Simpson's Diversity Index

10 Key excerpts on "Simpson's Diversity Index"

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  Philosophy of Ecology

  • Dov M. Gabbay, Paul Thagard, John Woods(Authors)
  • 2011(Publication Date)
  • North Holland

    (Publisher)

  individual community is negative, which would clearly be problematic.

  3. Simpson's Index

  The first index that included species richness and evenness as components of diversity found in the ecological literature was proposed by Simpson [1949] . Simpson claimed that the probability two individuals drawn at random (with replacement) from an indefinitely large collection are of the same group is , where n is the number of groups exhibited within the collection, and he called it a “measure of concentration.” Applied to biological communities, then measures the dominance (in terms of abundance) of species within the community (Pielou 1977 ) and is at its minimal value ( ) for a given species richness n when individuals of the community are equally distributed among the n species, i.e., when Vp = .

  On Simpson's interpretation, the complement of the concentration measure:

  (D)

  , 23 represents the probability two randomly selected individuals belong to different species, which is an intuitive measure of diversity. 24 D is at its maximal value for a given species richness when individuals are maximally equally distributed among species, i.e., when Vp = , and at its minimal value when individuals are maximally unequally distributed among species, i.e., when Vp = . 25

  23 The inverse of Simpson's concentration measure, , is also commonly used as an index of diversity [Williams, 1964; Levins, 1968; [Hurlbert, 1971] , [MacArthur, 1972] , [Hill, 1973] , [May, 1975] , [Pielou, 1977] , [Magurran, 1988]

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  The Cambridge Companion to the Philosophy of Biology

  • David L. Hull, Michael Ruse(Authors)
  • 2007(Publication Date)
  • Cambridge University Press

    (Publisher)

  The second interpretation is more interesting: (i) it suggests that these intuitions be examined to determine which are less dispensable than others, and (ii) it leaves open the option to prefer a measure of diversity on the ground that it is connected to ecological processes. Table 21.2 shows that the Simpson and Shannon measures perform better than all other measures with respect to the number of conditions they meet, a point that has often been ignored when new measures are proposed. Efforts to find connections between diversity measures and eco- logical processes have also foundered. With respect to the ecological determinants of biodiversity, Patil and Taillie (1976, 1979, 1982) initiated an ambitious program of deriving diversity, interpreted as average abundance rarity, from probabilistic models of interspecific and intraspecific encounters between organisms. The Shannon and Simpson measures then emerge as different ways to compute the From Ecological Diversity to Biodiversity 395 average. 7 Most commentators since the 1980s have ignored these efforts. The trouble is that these models have no obvious bearing on any other question in ecology. Consequently, any diversity index emerging from these models may not be relevant in other ecological contexts; in the absence of such connections, diversity will play no significant theoretical role in ecology. However, if diversity is interpreted as richness, any attempt to derive species-area curves from fundamental principles is a model of diversity. Similarly, the theory of island biogeography (MacArthur and Wilson 1967) is also such a theory of diversity qua richness, as, more explicitly, is Hubbell’s (2001) neutral theory of biodiversity. Hutchinson’s justly famous query about the source of variation of species richness with latitude also constitutes a call for such a theory of diversity qua richness. But, as noted earlier, there is more to diver- sity than richness.

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  Metagenomics

  Perspectives, Methods, and Applications

  • Muniyandi Nagarajan(Author)
  • 2017(Publication Date)
  • Academic Press

    (Publisher)

  (8.2) , there might be a threshold for the independence to occur, and further the threshold may depend on the sampling efforts in the diversity investigation. This kind of subtle difference may be a burden for applied ecologists without gaining commensurable insights from the efforts.

