplotypes differ in allelic state. This measure will be called the minimum genic difference in the following. Analogously, the difference between two genotypes in number of allelic states is determined for each gene locus by
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the number of alleles present in one but not in the other genotype plus that number present in the other but not in the first genotype. These numbers are summed over all loci, and the sum is divided by the total number of allelic types present at all loci in both genotypes together. The result is akin to what is commonly known as Jaccard’s index. Note that this definition again applies equally to all degrees of ploidy. Both measures address numbers or proportions of genes by which two individuals or genotypes differ, but they do so to different extents. This is demonstrated by the representations in Fig. 1, which are based on data for six nuclear microsatellite loci obtained in three stands of wild cherry (data from the dis¨ sertation of Holtken, 2005). Generally, Jaccard differences lead to larger relative and absolute diversities than minimum genic differences (of course with the exception of = 0). This goes along with distinctly more steps for the Jaccard differences, which is a consequence of the fact that the normalization of genic differences varies strongly with genotypes. Although for high resolution (small ) relative diversities (upper graphs in Fig. 1) are large and very similar for all three populations, they differ distinctly for their corresponding absolute diversities, i.e., effective numbers (lower graphs). Whereas at intermediate resolutions (0.40 < < 0.75 for minimum genic differences, 0.55 < < 0.85 for Jaccard differences) the three populations are most strongly distinguished for their relative diversities, their absolute diversities already converge after a strong decline. This observation confirms the above remarks that relative and absolute measures of diversity lead to different statements on diversity. In particular it demonstrates that the three populations display their differences in genetic diversity most clearly for their effective numbers of genotypes at low resolution. For relative diversities their differences become more evident for intermediate resolutions. Concerning the diversity ranking of the three populations, it turns out that Settmarshausen is uniformly more diverse than Wibbecke for both the minimum genic and the Jaccard differences. In contrast, Roringen changes diversity ranking with respect to both of the other two populations, starting with the largest diversity at high resolution and falling below Settmarshausen and Wibbecke in succession with decreasing resolution. Hence, in comparison to the other populations, the assessment of the genetic diversity of Roringen depends on the chosen resolution. In fact, genotypes that are distinguished from other genotypes by many genes can be expected to normally occur at low frequencies. To increase the frequency of such genotypes and to thus increase D() as well as ne () for larger values of , systems of reproduction are required that increase genic associations. Among such systems is clonal propagation, which is known to occur regularly in wild cherry by root suckering. Interestingly, it was shown ¨ in the work of Holtken (2005) that the degree of cloning was lowest in Roringen, distinctly larger in Settmarshausen, and largest in Wibbecke. Thus, the presumably smaller genic associations in Roringen would explain its high genotypic diversity values for high resolution and its comparatively strong decline with decreasing resolution. These findings call for comparison with the results of a classical assessment of genetic diversity of the three populations. As is seen in table of Fig. 1, even for complete
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resolution ( = 0) the ranking of the populations differs from that obtained for both relative average allelic (Da ) and average genotypic (Dg ) diversity (the same holds for the corresponding effective numbers nea and neg ). While the ranking of Settmarshausen and Wibbecke is maintained, Roringen shows the lowest average allelic diversity. However, it ranks well between Settmarshausen and Wibbecke for its average genotypic diversity. Consequently, there is little correspondence between the diversity rankings obtained for multiple-locus diversity functions and average single-locus diversities. A straightforward explanation is suggested by the negligence of gene associations in classical diversity analyses. Finally, there is no indication of distinct steps in the diversity functions such as would be expected for hierarchical organization of genetic variation. As was explained above (last paragraph of the section on grouping for quantitative differences), the existence of genetically differentiated subpopulations (or demes) should be recognizable in this way at lower levels of resolution, since genetic differences within subpopulations should be distinctly smaller than between subpopulations. The population substructure due to clonal reproduction does not show up in the diversity functions, since it is limited to = 0. In fact, the small population sizes as well as the supposed selective neutrality of the microsatellite gene markers give little reason to expect hierarchical structure. Software for computing and drawing diversity functions D() and choropleths for illustration of difference matrices is available from the co-author.
Acknowledgements The authors are thankful for the suggestions of anonymous reviewers that helped to improve several of our formulations.
references
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