Parameter dependence of correlation between the Shannon index and members of parametric diversity index family

Ecological Indicators 7 (2007) 181–194 This article is also available online at: www.elsevier.com/locate/ecolind

Parameter dependence of correlation between the Shannon index and members of parametric diversity index family Ja´nos Izsa´k * Department of Zoology, Berzsenyi College, Ka´rolyi Ga´spa´r te´r 4, Szombathely 9701, Hungary Received 20 May 2005; received in revised form 1 December 2005; accepted 2 December 2005

Abstract We present detailed data on correlation of the Shannon diversity index with members of Hill’s and Hurlbert’s parametric index family. The correlation reaches its maximum in both cases at a definite parameter value. These specified data on correlation are exploitable when applying the above diversity indices. Namely, in case of their parallel use with the Shannon index a strong correlation between the indices should be avoided. The investigations were carried out on sets of moth and fly collections. Moreover, we hypothesized that a smaller distance between so-called sensitivity profiles of diversity indices predicts a larger correlation between the indices concerned. Based on investigations of the sensitivity profile distance between the Shannon index and members of parametric diversity index family as a function of index parameter, we point out that on the whole the above hypothesis is true. This makes possible to estimate the correlation of two diversity indices omitting troublesome index calculations on a large set of abundance lists. # 2005 Elsevier Ltd. All rights reserved. Keywords: Correlation; Diversity indices; Hill’s indices; Hurlbert’s indices; Index sensitivity; Shannon index

1. Introduction Diversity indices are classic scalar ecological indicators. The application of these indices is common in ecological analysis. A frequent finding is that species diversity indicates the status of the ecosystem or community (Ferna´ndez-Ala´ez et al., 2002; Diserud and Aagaard, 2002; Park et al., 2003; Salas et al., 2005) and, in a wider sense, the quality of the living * Tel.: +36 94 504 300; fax: +36 94 504 404. E-mail address: [email protected].

environment (Gaston and Spicer, 2004). Moreover, a high species diversity contributes to the stability of the ecosystem (Sankaran and McNaughton, 1999; Naeem and Baker, 2005; Kiessling, 2005; Moore, 2005). In addition to hundreds of articles, several books discuss diversity aspects in ecology and the methodology of diversity measures (Pielou, 1975; Grassle et al., 1979; Patrick, 1983; Magurran, 1988, 2004). The parallel use of some diversity indices is a general praxis. However, the well-known correlation between most diversity indices is considerable. As Ricklefs (1990) writes: ‘‘. . . the results of most studies

1470-160X/$ – see front matter # 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ecolind.2005.12.001

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182

are relatively insensitive to which index of diversity is applied . . .’’. If one interprets this correlation as the sign of an abstract diversity property of the community, so this correlation is reassuring. However, if the correlation expresses the fact that diversity indices inform almost uniformly on the level of diversity, then from a practical point of view the introduction of more and more new diversity indices seems to be questionable. In her popular book Magurran (1988) summarized data on diversity index correlation. In the recent past Bryja and Kula (2000) analyzed Spearman’s rank correlations of numerous diversity indices on bug collections. However, on the whole the investigations of diversity index correlation are sporadic and phenomenological. Our investigations are aimed to make a first step to a more systematic and formalized treatment of this topic. The correlation of diversity indices can be reduced to the property that the usually applied diversity indices measure mainly the dominance structure of the communities. Realizing this one-sidedness, some authors suggested the use of diversity indices more sensitive to changes in the range of the middle and small species abundances. Such indices are, for example, some members of Hill’s and Hurlbert’s index family. Hill’s index family can be associated with the wellknown Shannon diversity index or multinomial entropy defined by the formula: H 0 ð p1 ; . . . ; ps Þ ¼ 

s X

pi ln pi ;

i¼1

where s is the number of the occurring species in the sample and pi is the theoretical probability of occurrence of species i; p1 + p2 +    + ps = 1 (Pielou, 1975). The quantity H0 ( p1, . . ., ps) expresses the information gain when one is informed on the distribution ( p1, . . ., ps) after the a priori hypothesis on the uniform distribution (1/s, . . ., 1/s) in the sample. A generalization of index H0 is Re´nyi’s generalized entropy index or a-order entropy family with the formula P ln si¼1 pai Ra ¼ ; a 6¼ 1; R1 ¼ lim1 Ra ¼ H 0 1a (Re´nyi, 1961). Hill’s diversity index family (Hill, 1973), which is well-known in the literature of eco-

logical diversity, is defined by the formula Na ¼

X s

pai

1=ð1aÞ ;

i¼1

a 6¼ 1; N1 ¼ lim1 Na ¼ expðH 0 Þ: Thus, Na equals to exp Ra. Clearly, the Na indices also have an information theoretical meaning, as they are directly related to Re´nyi’s indices. A basic property of Hill’s indices is that by decreasing the (positive) parameter a the index stresses more and more the distribution conditions in the range of the small species abundances. An extreme illustration can be that N0 = lim1 Na = s for any abundance distribution. That is, upon the emergence of a new species in the population, represented if only by a single individual, the value of N0 increases from s to s + 1, regardless of the size of other abundances. On the other hand, it is easy to show (see Appendix A) that N1 = lim1 Na = 1/pi,max, which is the reciprocal of the Berger–Parker concentration index. Apparently, N1 is sensitive only to changes in the maximum species probability. The application of Na indices, mainly N1, N2 is frequent in the statistical ecology, although the application of NP context. 2 emerges often in Panother Namely, N2 ¼ ð p2i Þ1=ð12Þ ¼ 1= p2i , which is the well-known reciprocal Simpson index. The sporadic application of Hill’s indices with further parameter values can be reduced to insufficiency of data about the relation between the statistical behavior of Na and frequently used diversity indices, such as H0 index or Fisher’s alpha index, etc. The diversity measures s(m), m = 2, 3, . . . (Hurlbert, 1971) were introduced originally as socalled rarefaction indices (Simberloff, 1979). Let us choose randomly m individuals from the population with species probabilities p1, p2, . . ., ps ( p1 + p2 +    + ps = 1). Let Sm be the probability variable relating to the number of represented species in the sample. It is easy to show that the expectation of the number of species in the sample is ESm = s(m). Particularly, s(2) is the mean of the number of species represented in a sample consisting of two randomly chosen individuals. The quantity s(2) bears a relation also to the probability that two randomly chosen individuals belong toP different species. Namely, this probability is 1  p2i (Gini-Simpson

