OIKOS 76: 70-82. Copenhagen 1996 A consumer's guide to evenness indices Benjamin Smith and J. Bastow Wilson Smith. B. and Wilson, J. B. 1996. A consumer's euide 10 evenness indices. - Oikos 76: 70-82. Many indices have been proposed for measuring species evenness in ecological communities, bul there is no consensus on which arc best. We assemble criteria for an appropriate index for the evenness of a biological sample. The most important criterion is that evenness should be independent of species richness. Twelve previously proposed indices and variants are considered, and two apparently new indices. Four indices are recommended as joint best buys: A) If symmetry between minor and abundant species is not important, or if it is required that the index be less alfectcd by minor species: 1) If it is essential that the index be able to reach a minimum of zero with any particular number of species, or if the shape of the index response to an evenness gradient is important: £, D (based on a common form of Simpson's index). 2) If good mid-range behaviour is desired: E' (proposed by Camargo). B) If equal sensitivity to minor and abundant species is required: 1) If the shape of the index response to an evenness gradient is not important, the clear winner is: Eg (a new index). 2) If the shape is important: £,.„, (another new index). The overall recommendation for general use is Ecar. B. Smith and J. B. Wilson (correspondence). Bor any Depi, Univ. of Orago, P. O. Box 56, Dunedin, New Zealand. A basic feature of biological communities is the distribution of abundance among species. There are many aspects of this distribution that can be measured, but the simplest feature is evenness'. A community in which each species present is equally abundant has high evenness: a community m which the species differ widely in abundance has low evenness. The concept is related to that of species diversity. It is generally understood that species diversity can be split into two components: species richness (the number of species in the sample) and species evenness (Pielou 1977). To make sense, these two components have to be independent, i.e. evenness has to be unaffected by richness (Heip 1974). The term 'equitability' has sometimes been used as if it were a synonym of "evenness'. As Cotgreave and Harvey (1994) point out. this is a solecism. Many indices of evenness have been proposed. For the measurement of species diversity, there is some consensus towards using the Shannon-Weiner index H' (Shannon and Weaver 1949) or a variant of the Simpson index D (Simpson 1949). In contrast, no consensus has emerged on which index of evenness should be used, and new indices continue to be proposed (Molinari 1989, Camargo 1993, Nee et al. 1992, Bulla 1994). Lack of knowledge of the properties of the various indices has been suggested as a limitation to their ecological usefulness (Alatalo 1981). This lacuna is surprising because, unlike diversity, there are some clear criteria which an evenness index should meet. We here test fourteen evenness indices against a range of requirements and of desirable features. We do not attempt to test all the indices ever proposed. Rather, we include indices in common use, those recently proposed as improvements over commonly used Accepted 30 October 1995 Copyright C OIKOS 1996 ISSN 0030-1299 Printed in Ireland - all rights reserved 70 OIKOS 76:1 (1996) ones, and some thai appear to have particular desirable features. We propose two new indices. Indices Let: 5 = the number of species in the sample xs = the abundance of the ith species E*= I x. Ps=Xs/Y,X H' = - I pt Info) s= i [The latter is the Shannon-Weiner diversity index. Pielou gave two diversity indices: H for 'completely censused communities', but of individuals, and H' for a sample. According to Pielou, the latter should ideally be corrected for the 'total number of species in the whole community'. This does not seem realistic. We therefore follow normal usage, calculating H' as above.] ß= £/>; s= 1 [This is the Simpson 'dominance' index.] Wc consider these indices: /=- H' HŠ) Pielou (1975) based this index on the Shannon-Weiner diversity index, H'. The division by ln(5) is intended to compensate for the effect of species richness on H'. c-IMp ~ e'r-l 5-1 