  Simpson's [8 ] is another commonly used diversity index:

  D =

  ∑

  i = 1

  R

  p i 2

    (8.3)

  D is inversely correlated with diversity and a better alternative derived from D is PIE (probability of an interspecific encounter):

  PIE =

  n

  n − 1

  1 −

  ∑

  i = 1

  R

  p i 2

  =

  n

  n − 1

  1 − D

    (8.4)

  PIE is the probability that two randomly chosen individuals from a community should represent different species, and it is positively correlated with diversity. In economics, PIE without the adjusting factor n /(n −  1) is the well-known Gini coefficient , i.e., Gini index = 1 − D .

  The Shannon index is considered to be a better index due to its apparent connection with Shannon's entropy in his information theory. In fact, many probability distributions have an entropy defined; for example, the Weibull distribution, for which the author [1 ] discovered wide applicability in modeling diversity and richness, has the following information entropy:

  H = γ

  1 −

  1 β

  + ln

  1

  β λ

  + 1

    (8.5)

  where γ is the Euler-Mascheroni constant, and is equal to 0.5772156649015328606065…. β and λ

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  Surveying Natural Populations

  Quantitative Tools for Assessing Biodiversity

  • Lee-Ann Hayek, Martin Buzas(Authors)
  • 2010(Publication Date)
  • Columbia University Press

    (Publisher)

  Throughout this book we have not attempted to catalog all the approaches used for sampling natural populations (for a review see, e.g., Hayek 1994). Instead, we have presented some that, in our judgment, have a sound statistical or mathematical foundation under certain conditions and, at the same time, provide us with enough of the best tools for sampling natural populations. We will follow that approach here. Researchers use indices in most, if not all, scientific fields, each with slight varia-tions in meaning. A fact that is not commonly known but is important to understand is that a diversity index is also a statistic, not merely a numerical value. That is, the di-versity measure calculated from a sample can provide an estimate or approximation for the related population quantity if used correctly. In turn, the true value of the in-dex for a community or assemblage of interest, which is always unknown, is a param-eter. When used properly, the calculated diversity measure can tell us many interest-ing aspects and characteristics of our assemblage. Diversity of species and communities has always been a fundamental question of concern in ecology. If the investigator counts both the species, S , and the total number of all individuals observed, N , this information can provide useful data from the sample. A measure that uses both S and N can result in a measure of S that is charac-teristic of an area, without the necessity of standardizing N . Let us think about how to describe the diversity of an assemblage when the data recorded is total N and total S over some space or time. First, it is important to realize that whenever we consider 2 or more populations with distinctly different numbers of individuals within each species, there will be recognizable shortcomings when at-tempting to measure diversity with only total values.

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  Measuring Biological Diversity

  • Anne E. Magurran(Author)
  • 2013(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  How should inconsistencies in ranking be dealt with? One option is to compare only those assemblages that are ranked consistently when different orders of diversity are used. The methods described by Rényi (1961), Hill (1973), and Tóthméresz (1995) can be used to accomplish this. Indeed, Southwood and Henderson (2000) argue that such diversity ordering must be undertaken if the intention is to compare communities using a single “nonparametric” measure. In practice, however, most investigators omit this step. This is acceptable as long as it is clear that the aspect of diversity measured relates only to the index used to measure it, and there is no claim or suggestion that diversity in any broader sense is being measured.

  Figure 5.7 Different measures of diversity do not always rank assemblages in the same way. In this example of soft-sediment macrobenthos from 16 localities in the southern part of the Norwegian continental shelf, there is little concordance between the Shannon index and species richness (

  rs

  = 0.25, P > 0.05). The Shannon and Simpson measures, by comparison, produce highly concordant rankings of sites (

  rs

  = 0.95, P < 0.01). The exponential form of the Shannon index and reciprocal form of the Simpson index are shown. P values have received Bonferroni correction. (Data from table 1, Ellingsen 2001.)