J. Izsa´k / Ecological Indicators 7 (2007) 181–194

or PIE index; PIE is the abbreviation of ‘‘probability of interspecific encounter’’ (Hurlbert, 1971)). On the other hand, it holds X X X sð2Þ ¼ ð1  ð1  pi Þ2 Þ ¼ 2 pi  p2i X ¼1þ1 p2i ¼ 1 þ Gini-Simpson index: Note P that s(2) is linked also with index N2, as N2 ¼ 1= p2i ¼ 1=ð2  sð2ÞÞ. The applied estimating formula, yielding a minimum variance unbiased estimation is  1 0 n  ni s B X C m B1    C; m  n: sˆðmÞ ¼ @ A n i¼1 m By increasing parameter m, the index stresses more and more the distribution conditions in the range of small species abundances.PAn extreme illustration can be that lim1 s(m) = 1 = s. The situation is similar to that mentioned in connection with N0. On the other hand, s(2) is rather insensitive to changes in small abundances (Magurran, 1988, p. 39). The most frequently used member of the s(m) index family is s(2). Examples of using s(m) index with large m can be found e.g. in Natuhara and Imai (1998) and Bryja and Kula (2000). The cause of the scarce application of various members of this index family except of s(2) can be reduced to similar causes mentioned above with the description of the Na indices. Clearly, the possibility of using members of the Na and s(m) index family expands the range of the diversity investigations. However, since the practical statistical and numerical properties of diversity indices are mostly unexplored, ecologists are often puzzled as to how many and what kind of diversity indices should be applied simultaneously for a better description of diversity conditions. The parallel application of diversity indices goes back to several points. The first point is that diversity is an ambiguous concept and that, in essence, the number of diversities is equal to the number of diversity indices (Peet, 1974). Another motivation to search for data on diversity index correlation emerges when one intends to perform a multiscale grouping of ecosystems by their diversity properties, so the introduction of a new diversity

183

index, in other words a new diversity axis, makes it possible to make a further segregation of the ecosystems grouped formerly. This results in additional information on the diversity conditions in the set of the ecosystems. However, the larger is the correlation between the indices, the less the efficiency of the segregation process. Consequently, by the parallel use of diversity indices information on their correlation is crucial. This is the case also when we use index H0 and members of the Na or s(m) index family. Consequently, when using H0 , information on the correlation of the H0 index with indices Na or s(m) is crucial. The main aim of the recent paper is to publish experimental data on how this correlation depends on parameter a and m. Naturally, correlation properties depend also on the concrete set of abundance vectors z(i) (i = 1, 2, . . ., r) on which the (H0 (z(i)), Na(z(i))) or (H0 (z(i)), s(m,z(i))) pairs of diversity values are taken into account. Moreover, to establish a mathematical approach of index correlation properties we proposed a working hypothesis, supposing that the index correlation is closely related to the distance of the so-called sensitivity profiles. 1.1. Sensitivity profiles Little is known on the index sensitivity to partial and relatively small changes of abundances. The issue is mentioned earlier by Peet (1974), Taillie and Patil (1979) and Patil and Taillie (1982). Schmid (1991) developed a sensitivity concept closely related to the transfer principle for inequality measures used in economics. Boyle et al. (1990) reported results of numerical experimentation on several diversity indices. Some results, based on the so-called sensitivity profiles or index response profiles were published by Izsa´k (1991, 1996). These profiles are defined as the series of sensitivity values relating to the decreasingly enumerated series of the differing abundances. Sensitivity profiles provide much information on the diversity index properties (Izsa´k, 1996; Izsa´k and Papp, 2002). To make a comparison of sensitivity differences more realistic, in the present article we use sensitivity/index value ratios. Taking diversity indices D1 and D2, their squared profile

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184

distance with respect to a concrete abundance list z is s0  X sensðD1; k; zÞ k¼1

D1ðzÞ

sensðD2; k; zÞ  nk   nk D2ðzÞ

2 ;

(1)

where s0 is the number of the differing abundances in z. For example, in case of H0 the index response belonging to the kth differing element zk of the abundance vector z is sensðH 0 ; k; zÞ  1 zk log ¼  sum0 z sum0 z 1

s0 X

zj þ z j log 2 0 sum0 z ðsum zÞ j¼1

  zk

ðk ¼ 1; 2; . . . ; s0 Þ where sum0 z is the sum of the differing abundances in z (Izsa´k and Papp, 2002). The (squared) distance of the sensitivity profiles is applicable to express the similarity of sensitivity properties of the diversity indices concerned. Sensitivity profile distances between H0 and Na or H0 and s(m) relating to a fixed abundance list depend on the index parameter a and m, respectively. By plotting the profile distances against the parameter values we obtain so-called sensitivity profile distance graphs or ‘‘curves’’. As a working hypothesis, we assumed that when taking a fixed set of abundance lists, a parallelism exists between index correlation and sensitivity profile distance. It should be noted that index correlation is bound to the whole set of the abundance lists, while profile distance is defined for a single abundance list. An additional assumption is the relative constancy of the shape and location of the profile distance curves in case of different abundance lists. If the mentioned parallelism exists, one can reduce the study of index correlation properties to a study of index sensitivity properties, which are analyzable by mathematical methods. Further on, a single profile distance curve would serve for a proxy to gain information for index correlation properties, which could be determined only on the whole set of the abundance lists.