Heip (1974) proposed this index to overcome a problem with previous indices, of dependence on S and failure to attain a low value when evenness is obviously low. Heip gave it no symbol, referring to it as "the proposed new index'". Since he uses the symbol E for all evenness indices, we use the name EHelp. Ei-d — \___D_ 1 - 1/5 The widely-used index of species 'dominance' D (Simpson 1949) is a natural starting point for an index of evenness. Simpson's index has been converted into an index of species diversity in various ways. The most common transformation is to use the complement of D (i.e. 1 - D) as the diversity index (Pielou 1969). Krebs (1989) points out that for continuous data, or with a very large number of discrete records, the maximum value of D is 1/5. This leads to the evenness index above. E -O» Williams (1964) used \jD to convert Simpson's index of 'dominance' into an index of diversity. The equivalent measure of evenness is as above. In/) In S Pielou (1977) advocates the use of —In D as a diversity index. The corresponding index of evenness is as above. fn- l./ZJ— 1 Hill (1973) proposed evenness index £2.i (not considered here). Fu is a modification of it, intended to give a better approach to intuitive evenness (Alatalo 1981). G2_, = If F2J > yi/2 then F2_, 0.636611 arcsine F2 , Else: Fl, [arcsine is assumed to provide an angle in radians.] Molinari (1989) proposed this index to give a linear response to an artificial 2-species dataset proposed by Alatalo (1981). 0= Y, minimum^, 1/5) i- i This index had been used as a measure of community similarity or of niche overlap. Bulla (1994) proposed using it as a measure of evenness. E = - O-MS 1 - 1/5 where O is as above. The index was proposed to bring O to a 0-1 range (Bulla 1994). E/Hcl - / s Yx- t x? E.v-X*\/5 Pielou (1969) proposed this as a conversion of Mcintosh's (1967) index of species diversity into an index of species evenness. OriCOS 76:1 (1996) 71 E'=i- i í Ifc,-Aal/S >i - i O _tl t i Proposed by Camargo (1993). £„„=1-2 i ti arctan j £ (\a(xs)- f ln(;t,)/s) I S> [arctan is assumed to provide an angle in radians.] We propose this as a new index. It is based on the variance in abundance over the species, an intuitive way to measure evenness. This variance is taken over log abundances, to examine proportional differences, and to ensure the index is not dependent on the units used. The variance is then converted by —I'm arctan() to a 0-1 range, with 0 representing the minimum evenness, and 1 the maximum. NHC=b where: b = the slope of log abundance on the rank of abundance, fitted by least-squares regression (i.e. the slope of the Dominance, Diversity curve: Wilson 1991a). The index, proposed by Nee et al. (1992). ranges from -co to 0. Therefore, whilst higher (i.e. less negative) numbers indicate greater evenness, the values are all negative, save zero for perfect evenness. It would be possible to improve the index by transformation to a 0-1 scale. We do not make such a transformation, because the index has other problems (see below). Eg= —2/n arctan(ft') where: b' = the slope of the scaled rank of abundance on log abundance, fitted by least-squares regression, [arctan is assumed to provide an angle in radians.] The ranks are scaled before the regression is fitted, by dividing by the maximum rank, thus giving rank 1.0 for the most abundant species, and rank l/S for the least abundant. The slope is the inverse of the usual dominance/diversity slope, because only thus is independence from species richness achieved. The transformation converts it to a 0-1 range. It would be possible to improve the response to a gradient of evenness (Feature 12, below) by taking the square root of the index, but this has disadvantageous side effects (see Conclusions). We propose this index because it combines the ready graphical interpretation of index .