  A related problem was noted by Lande et al. (2000), who observed that species accumulation curves may intersect (see also the discussion in Chapter 3). This means that rankings of assemblages can differ as a function of sample size. Lande et al. (2000) recommend the Simpson index for its ability to consistently rank assemblages when sample size varies. Moreover, the probability that the observed (estimated) Simpson diversity accurately reflects the true Simpson diversity increases rapidly with sample size. In their example a sample of 100 individuals was sufficient to correctly rank butterfly assemblages using the Simpson diversity index. The required sample size rose to 2,000 individuals if species richness was used to rank them (see Figure 3.8 ). The Shannon index was rejected due to its high bias in small samples (see also Lande 1996). Platt et al. (1984) have also argued that the diversity of two or more assemblages can only be unambiguously compared when k -dominance plots do not overlap (see Figure 2.6

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  Spatial Analysis

  A Guide For Ecologists

  • Mark R. T. Dale, Marie-Josée Fortin(Authors)
  • 2014(Publication Date)
  • Cambridge University Press

    (Publisher)

  Therefore, the α-diversity measured is affected by spatial scale, both as grain of sampling and as extent (see Figure 10.1). Turning to the measurement of diversity, there are several questions that arise in developing or choosing a measure of diversity at this level. Is this a complete census of the organisms, or truly a sample of those that are present? If it is a sample, what is the relationship between the area of the site about which inference is to be made and the total area covered by the samples? Are there measures of abundance for the classes that are being used or just presence : absence data? How certain is the identifi- cation of the organisms and the taxonomy of the classification used? In general, the assumptions for evaluating diversity at a single site include at least (1) that all identified taxa are treated as equivalent, (2) that the measure of abundance is appropriate for the kinds of organisms under study, and (3) that where ‘individuals’ are used as the basis for calculation, all individuals are treated as equivalent (Peet 1974). Many measures of diversity combine both the number of taxa, known as ‘richness’ and the equality of their representation in the data, known as ‘evenness’. The many measures available combine the two with various degrees of relative influence on the final value of the index and the relative influence of rare versus common taxa on the outcome (Magurran 2004). The two most common measures that combine richness and even- ness for abundance data are Simpson's index, D S , based on probability, and the Shannon–Weaver index, H 0 , based on information theory, both of which use the proportion of the total abundance, however measured, that belongs to the ith category of the s in the classification, p i : D S ¼ 1  X s i¼1 p 2 i and ð10:1Þ H 0 ¼ X s i¼1 p i log e p i : ð10:2Þ Simpson's index, D S , is usually formulated as the probability that two randomly chosen individuals (however defined) belong to different species.

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  Ecological Methods

  • Peter A. Henderson, T. R. E. Southwood(Authors)
  • 2016(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  13.6 a). The Simpson–Yule index is strongly influenced by the few dominant species, although bearing this limitation in mind it could be of value as an indicator of interspecific encounters. As is shown in Fig 13.7 for the Hinkley Point data, all of the non-parametric indices showed the same temporal pattern as species number and gave no additional insight. Thus, species number is often a straightforward measure for comparing diversity between samples collected in similar fashion. If the comparison is to be made between samples which differ in sampling effort, then estimates of total species richness, S MAX, can be compared or rarefaction undertaken to produce a species richness for a standardised effort (see section 13.1.2.1). Figure 13.7 A comparison of a variety of diversity measures applied to monthly fish samples collected at Hinkley Point, England. In this case the data can be considered to be well ordered in that all the indices plotted show the same temporal pattern. Note that the Shannon–Wiener index is the least sensitive to change in the community. Non-parametric diversity indices become useful when used for diversity-ordering (Tóthmérész, 1995). By generating a family of diversity indices it is possible to recognise non-comparable communities. If the objective of a study is to compare communities using a single non-parametric diversity measure, then diversity ordering must be undertaken. 13.1.6 Comparing communities – diversity ordering Different diversity indices may differ in the ranking they give to communities (Hurlbert, 1971; Tóthmérész, 1995). An example from the latter report illustrates the point

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  Handbook of Spatial Point-Pattern Analysis in Ecology