2. Study material and methods The study material was a set of 18 species abundance lists (collections) published in a study on moths (Usher and Keiller, 1998, Table A1) and another set of 15 species lists published in a study on flies, collected at Forra´spuszta (Hungary) (Papp, 1995; Table 1). In the latter case, of the original set of 21 collections, we did not include those having less than 50 specimens and an extremely large collection. The abundance lists in the referred tables are denoted by vðiÞ (i = 1, 2, . . ., 18) and wð jÞ (j = 1, 2, . . ., 15), respectively. Sample size, species number and some diversity values for each community examined are given in Table 1. For further details such as dominance conditions in the communities can be concluded from the mentioned tables in Usher and Keiller (1998) and Papp (1995). To investigate index correlation, we computed first the H0 , Na and s(m) index values for the mentioned abundance lists and for each a and m parameter given in Tables 2 and 3. By calculations we applied with H0 and the Na indices standard estimation, that is probability pi was substituted be ni/n, where ni is the abundance of species i and n (=n1 + n2 +    + ns) is the sample size. An exception isPthe estimation of N2; this is estimated by i ðni 1Þ 1 ð nnðn1Þ Þ . Computations were carried out by a Dbase program using the program package DIVERSI 2.1 at no cost (Izsa´k, 2003). (A new version is DIVERSI 2.2 (Izsa´k, in press).) To make clear the notations used, e.g. N0:10 ðvð3ÞÞ denotes the value of the diversity index N0.10 computed for the third column of the referred Table A.1, coded there by NS. Based on the computed diversity index values, linear correlation coefficients were calculated for the set of 18 pairs of indices ðH 0 ðvðiÞÞ; Na ðvðiÞÞÞ and ðH 0 ðvðiÞÞ; sðm; vðiÞÞÞ (i = 1, 2, . . ., 18) for each a and m parameters given in Tables 2–5. Similarly, based on the computed diversity index values, linear correlation coefficients were calculated for the set of 15 pairs of indices ðH 0 ðwð jÞ; Na ðwð jÞÞÞ, and ðH 0 ðwð jÞ; sðm; wð jÞÞÞ (j = 1, 2, . . ., 15) for each a and m parameter. We preferred using linear correlation coefficient because this is more suitable for possible further mathematical analysis than rank correlation coefficient.

Table 1 Basic statistics and some diversity values relating to the 18 community samples of moths and 15 community samples of flies vð1Þ

vð2Þ

vð3Þ

vð4Þ

vð5Þ

vð6Þ

vð7Þ

vð8Þ

vð9Þ

vð10Þ

vð11Þ

vð12Þ

vð13Þ

vð14Þ

vð15Þ

vð16Þ

vð17Þ

vð18Þ

Sample size Number of species H0 N0.1 N0.50 N90 s(2) s(9) s(50)

wð1Þ

wð2Þ

wð3Þ

wð4Þ

wð5Þ

wð6Þ

wð7Þ

wð8Þ

wð9Þ

wð10Þ

wð11Þ

wð12Þ

wð13Þ

wð14Þ

wð15Þ

78 24 2.61 22.85 18.37 3.96 1.89 6.39 19.19

175 22 1.9 19.58 11.82 2.13 1.73 4.58 11.56

68 19 2.12 17.76 12.99 2.29 1.78 5.27 15.88

543 25 1.68 20.91 10.08 2.27 1.72 4.02 8.59

183 24 1.68 21.27 11.87 1.59 1.59 4 12.41

212 19 1.73 16.79 9.93 2.08 1.71 4.19 10.24

472 20 1.53 16.61 8.07 2.2 1.69 3.76 7.23

980 20 1.55 15.89 7.33 2.79 1.73 3.9 6.4

751 36 1.35 29.83 11.83 1.36 1.45 3.18 9.92

107 23 2.12 21.25 14.53 2.16 1.76 5.04 15.71

226 24 2.11 21.57 13.7 2.54 1.79 5.04 12.55

135 18 1.87 16.24 10.5 2.62 1.76 4.53 11.31

339 21 1.57 17.74 8.9 1.92 1.67 3.88 8.19

182 23 20.3 20.62 12.89 3.07 1.8 4.81 12.28

343 23 1.67 19.37 9.63 2.2 1.72 4.04 8.43

J. Izsa´k / Ecological Indicators 7 (2007) 181–194

Sample size 1052 491 889 756 386 828 762 1325 1020 691 871 708 1326 696 974 1004 1354 1011 Number of 102 76 92 95 67 91 80 100 103 85 101 75 129 94 96 111 113 94 species 3.57 3.66 3.71 3.87 3.49 3.58 3.44 3.71 3.53 3.69 3.71 3.12 4.14 3.73 3.62 4.04 3.4 3.45 H0 N0.1 91.96 70.89 84.26 88.51 62.48 83.37 73.11 90.61 92.92 78.35 92.56 68.12 118.7 86.72 87.05 103.1 100.2 84.87 N0.333 71.51 60.15 68.63 74.94 52.76 67.32 58.71 72.04 71.99 64.76 74.88 53.09 98.07 71.38 68.79 86.85 73.97 65.93 N90 6.17 10.5 10.53 9.82 7.59 5.63 5.39 9.5 6.86 10.13 6.59 3.43 13.79 10.51 9.24 16.98 4.4 5.63 s(2) 1.95 1.96 1.96 1.97 1.95 1.94 1.94 1.96 1.94 1.96 1.95 1.88 1.98 1.96 1.96 1.97 1.92 1.93 s(10) 8.12 8.55 8.5 8.75 8.24 8.14 7.92 8.49 7.97 8.53 8.35 7.06 9.06 8.5 8.34 8.97 7.61 7.83 s(50) 25.62 27.31 27.33 29.62 25.58 26.19 24.72 26.98 25.11 27.35 27.36 22.17 32.4 27.61 25.61 31.23 23.69 24.43 s(150) 46.98 48.16 48.69 53.74 45.36 48.14 44.34 47.96 47.17 47.84 50.89 42.12 59.44 50.75 46.26 56.98 44.86 45.34

185

186 Table 2 Parameter values (first column), H0 vs. Na correlation coefficients (second column) and Na vs. H0 sensitivity profile distances multiplied by 100 for the 18 abundance lists of moths r

vð1Þ

vð2Þ

vð3Þ

vð4Þ

vð5Þ

vð6Þ

vð7Þ

vð8Þ

vð9Þ

vð10Þ

vð11Þ

vð12Þ

0.050 0.100 0.150 0.200 0.250 0.300 0.333 0.350 0.400 0.500 0.667 1.000 1.333 1.667 2.000 2.500 5.000 10.000 30.000 50.000 80.000 90.000

0.623 0.656 0.690 0.724 0.759 0.792 0.813 0.824 0.853 0.902 0.954 0.985 0.983 0.974 0.963 0.956 0.923 0.898 0.888 0.887 0.886 0.886

0.674 0.515 0.372 0.249 0.151 0.085 0.060 0.055 0.068 0.249 1.124 5.599 13.672 23.940 33.345 48.782 76.663 74.800 65.154 63.240 62.198 62.008