\'HC, with independence from species richness, and a 0-1 range. Criteria, tests and results We assemble, from the literature and from first principles, a number of criteria that an evenness index should meet. Some criteria are more important than others. We divide ours into Requirements (i.e. essential fea- tures). Features (i.e. those that are desirable), and one that has been advocated but which we reject. Some authors (e.g. Pielou 1975) have distinguished between a "fully censused community' and a "sampled community". We believe the concept of a sample from a community is unrealistic, because it assumes that communities have reality as discrete units, which very few ecologists believe (Underwood 1986, Wilson 1991b. 1994. Palmer and White 1994). We prefer to see the quadrat or sample as a small, fully-censused piece of biotic space at a particular scale. Those who prefer to make the distinction should take our study as referring to the evenness of samples, not communities. Requirement 1: Independent of species richness Requirement: The index must be independent of species richness. Splitting diversity into two components — richness and evenness (Pielou 1977) - logically requires that the value of evenness is not affected by the species richness of the community (i.e. there should be a horizontal line in Fig. I). Test: 1.1a: 1479 1 I I (representing abundance values for four different species) should have the same evenness as all multiples of itself, e.g.: 1.1b: 1479 1 1 1 1479 1 1 I 1.1c: 1479 1 1 1 1479 1 1 1 1479 1 1 1 etc. (We test for 2, 3, 5, 10 20 and 40 repetitions of the 4-species sequence). Similarly: 1.2a: 800 400 200 100 should have the same evenness as all multiples of itself, as should: 1.3a: 378 376 374 372 Results: EMcl and NHC fail badly, the value of the indices changing considerably as S changes (Fig. 1). £//<•//>• &u> G2.1 and £ fail below about 5= 10. /', £"[ _2> and E_taD fail up to 5 = 25 or higher. Only indices El/D, O, £", Eair and EQ pass. Requirement 2. Decreased by reducing marginally the abundance of the most minor species Requirement: The index must decrease when the abundance of the most minor (i.e. least abundant) species in a community is marginally reduced (this is requirement "Rl" of Routledge 1983). 72 OlKOS 76:1 (19961 1.0 0.8 o •i 0.6 > ■g 0.4 0.2 0 0 J' 25 50 75 1.0 0.8 0.6 0.4 0.2 0 Enelp 25 50 75 --In D 2,1 25 50 75 -Mel NHC 25 50 Number ot species (S) 25 50 Number ot species (S) 75 OIKOS 76:1 (1996) 73 -1-D -1/0 1.0 0.8 0.6 0.4 0.2 0 25 50 75 1.0 0.8 0.8 0.4 0.2 0 25 50 75 "2,1 1.0 0.8 0.6 C.4 0.2 0 E' 0 25 50 Number of species (S) 75 1.0 0.8 0.8 0.4 0.2 C 25 60 25 50 Number of species (S) 75 75 Key: Series 1.1 Series 1.2 Series 1.3 Fig. 1. Change in evenness index value with increase in species richness, true evenness being held constant, for 74 C Test: 2a: 80 40 20 10 1 should have a higher index value than: 2b: 80 40 20 10 0.5 Results: F2A and G2A fail because 2a has a lower value (Table 1). All other indices pass. Requirement 3. Decreased by lite addition of a very minor species Requirement: The index must decrease when a very minor species is added to a community (this is requirement 'R2' of Routledge 1983). Test: 3a: 80 40 20 10 should have a higher value than: 3b: 80 40 20 10 0.5 Results: All indices pass (Table 1). Requirement 4. Unaffected by the units used Requirement: The index must be unaffected by the units (e.g. kg, g, mg) used to measure abundance, i.e. it should examine proportional differences, not absolute ones. [Of course, a different measure of abundance, e.g. frequency instead of biomass. will usually give a different index value.] Test: 4a: 1 2 3 should have the same value as: 4b: 100 200 300 Results: All the indices pass this test (results not shown). Feature 5. Maximal when the species abundances are equal Feature: The index should be maximal when the species abundances are all equal. Test: 5: 375 375 375 375 should give the maximum value Results: All the indices considered pass this test (Table 1; cf. Fig. 2). Feature 6. Maximum value 1.0 Test: 6: 375 375 375 375 should give the value 1.0 Results: All indices except NHC have maximum values of 1.0 (see the values under 'Feature 5' in Table 1). NHC has a maximum of 0, which is at least well defined. Feature 7. Minimal, for any number of species, when the species abundances are as unequal as possible Feature: The index should be minimal, for any particular number of species, when the species abundances are as unequal as possible. [If we are dealing in a measure of abundance that has integer values (e.g. number of individuals), it is possible to consider a finite minimal evenness. In the more general case of