  • Thorsten Wiegand, Kirk A. Moloney(Authors)
  • 2013(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  context (but see Reardon and O’Sullivan 2004 for its use in a sociological context)� The spa-tially explicit point-pattern extensions of the classical diversity indices pro-vide basic information on multispecies spatial patterns and spatial variation of species diversity� They consider all pairs of points at a given distance and measure the overall local diversity within a distance r in a point centric way� They also characterize the overall ecological dissimilarity at distance r � In Section 3�1�5 we provided examples for the Simpson index�

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  Statistical Inference Based on Divergence Measures

  • Leandro Pardo(Author)
  • 2018(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  2 Entropy as a Measure of Diversity: Sampling Distributions 2.1. Introduction The term diversity is usually synonymous of “variety” and is simply an in-dication of the number of di ff erent ways a characteristic is present in a group of elements, taking in account the total of elements with each value of the charac-teristic. For example, we often speak of a “diversity of opinions”. While simply accounting for the number of di ff erent types of opinions on a topic one can give a rough idea of the “diversity of opinions;” the total of people with the same opinion must be taken into account to get the true sense of the diversity. The concept of diversity appears in a great number of research areas: ecology, biology, genetics, economics, linguistics, etc. It is in some sense the degree of heterogeneity of the individuals with respect to characteristics under study. If diversity is de fi ned as “the presence of a great number of di ff erent types of industries in a geographical area” (Economics) or “the linguistic di ff erences between the inhabitants of neighboring regions” (Linguistics) or “the number of species in a place as well as the abundance of those species” (Biology), then it would be useful to have a summary statistic to describe the diversity of a characteristic in an area and compare it to that of other areas. 56 Statistical Inference based on Divergence Measures A diversity measure should satisfy certain intuitive conditions which are satis-fi ed by an entropy measure. Later this point will be clari fi ed. Shannon’s entropy measure as well as Gini-Simpson index (The expected distance between two in-dividuals drawn at random when the distance is de fi ned as zero if they belong to the same category and unity otherwise, see Exercise 1 of this chapter) have been used as indexes of diversity. We can observe in Exercise 1 that Gini-Simpson index is the φ α -entropy of Havrda and Charvat with α = 2 and sometimes is called quadratic entropy.

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  Economic Valuation of Biodiversity

  An Interdisciplinary Conceptual Perspective

  • Bartosz Bartkowski(Author)
  • 2017(Publication Date)
  • Taylor & Francis

    (Publisher)

  Chapter 5 with measurable data.

  Many different measures of biodiversity have been proposed in various contexts: ‘At a global scale, there are roughly 40 potential measures being developed for the Convention on Biological Diversity (CBD) and about 26 indicators being considered in the Streaming Biodiversity Indicators in Europe 2010 process’ (Ding and Nunes, 2014, p. 61; see also Polasky et al., 2005). Most diversity measures focus on different components of it. For instance, genetic diversity can be measured by comparing the genotypic differences between species or individuals. Species diversity is mostly measured by means of rather simple indices, such as species richness (number of species) or different indices originating mostly from information theory, in which species numbers are weighted by relative abundances (i.e. they combine variety and balance information according to Stirling, 2007). A typical example is the Shannon index, which is defined as follows:

  (3.1)

  with pi being the relative abundance of species i, i.e. pi = ni /N, where ni is the number of individuals of i and N is the overall population.

  Phylogenetic diversity can be measured by combining information on species numbers with genetic dissimilarity between them (phylogenetic distance trees/dendograms), thus taking into account Stirling’s variety and disparity categories (Gotelli and Chao, 2013). Functional diversity measures are closely related to phylogenetic diversity, with the difference that the unit of comparison of species are functional traits (such as, e.g., nutrient capture) instead of genetic information (Petchey and Gaston, 2006). Functional diversity is an attempt to more directly capture the influence of biodiversity on ecosystem functioning (on this, see next section). Table 3.1

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