0.197 0.136 0.086 0.047 0.022 0.011 0.013 0.017 0.040 0.143 0.496 1.889 3.998 6.489 8.099 13.119 35.535 75.348 84.556 82.247 80.965 80.730

0.167 0.118 0.077 0.045 0.025 0.016 0.017 0.021 0.040 0.126 0.427 1.629 3.454 5.521 7.034 9.870 13.871 15.354 17.117 16.770 16.506 16.457

0.203 0.146 0.097 0.058 0.031 0.016 0.015 0.017 0.034 0.127 0.486 2.199 5.535 10.639 15.949 29.904 84.125 92.330 80.591 78.320 77.084 76.858

0.404 0.299 0.207 0.130 0.071 0.034 0.023 0.021 0.035 0.158 0.691 3.214 7.602 13.198 17.088 27.560 50.478 67.987 74.330 72.372 71.218 71.008

0.798 0.623 0.462 0.320 0.202 0.114 0.075 0.062 0.056 0.207 1.118 6.572 17.912 33.676 48.045 70.504 85.161 69.786 59.597 57.806 56.831 56.653

0.923 0.720 0.533 0.367 0.229 0.126 0.079 0.063 0.051 0.211 1.190 6.761 17.208 29.988 40.080 54.904 70.850 67.832 58.960 57.192 56.230 56.055

0.241 0.169 0.109 0.063 0.034 0.022 0.026 0.031 0.061 0.192 0.637 2.372 5.025 8.314 11.599 18.599 58.492 89.032 81.443 79.157 77.913 77.685

0.736 0.555 0.393 0.253 0.143 0.070 0.044 0.039 0.059 0.280 1.298 5.959 12.649 18.820 22.458 27.173 30.073 30.319 51.073 63.985 68.376 68.596

0.240 0.171 0.113 0.067 0.035 0.019 0.018 0.021 0.042 0.149 0.550 2.287 5.215 8.942 11.914 18.408 31.184 48.166 81.534 81.091 79.878 79.647

0.556 0.425 0.307 0.205 0.124 0.067 0.047 0.042 0.051 0.201 0.954 5.113 13.507 25.418 36.922 57.807 88.533 77.668 66.697 64.743 63.679 63.485

2.873 0.082 0.280 0.235 0.083 1.551 2.336 0.056 0.199 0.163 0.053 1.233 1.822 0.035 0.130 0.103 0.030 0.936 1.343 0.019 0.074 0.059 0.015 0.667 0.915 0.011 0.036 0.031 0.007 0.436 0.555 0.010 0.016 0.022 0.008 0.257 0.364 0.014 0.016 0.028 0.014 0.172 0.286 0.017 0.019 0.034 0.018 0.142 0.130 0.033 0.045 0.068 0.038 0.106 0.266 0.093 0.181 0.205 0.109 0.334 2.334 0.289 0.681 0.643 0.323 1.905 15.363 1.118 2.760 2.175 1.087 10.827 35.201 2.710 5.861 4.226 2.198 26.047 50.653 5.341 9.184 6.586 3.530 42.322 56.921 8.687 11.208 8.818 4.584 54.529 61.544 18.236 15.491 14.162 7.376 67.979 51.630 83.380 21.453 50.389 20.037 69.630 40.660 100.744 26.657 86.899 36.424 56.486 33.978 88.196 61.778 80.831 51.486 47.874 32.812 85.749 77.564 78.562 63.334 46.361 32.179 84.416 80.955 77.326 77.250 45.539 32.063 84.173 80.981 77.100 80.203 45.389

Minima are bold-faced.

vð13Þ

vð14Þ

vð15Þ

vð16Þ

vð17Þ

vð18Þ 0.923 0.709 0.514 0.343 0.204 0.105 0.065 0.054 0.060 0.276 1.392 6.903 15.474 24.186 30.360 39.187 57.744 67.524 62.534 60.694 59.690 59.507

J. Izsa´k / Ecological Indicators 7 (2007) 181–194

a

Table 3 Parameter values (first column), H0 vs. Na correlation coefficients (second column) and Na vs. H0 sensitivity profile distances multiplied by 100 for the 15 abundance lists of flies r

wð1Þ

wð2Þ

wð3Þ

wð4Þ

wð5Þ

wð6Þ

wð7Þ

wð8Þ

wð9Þ

wð10Þ

wð11Þ

wð12Þ

wð13Þ

wð14Þ

wð15Þ

0.05000 0.10000 0.15000 0.20000 0.25000 0.30000 0.33333 0.35000 0.40000 0.50000 0.66666 1.00000 1.33333 1.66666 2.00000 2.50000 5.00000 10.00000 30.00000 50.00000 80.00000 90.00000

0.111 0.024 0.166 0.308 0.441 0.560 0.629 0.660 0.742 0.857 0.948 0.979 0.957 0.927 0.897 0.868 0.796 0.761 0.742 0.740 0.739 0.739

1.575 1.252 0.954 0.687 0.458 0.271 0.174 0.135 0.054 0.086 0.806 4.983 11.760 18.754 18.788 29.876 41.975 52.195 49.275 47.773 46.953 46.804

7.444 6.238 5.086 4.009 3.028 2.167 1.669 1.445 0.883 0.294 0.987 7.017 14.204 19.270 20.998 23.506 18.817 13.564 10.691 10.203 9.940 9.892

6.391 5.358 4.365 3.429 2.567 1.801 1.353 1.150 0.635 0.098 0.961 9.385 21.392 29.757 28.321 32.497 21.165 14.815 11.657 11.119 10.829 10.776

4.798 3.879 3.046 2.317 1.705 1.219 0.968 0.864 0.634 0.509 0.972 2.685 4.757 7.296 9.997 14.508 27.017 26.853 22.641 21.843 21.411 21.332

19.265 16.503 13.781 11.147 8.656 6.370 4.991 4.353 2.669 0.521 0.972 10.062 12.768 9.172 5.226 2.357 0.133 0.368 0.733 0.819 0.868 0.877

7.431 6.178 4.996 3.907 2.931 2.088 1.609 1.395 0.865 0.315 0.806 4.705 8.655 11.526 13.190 16.197 19.734 16.528 13.466 12.934 12.647 12.594

4.935 4.023 3.203 2.487 1.885 1.400 1.141 1.029 0.765 0.506 0.613 1.676 3.228 5.053 6.897 10.028 22.015 26.959 23.669 22.864 22.427 22.348