continuous data (e.g. biomass), minimal evenness is an extreme that can never be reached, but the index can asymptote to it.) Test: The series: 7a: 999 1 7b: 900 100 7c: 800 200 7d: 700 300 7e: 600 400 7f: 500 500 should approach its lowest evenness value at the lower (7a) end. Results: AH the indices pass this test (Fig. 2). Feature 8. A value close to its minimum when the community is as uneven as we would be likely to meet Feature: The index should show a value close to its minimum when the community is as uneven as we would be likely to meet (i.e. unrealistically uneven communities should not be necessary before the index value is low). Test: 8: 1497 1 I 1 should have a value close to its minimum. For an index with a minimum of 0.0 we interpret 'close to its minimum' arbitrarily as less than 0.05. Results: The values for EllD, F2.i- O, £" E,.„r and NHC, are too large (Table 1). We count the result as 'poor', not 'fail', when the value is <0.1. It is difficult to use this criterion for NHC because it is not on a 0-1 scale; to be generous we count it as 'poor'. OIKOS 76:1 (1996) 75 Table I. Results for fourteen evenness indices on tests 2, 3. 5, 8-11. 13 and 14. Requirement 2 Requirement 3 2a 2b 3a 3b r 0.726 0.718 pass 0.820 0.718 pass E Help 0.555 0.544 pass 0.706 0.544 pass £| O 0.784 0.781 pass 0.830 0.781 pass EUD 0.536 0.533 pass 0.662 0.533 pass £-i.i n 0.613 0.609 pass 0.702 0.609 pass F2., 0.758 0.765 fail 0.778 0.765 pass C2.1 0.415 0.425 fail 0.441 0.425 pass O 0.605 0.603 pass 0.700 0.603 pass 1: 0.507 0.503 pass 0.600 0.503 pass "Mel 0.704 0.701 pass 0.771 0.701 pass E 0.502 0.498 pass 0.617 0.498 pass E„„ 0.264 0.200 pass 0.656 0.200 pass NHC -1.015 -1.154 pass -0.693 -1.154 pass Eq 0.113 0.095 pass 0.220 0.095 pass Feature 5 Feature 8 Feature 9 Feature 10 Feature 1 11a 0.923 J' 1 pass 0.012 pass 0.000 pass 0.0IM) pass ť'jleip [ pass 0.006 pass 0.000 pass 0.000 pass 0.865 £\—D 1 pass 0.005 pass 0.000 pass 0.000 pass 0.933 &UD 1 pass 0.251 fail 0.000 pass 0.000 pass 0.833 *^-ln/> 1 pass 0.003 pass 0.000 pass 0.000 pass 0.868 ^.i 1 pass 0.239 fail 0.000 pass 0.000 pass 0.899 C2.1 1 pass 0.014 pass 0.000 pass 0.000 pass 0.639 0 1 pass 0.252 fail 0.000 pass 0.500 poor 0.800 E 1 pass U.0U3 pas. 0.000 pass 0.000 pass 0.733 E Mel 1 pass 0.004 pass 0.000 pass 0.000 pass 0.905 E' 1 pass 0.252 fail 0.000 pass 0.500 poor 0.750 E„, 1 pass 0.063 poor 0.000 pass 0.000 pass 0.832 NHC 0 poor -2.193 poor — oo fail - / pass - 0.456 Ee 1 pass 0.043 pass o.ooo pass 0.000 pass 0.308 Feature 13 Feature 14 lib 13a 13b 14a 14b 14c 0.820 fail 0.794 0.017 fail 0.022 0.618 0.899 pass 0.706 poor 0.669 0.008 fail 0.006 0.614 0.898 pass 0.830 fail 0.889 0.008 fail 0.012 0.801 0.960 pass 0.662 poor 0.751 0.252 fail 0.008 0.405 0.802 pass 0.702 poor 0.793 0.004 fail 0.168 0.501 0.834 pass 0.778 Tail 0.997 0.251 fail 0.250 0.991 0.999 pass 0.441 pass 0.948 0.016 fail 0.016 0.907 0.964 pass 0.700 poor 0.750 0.253 fail 0.172 0.501 0.834 pass 0.600 pass 0.667 0.004 fail 0.006 0.401 0.800 pass 0.771 fail 0.846 0.006 fail 0.008 0.715 0.934 pass 0.617 pass 0.750 0.253 fail 0.172 0.501 0.834 pass 0.656 poor 0.071 0.071 pass 0.095 0.053 0.095 poor 0.69.3 pass -2.072 -2.072 pass -0.987 -1.776 -0.987 poor 0.220 poor 0.046 0.046 pass 0.046 0.046 0.046 pass "Help 999 900 800 700 600 500 400 300 200 100 1 1 100 200 300 400 500 600 700 800 900 999 999 900 800 700 600 500 400 300 200 100 1 100 200 300 400 500 600 700 800 900 --In D 2,1 999 900 800 700 600 500 400 300 200 100 1 I 100 200 300 400 500 600 700 800 900 999 1 On 999 900 800 700 600 500 400 300 200 100 1 1 100 200 300 400 500 600 700 800 900 999 "Mel 1.0 n 1.0 n 999 900 800 700 600 500 400 300 200 100 1 1 100 200 300 400 500 600 700 800 900 999 999 900 800 700 600 500 400 300 200 100 1 1 100 200 300 400 500 600 700 800 900 999 NHC 999 900 800 700 600 500 400 300 200 100 1 1 100 200 300 400 500 600 700 800 900 999 999 900 800 700 600 500 400 300 200 100 1 1 100 200 300 400 500 600 700 800 900 999 (IW6) Species abundances Species abundances E1-D E,/D i 900 800 700 600 500 400 300 200 100 1 1 100 200 300 400 500 600 700 800 900 999 0.8 - 3.2- 999 900 800 700 600 500 400 300 200 100 1 1 100 200 300 400 500 600 700 800 900 999 G2,1 O 999 900 800 700 600 500 400 300 200 100 1 1 100 200 300 400 500 600 700 800 900 999 '.C -0.8-0.6-0.4-0.2- 999 900 800 700 600 500 400 300 200 100 1 1 100 200 300 400 500 600 700 800 900 999 1 O-i 0.8-0.6-0.4-0.2- 1.0-1 999 900 800 700 600 500 400 300 200 100 1 1 100 200 300 400 500 600 700 800 9O0 999 Species abundances 999 900 800 700 600 500 400 300 200 100 1 1 100 200 300 400 500 600 700 800 900 999 Species abundances Fig. 2. Change in evenness index value along an intuitive gradient of evenness, for 14 indices. 