2.697 2.151 1.685 1.300 0.995 0.760 0.640 0.589 0.469 0.350 0.395 1.007 1.875 2.656 3.184 3.664 3.810 6.505 28.841 42.077 46.893 47.196

29.494 25.378 21.288 17.300 13.510 10.021 7.918 6.944 4.381 1.083 0.745 4.010 1.586 0.244 0.584 1.913 5.859 7.783 8.968 9.195 9.321 9.345

7.837 6.569 5.345 4.187 3.120 2.170 1.617 1.367 0.741 0.128 1.378 11.502 22.827 28.051 26.187 26.532 17.132 11.898 9.260 8.815 8.575 8.532

4.478 3.687 2.943 2.260 1.654 1.138 0.851 0.726 0.429 0.218 1.074 6.518 13.964 20.803 24.962 31.479 33.383 26.412 21.702 20.885 20.442 20.361

4.187 3.354 2.587 1.903 1.313 0.831 0.574 0.466 0.222 0.101 0.851 3.862 5.885 6.508 6.109 6.999 14.551 30.559 36.897 35.850 35.244 35.133

8.804 7.423 6.119 4.914 3.830 2.886 2.340 2.094 1.463 0.683 0.646 3.361 7.164 10.629 12.951 15.400 13.515 9.529 7.357 6.993 6.797 6.762

3.152 2.477 1.864 1.326 0.875 0.523 0.346 0.276 0.139 0.197 1.128 4.150 6.065 6.637 6.181 6.388 6.727 13.469 41.097 46.502 46.689 46.577

5.304 4.354 3.484 2.710 2.048 1.504 1.210 1.084 0.786 0.523 0.831 2.798 5.620 8.979 12.216 17.370 26.062 22.281 18.309 17.612 17.235 17.166

J. Izsa´k / Ecological Indicators 7 (2007) 181–194

a

Minima are bold-faced.

187

188 Table 4 Parameter values (first column), H0 vs. s(m) correlation coefficients (second column) and s(m) vs. H0 sensitivity profile distances multiplied by 104 for the 18 abundance lists of moths r

vð1Þ

vð2Þ

vð3Þ

vð4Þ

vð5Þ

vð6Þ

vð7Þ

vð8Þ

vð9Þ

vð10Þ

vð11Þ

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 22 24 26 28 30 35 40 45 50 60 70 80 90 100 150

0.915 0.923 0.930 0.936 0.942 0.947 0.952 0.956 0.960 0.963 0.966 0.968 0.971 0.973 0.974 0.978 0.980 0.982 0.983 0.984 0.985 0.986 0.986 0.986 0.985 0.984 0.982 0.979 0.976 0.973 0.970 0.955

54.177 33.791 20.467 12.111 7.193 4.612 3.579 3.541 4.112 5.028 6.114 7.255 8.378 9.442 10.425 12.114 13.450 14.489 15.295 15.927 16.429 16.837 17.593 18.127 18.532 18.844 19.246 19.413 19.405 19.280 19.086 18.037

20.114 14.885 10.772 7.591 5.187 3.426 2.197 1.404 0.965 0.813 0.891 1.149 1.548 2.053 2.638 3.954 5.358 6.755 8.088 9.320 10.433 11.418 13.329 14.539 15.195 15.438 15.140 14.301 13.271 12.220 11.224 7.574

16.729 12.340 8.915 6.292 4.335 2.927 1.969 1.375 1.074 1.007 1.123 1.380 1.743 2.184 2.679 3.762 4.878 5.960 6.968 7.880 8.690 9.398 10.759 11.645 12.188 12.492 12.657 12.493 12.160 11.741 11.280 9.086

19.467 13.980 9.838 6.773 4.565 3.034 2.037 1.453 1.189 1.168 1.330 1.624 2.014 2.469 2.966 4.017 5.069 6.067 6.985 7.811 8.546 9.191 10.463 11.332 11.893 12.219 12.368 12.089 11.579 10.968 10.336 7.824

35.174 23.110 14.619 8.848 5.124 2.920 1.826 1.527 1.780 2.404 3.259 4.245 5.288 6.336 7.352 9.202 10.742 11.960 12.881 13.547 14.005 14.299 14.544 14.400 14.098 13.747 13.055 12.424 11.842 11.296 10.788 9.074

59.025 34.126 18.992 10.332 5.861 4.011 3.730 4.324 5.350 6.530 7.704 8.784 9.729 10.526 11.183 12.130 12.714 13.069 13.297 13.466 13.619 13.776 14.243 14.796 15.375 15.923 16.823 17.426 17.783 17.968 18.048 18.175

68.125 39.021 21.206 10.938 5.608 3.418 3.146 3.979 5.385 7.026 8.694 10.267 11.682 12.911 13.949 15.488 16.427 16.917 17.090 17.054 16.887 16.649 15.973 15.411 15.061 14.914 15.055 15.486 15.984 16.447 16.853 18.534

24.107 17.722 12.804 9.076 6.313 4.327 2.965 2.100 1.629 1.469 1.550 1.818 2.228 2.744 3.336 4.662 6.073 7.484 8.846 10.129 11.319 12.408 14.691 16.393 17.602 18.408 19.127 19.064 18.562 17.835 17.012 13.218

62.553 40.806 25.627 15.396 8.847 4.994 3.082 2.536 2.922 3.920 5.293 6.872 8.535 10.199 11.810 14.745 17.210 19.197 20.753 21.941 22.825 23.462 24.272 24.354 24.014 23.430 21.925 20.332 18.863 17.599 16.558 13.911

23.160 16.522 11.452 7.673 4.946 3.072 1.881 1.233 1.010 1.113 1.463 1.993 2.650 3.389 4.177 5.793 7.346 8.749 9.962 10.972 11.785 12.417 13.344 13.610 13.496 13.193 12.436 11.755 11.217 10.792 10.439 9.120

45.087 169.015 8.950 27.625 23.802 9.661 104.519 73.005 28.372 79.337 6.975 20.176 17.585 7.829 56.005 44.821 17.351 35.501 5.374 14.362 12.808 6.270 29.159 26.485 10.370 15.831 4.091 9.913 9.192 4.954 15.303 15.035 6.214 8.303 3.078 6.597 6.512 3.854 8.962 8.329 3.995 6.503 2.293 4.215 4.578 2.948 6.765 4.835 3.073 7.156 1.701 2.599 3.241 2.213 6.703 3.466 2.993 8.678 1.269 1.603 2.375 1.630 7.613 3.465 3.436 10.358 0.972 1.106 1.881 1.181 8.859 4.304 4.179 11.916 0.787 1.005 1.677 0.851 10.115 5.628 5.073 13.270 0.694 1.213 1.698 0.624 11.235 7.195 6.020 14.422 0.677 1.660 1.892 0.489 12.169 8.851 6.958 15.400 0.720 2.284 2.217 0.432 12.921 10.496 7.849 16.237 0.813 3.037 2.639 0.445 13.514 12.069 8.674 16.961 0.945 3.879 3.133 0.517 13.981 13.538 10.094 18.156 1.292 5.703 4.254 0.805 14.657 16.105 11.219 19.100 1.709 7.563 5.463 1.241 15.151 18.176 12.096 19.855 2.158 9.340 6.688 1.778 15.583 19.804 12.779 20.461 2.614 10.967 7.887 2.381 16.019 21.062 13.317 20.951 3.062 12.409 9.038 3.021 16.484 22.019 13.749 21.351 3.491 13.653 10.129 3.677 16.984 22.735 14.104 21.687 3.896 14.703 11.153 4.332 17.515 23.257 14.769 22.364 4.784 16.561 13.424 5.893 18.931 23.948 15.236 22.972 5.503 17.541 15.303 7.265 20.376 24.054 15.574 23.608 6.077 17.905 16.836 8.408 21.762 23.804 15.815 24.305 6.536 17.865 18.067 9.322 23.040 23.337 16.077 25.859 7.201 17.133 19.786 10.548 25.178 22.078 16.138 27.558 7.628 16.052 20.730 11.158 26.747 20.703 16.075 29.358 7.889 14.921 21.109 11.353 27.822 19.417 15.937 31.259 8.025 13.851 21.081 11.283 28.518 18.307 15.753 33.277 8.063 12.880 20.766 11.052 28.949 17.404 14.703 45.303 7.470 9.422 17.394 9.161 29.694 15.526

Minima are bold-faced.

vð12Þ

vð13Þ vð14Þ

vð15Þ

vð16Þ

vð17Þ

vð18Þ

J. Izsa´k / Ecological Indicators 7 (2007) 181–194

m

Table 5 Parameter values (first column), H0 vs. s(m) correlation coefficients (second column) and H0 vs. s(m) sensitivity profile distances multiplied by 104 for the 15 abundance lists of flies r

wð1Þ

wð2Þ

wð3Þ

wð4Þ

wð5Þ

wð6Þ

wð7Þ

wð8Þ

wð9Þ

wð10Þ

wð11Þ

wð12Þ

wð13Þ

wð14Þ

wð15Þ

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 22 24 26 28 30 35 40 45 50

0.827 0.881 0.922 0.952 0.972 0.985 0.992 0.995 0.995 0.994 0.991 0.988 0.984 0.980 0.976 0.967 0.959 0.952 0.944 0.938 0.932 0.926 0.914 0.904 0.896 0.889

100.215 48.690 21.363 8.199 3.004 2.039 3.108 4.967 6.949 8.722 10.154 11.215 11.931 12.355 12.544 12.439 11.996 11.479 11.049 10.795 10.757 10.946 12.340 14.813 18.062 21.852

332.987 129.893 54.355 24.921 12.381 7.022 5.621 6.885 10.064 14.588 19.995 25.927 32.115 38.372 44.571 56.518 67.678 78.044 87.709 96.784 105.370 113.544 132.493 149.645 165.198 179.283

281.329 100.659 34.866 12.074 4.234 1.408 0.455 0.513 1.373 2.954 5.174 7.935 11.133 14.673 18.472 26.590 35.110 43.826 52.636 61.501 70.409 79.363 101.973 124.908 148.084 171.362

260.241 128.070 69.248 39.600 23.496 15.525 13.357 15.458 20.549 27.555 35.626 44.135 52.643 60.868 68.636 82.483 93.977 103.306 110.789 116.766 121.549 125.403 132.290 136.907 140.434 143.424

832.079 341.626 159.952 86.201 52.685 35.796 26.631 21.503 18.740 17.523 17.432 18.234 19.798 22.045 24.922 32.417 42.032 53.545 66.735 81.382 97.270 114.195 159.839 208.316 257.775 306.934

363.064 159.817 74.668 36.024 18.373 12.075 12.579 17.121 23.819 31.400 39.057 46.324 52.975 58.933 64.215 73.023 80.069 85.987 91.248 96.163 100.919 105.620 117.409 129.320 141.243 153.061

263.260 123.621 62.149 34.169 22.141 18.576 19.668 23.194 27.824 32.815 37.814 42.707 47.505 52.271 57.078 67.026 77.559 88.592 99.882 111.140 122.105 132.570 155.754 174.220 188.323 198.851

163.828 75.685 33.067 14.428 7.227 4.791 3.962 3.505 3.126 2.918 3.077 3.776 5.124 7.160 9.865 17.036 25.967 35.925 46.275 56.538 66.390 75.633 95.592 111.233 123.438 133.166

1469.186 773.674 477.046 326.138 238.004 181.829 144.129 118.055 99.726 86.779 77.701 71.493 67.466 65.141 64.168 65.313 69.491 75.828 83.743 92.830 102.788 113.393 141.701 171.299 201.261 231.129

354.380 135.785 52.678 20.597 7.563 2.201 0.530 1.027 2.951 5.829 9.305 13.109 17.040 20.953 24.755 31.825 38.096 43.656 48.688 53.400 57.980 62.580 74.855 88.899 104.892 122.619

220.862 90.805 38.469 17.512 8.655 4.614 2.902 2.725 3.779 5.852 8.734 12.209 16.080 20.177 24.363 32.623 40.358 47.376 53.655 59.259 64.282 68.820 78.588 86.837 94.124 100.734

261.133 127.661 57.382 22.478 7.198 2.494 3.134 6.119 9.787 13.274 16.191 18.414 19.963 20.923 21.405 21.376 20.637 19.725 18.978 18.589 18.656 19.214 22.684 28.645 36.380 45.241

378.020 154.692 76.029 45.081 32.085 28.361 30.868 37.893 48.092 60.362 73.827 87.831 101.900 115.712 129.052 153.855 175.863 195.099 211.786 226.209 238.655 249.387 270.178 284.514 294.395 301.219

211.125 109.133 52.917 23.789 10.041 4.633 3.514 4.452 6.281 8.427 10.633 12.796 14.884 16.893 18.826 22.473 25.813 28.812 31.441 33.691 35.575 37.120 39.776 41.256 42.143 42.848

262.112 119.972 62.515 35.756 21.919 15.672 15.095 18.911 25.907 34.953 45.097 55.604 65.950 75.794 84.934 100.773 113.397 123.238 130.870 136.841 141.608 145.526 153.162 159.348 164.958 170.218

J. Izsa´k / Ecological Indicators 7 (2007) 181–194

m

Minima are bold-faced.