78 OIKOS 76:1 (1996) Feature 9. Minimum value O Feature: The minimum possible index value (not necessarily with a particular number of species) should be 0. Test: Increasingly extreme uneven communities were used, to see whether the index approached 0. Results: All indices pass, except for NHC which has a minimum of — x (Table 1). Feature 10. Minimum attainable with any number of species Feature: The minimum value of the index should be reachable with any particular number of species. Test: 10: x 1 should give the value 0.0. Results: O and E' fail this test, all others pass (Table 1). Since we are not sure that it should conceptually be possible to obtain minimum evenness with only two species, we count these as 'poor', not 'fail'. Feature 11. A value in the middle of the scale for communities that we would intuitively consider intermediate Feature: The index should show a value in the middle of the scale for values that we would intuitively consider intermediate. Test: 11a: 600 450 300 150 response. His ideal is a straight-line response up to maximum evenness, and down again (e.g. index G2,, in Fig. 2). However, this entails a discontinuity at maximum evenness, where changes in species abundance that we feel should affect evenness little have a considerable effect on the index. We therefore consider a convex curve more desirable (e.g. index J' in Fig. 2). We shall refer to the shape of the curve in Fig. 2 as the 'Molinari shape". Test: For the sequence of Molinari (1989): 12a: 999 1 12b: 900 100 12c: 800 200 12d: 700 300 12e: 600 400 12f: 500 500 a convex curve (e.g. Fig. 2, J') seems preferable to a broken linear response (e.g. Fig. 2, G2a). Results: Ea has poor response to Molinari's sequence (Fig. 2). G2.i, O, E and £" have shapes that Molinari himself would consider excellent, but which we believe arc undesirable. NHC changes too rapidly at low evenness, and also comes to a peak. J', Elh.lp, E, D, EhD, E-)nD< Fi.u EMci and Evar have shapes that are ideal by our judgement. lib: 800 400 200 100 should both have intermediate values. We arbitrarily take 'intermediate values' to mean in the range 0.25 to 0.75. Results: J'. E{_D, F2A and EMc/ and have values that are too high for both 11a and lib; we count these results as 'fail' (Table 1). EHl,ip, Ei/D, £_,„,>, O and Evar, are too high only for 11a, and EQ is too low for lib; we count these as 'poor'. It is difficult to define intermediate values with \'HC. because it does not have a 0-1 range. Feature 12. Respond in a reasonable way to a series of communities that intuitively changes in evenness Feature: The index should respond in a reasonable way to a series of communities that changes in evenness (Fig. 2). We use a series proposed by Alatalo (1981). This is Molinari's (1989) major requirement. However, we disagree with Molinari's concept of the perfect Feature 13. Symmetric with regard to minor and abundant species Feature: The index should be symmetric with regard to minor and abundant species; i.e., a community with several abundant species and one minor one should have the same evenness as one with several minor species and one abundant one. This criterion was given by Pielou (1975). who said: "To estimate evenness entails giving as much weight to observations on minor species as to observations on abundant ones". It is possible to argue that: 1000 1000 1000 1 is more even than: 1000 1 1 1 or that it is less even. We do not accept either argument, and we therefore take symmetric behaviour as desirable. OIKOS 76:1 (1996) 79 Test: 13a: 1000 1000 1000 I should give the same value as: 13b: 1000 1 1 1 Results: Most indices fail this test (Table 1). Only Evll„ NHC and Eg pass. Feature 14. Skewed distributions- should give a lower value Feature: The index should take a low value when the distribution