189

190

J. Izsa´k / Ecological Indicators 7 (2007) 181–194

Fig. 1. Graphs showing parameter dependence of correlation between the index value of H0 and Na. Further two graphs relate to parameter dependence of sensitivity profile distances between H0 and Na. Distance data relate to abundance list vð1Þ and wð1Þ. For a better presentation the original distances were transformed uniformly to 0.14 (log(original value) + 7.5).

3. Results 3.1. Parameter dependence of the index correlation The correlation coefficients are given in the second columns of Tables 2–5. In Fig. 1 correlation coefficients between indices H0 and Na are plotted against parameter a for both sets of abundance lists. For practical reasons the horizontal axis is scaled logarithmically. One can observe (see Tables 2 and 3) that by increasing a the correlation between Na and H0 monotonically increases in both cases up to a = 1. Taking into account that the range of the occurring H0 values is relatively small (minimum: 3.119, maximum: 4.137 with moths and minimum: 1.532, maximum: 2.612 with flies), the site and the high value of the maximum correlation is not surprising. Namely, in a small range of H0 values a change in N1 (=exp(H0 )) can be well approximated by a linear function of difference in H0 . According to the results, a relatively good choice of a complementary Na index to be used parallel with H0 may be, say, N0.1, N0.2, N0.3, etc. By no means should be used Na with a above 0.5. Above the value a = 1 the correlation slowly declines. In the range of high parameter values this decline ceases (Fig. 1). For the interpretation of this latter finding see Section 4.

As a by-product of our analyses, we observed that the correlation between N0.5 and H0 is – at least with our study material – quite large, namely 0.902 at the moths collections and 0.857 at the flies. This condition is to be taken into consideration in the practice, even if some authors found that varieties of the diversity index log N0.5 (i.e. Re´nyi’s generalized entropy with parameter 0.5) has some favourable properties (Yue et al., 1998, 2003). In Fig. 2 correlation coefficients between indices H0 and s(m) are plotted against the logarithm of parameter m for both sets of abundance lists. One can observe (see also Tables 4 and 5) that by increasing m the correlation at first monotonically increases, reaching a maximum at m = 35 (moths) and m = 10 (flies). Above those parameter values the correlation monotonically declines. For an interpretation of this latter finding see Section 4. According to the results, a better choice of a complementary s(m) index to be used parallel with H0 may be s(2) than, say, s(10) or s(30), etc. It should be mentioned that data on the correlation between the H0 index and s(m) indices with different m values were published long before by Smith et al. (1979). These authors found a nearly linear relationship between H0 and s(10) and a weaker correlation between H0 and s(200). Our results on the moth statistics confirm this observation, inasmuch as correlation between H0 and s(10) is

J. Izsa´k / Ecological Indicators 7 (2007) 181–194

191

Fig. 2. Graphs showing parameter dependence of correlation between the index value of H0 and s(m). Further two graphs relate to parameter dependence of sensitivity profile distances between H0 and s(m). Distance data relate to abundance list vð1Þ and wð1Þ. For a better presentation the original distances were transformed to 0.026(log(original value) + 7.95) + 0.915 with vð1Þ and 0.035(log(original value) + 8.2) + 0.87 with wð1Þ.

0.960, correlation between H0 and s(150) is 0.955 and the correlation is rather decreasing in the range of high m values. (With the fly collections the use of m = 150 or 200 was impossible because of the smaller number of species.) Note: As both the Na versus H0 and s(m) versus H0 correlation graphs are maximum curves, one could probably search for direct parallelism in their shape, notwithstanding the absence of a direct correspondence between the considered a and m interval. However, one has to keep in mind that in contrast to the s(m) index, in case of Na, going right on the horizontal or parameter axis the index sensitivity decreases (see Section 1). This is an important point of view when discussing the curves’ shape. 3.2. Dependence of the sensitivity profile distances on the index parameter We computed by an appropriate computer program the squared distances between the H0 and Na sensitivity profile given by formula (1) with different values of a for each vðiÞ (i = 1, 2, . . ., 18) and wð jÞ (j = 1, 2, . . ., 15) abundance list. Similarly, we computed the squared distances between H0 and s(m) sensitivity profiles with different values of m also for all lists. The H0 versus Na sensitivity profile distances relating to numerous a parameters are given

in Tables 2 and 3. With the vðiÞ lists (moths) a minimum distance can be observed between a = 0.250 and 0.400 (see Table 2), with a mean 0.339. With the wð jÞ lists (flies) the minimum is located between a = 0.400 and 0.667 (see Table 3), with a mean 0.509, see also Fig. 1. With respect to the numerous abundance lists of flies the relative constancy of the minimum sites is striking. The H0 versus s(m) sensitivity profile distances relating to numerous m parameters are given in Tables 4 and 5. With the vðiÞ lists (moths) a minimum distance can be observed between m = 7 and 14 (see Table 4), with a mean 9.944. With the wð jÞ lists (flies) the minimum is located between m = 7 and 16 (see Table 5), with a mean 8.733. Taking into account the considerable heterogeneity of the abundance lists, the relative constancy of the minimum sites, mainly with the s(m) index, is noteworthy. The minimum distance of the sensitivity profiles is reached when the index sensitivity to changes in small abundances is considerable (Na indices) or moderately large (s(m) indices). In fact, this observation makes it possible to range the sensitivity properties of the Shannon index. For direct observations on these latter see Izsa´k and Papp (2002). For a visual demonstration of the parameter dependence of sensitivity profile distances we plotted after appropriate linear transformation (see the text at

192

J. Izsa´k / Ecological Indicators 7 (2007) 181–194

Fig. 1) the H0 versus Na sensitivity profile distances belonging to the abundance list vð1Þ and wð1Þ against the log-transformed parameter values a. We can observe that despite considerable differences between the abundance lists the basic characteristics of the sensitivity profile distance curves, not influenced by the transformations, are very similar. It is noteworthy that by a horizontal translation of the curves on the log-transformed parameter scale one arrives at curves which are practically counterparts of the correlation curves, expressing a negative parallelism of the parameter dependence of index correlation and (translated) profile distances. Numerous further properties of the profile distance curves can be read off from Fig. 1; Tables 2 and 3. Similar remarks pertain to the H0 versus s(m) profile distance curves belonging to the same two abundance lists. For the transformations before plotting see the text at Fig. 2. The distance curves seem to be more different than the transformed profile distance curves in Fig. 1. However, this is due only to the difference in the mentioned transformation relating to the vð1Þ and wð1Þ data. Yet, the basic characteristics of the H0 versus s(m) sensitivity profile distance curves are very similar. The single exception is the second local minimum in case of the distance curve relating to wð1Þ. Similar second minima frequently occur. This is observable in a reduced form also at the 15th point of the H0 versus Na profile distance curve belonging to wð1Þ; see Fig. 1. For further examples see e.g. H0 versus Na profile distance data belonging to the abundance lists wð5Þ; wð9Þ or H0 versus s(m) profile distance data belonging to the vð14Þ abundance list, etc. (see Tables 3 and 4). We do not discuss here the cause of such minima. Similarly, also the H0 versus s(m) profile distance curves can be translated horizontally to obtain curves which express the above mentioned negative parallelism. Numerous further properties of the H0 versus s(m) profile distance curves can be read off from Fig. 2 and Tables 4 and 5.

4. Discussion Parallel use of diversity indices in ecological studies is a general practice. A typical case is the parallel application of the Shannon index H0 and an index from Hill’s or Hurlbert’s parametric index

family. In these cases it is advantageous if the index chosen correlates weakly with the H0 index. Namely, this condition makes it possible to obtain considerable additional information on the diversity of the community. Therefore it is useful to know how the index correlation depends on the index parameter. We reported above a clear-cut dependence of index correlation with H0 on index parameter a and m. Particularly, by plotting the value of index correlation against parameter a or m, one arrives at typical maximum curves. This finding and other details are useful when searching for appropriate index parameter. We hypothesized negative parallelism between the parameter dependence of sensitivity profile distances and index correlation. According to the results, this negative parallelism would appear only after a translation along to the (log-transformed) parameter scale. The stability of the distance minima is desirable when using profile distance curves to gain information on diversity index correlations. This stability can be observed chiefly at the H0 versus Na profile distance curves. It is instructive to discuss a conspicuous property of the H0 versus Na and H0 versus s(m) correlation curves. In case of the Na index all these curves have horizontal asymptotes if a tends to zero or even infinity. In case of the s(m) indices the similar holds, if m tends to infinity. The signs of this property can be observed in Figs. 1 and 2. Similar horizontal asymptotes exist also in the case of profile distance curves. The mathematical background of these phenomena is outlined in Appendix A. Coming back to the correlation curves, we can determine the parameter values a0 and m0, for which the H0 versus Na and H0 versus s(m) correlation reaches its maximum. From the viewpoint of index correlation the choice of Na0 and s(m0) by a parallel use with H0 is contraindicated. As regards the optimal choice of index parameters, let us consider first the principle that parallel use of one of Hill’s or Hurlbert’s indices with the H0 index is optimal when correlation with H0 is the lowest. On the basis of the shape of the correlation curves, one should keep in mind that both s and s(2) are extreme elements of the index family concerned. In addition, s can be taken for a common element of these index families being lim1 s(m) = s = N0. Further on, the application both of s and s(2) emerges also in a different way.

J. Izsa´k / Ecological Indicators 7 (2007) 181–194

Specifically, s is the species number, that is the most frequently used richness index, and s(2) equals 1 + Gini-Simpson index (see Section 1). Another observation is that the index least correlating with H0 is fully insensitive (the case of N0 = s) or largely insensitive (the case of s(2)) to changes in the abundance distribution. This is characteristic of the index properties of H0 as well. If one prefers the parallel use of indices for which the sensitivity profile is closest to that of H0 , one should choose index lim1 Na = 1/Berger–Parker index or s(2) or, possibly, N0 = s = lim s1(m) (see the graphs in Figs. 1 and 2). This result and the conclusions do not differ significantly from that applying the earlier mentioned principle. We hope that the reported findings will serve as starting points for further methodological investigations.

Acknowledgement The author is grateful to an anonymous reviewer for numerous useful remarks.

Appendix A The existence of horizontal asymptotes informs on certain index properties. Well-known limiting relations are the following: lim Na(a ! 1) = sum z/ z max (=1/Berger–Parker index), where z max is the maximum abundance, and lim Na(a ! 0) = lim s (m) (m ! 1) = number of species (richness index). For illustration, in case of the vð1Þ abundance list the reciprocal of the Berger–Parker index is 1052/ 174 = 6.06 and s = 102 (see Table A1 in Usher and Keiller (1998)). In this case N90.0 = 6.17, N0.05 = 96.9 and s(999) = 100.2. Due to the mentioned limit relations, if a tends to zero or infinity, the index value becomes more and more insensitive to changes in the abundances, except z max in case when a tends to infinity. Then (H0 (z(i)), Na(z(i))) ! (H0 (z(i)), sum z/z max) (a ! 1) and (H0 (z(i)), richness(i)) (a ! 0), respectively, where richness(i) is the species number belonging to the ith member of the abundance set {z(1), z(2), . . .}. Similarly (H0 (z(i)),s(m,z(i))) ! (H0 (z(i)), richness(i))

193

(m ! 1). This stabilizes the H0 versus Na correlation in cases a ! 0 or a ! 1 and H0 versus s(m) correlation in case m ! 1. Further on, due to the mentioned trend of insensitiveness in cases a ! 0, a ! 1 and m ! 1, all partials tend to zero and the sensitivity profile tends to a zero-vector. Consequently, the profile distance tends to 2 s0  X sensðH 0 ; k; zÞ k¼1

H 0 ðzÞ

 z2k ;

see formula (1) in the text. This is why the profile distance curves tend to corresponding horizontal asymptotes.

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