of species abundances is very skewed. Test: 14a: 1000 11111 14b: 1000 1000 1000 1 1 1 14c: 1000 1000 1000 1000 1000 1 Our perception is that 14b is more even than 14a and 14c. However, we are willing to accept contrary opinion. Results: Our expectation that the evenness index for 14b should be greater than that for 14a is true for /', Eh„p, E, -p, E\id, E_inD, F2A, G2.i, O, E, EMcl and E' (for all of these, the conclusion is opposite comparing 14b with 14c, because of the response to minor and abundant species discussed in Feature 13 above). Ecar and NHC indicate 14b as having the lower evenness, compared to either 14a or 14b: we rate this as 'poor" rather than 'fail' because it is arguable which is the correct response. The three communities give identical values of E0, a response we can readily accept. There is one criterion that has been mentioned in the literature, but which we exclude: Routledge (1983) suggested that "Evenness should depend continuously on the proportional abundance of any species". He meant that when a very minor species was removed from a community, the evenness index should be little changed. We do not accept this argument. As Pielou (1975) observed, minor species are of equal relevance for evenness as abundant ones. An evenness index is calculated from the species that are present. The loss of the least abundant species changes the distribution of abundances among those species (e.g. it changes the range and the mean), and therefore inevitably changes the index of evenness. We therefore give no weight to this criterion, and do not test it. Conclusions The evenness indices that we compared often produce very different values. For example our test data 14c produced values from different indices ranging from 0.046 to 0.999 (on a 0-1 scale). Heip (1974) commented: •"Many [evenness] indices have been proposed, to such an extent that the choice of a suitable index became somewhat of a problem". Several further indices have been proposed since Heip wrote, and there has been little guidance on which to use. We hope we can provide that. Our foremost criterion has been that the index be independent of species richness. It is not absolutely clear how an index should behave to qualify for being "independent of species richness', but we believe our Requirement 1 is the only sensible criterion. Cotgreave and Harvey (1994) seemed quite unconcerned about the need for independence from species richness: "It is not, therefore, surprising that species number is a significant correlate of evenness". Their conclusion arose because all three of the indices of "evenness" that they used, D, Q and NHC, were actually indices of diversity (or, in the case of D, its complement). D of Simpson (1949) has never before been claimed to be a index of evenness. Q of Kempton and Taylor (1976) was proposed as an index of diversity, not evenness, and it is in fact an index of diversity, not evenness. Nee et al.'s (1992) new index ('NHC'), although proposed as an index of evenness, is really an index of diversity; indeed it is quite similar to the diversity index Q. The effect of species richness on evenness indices has been discussed before, with some workers observing the problem but being unsure how to solve it. DeBenedictis (1973) showed that there is a necessary mathematical connexion between the widely-used J' and species richness. Several other indices also fail the requirement of independence of species richness, including the recently-recommended Fu, G2,i and E. In spite of being proposed to overcome dependence of evenness on species richness (Heip 1974), E,,eip is still dependent on it. Kvälseth (1991) claimed that FZA was "quite unaffected by [species richness]", but according to our analyses that is not true. Only five indices pass Requirement 1, independence of richness: Emr, O, Eq, E' and El/D (Fig. 1, Table 2). All five meet the other Requirements (2-4). All have some other characteristics that we rate as problems (Table 2). O has many good features, but all these are matched by £" (Table 2), which in addition has intermediate index values for intermediate evenness (Feature 11). E' has very similar strengths and weaknesses to E,,D, the weaknesses being that they fail to reach values close to zero for quite uneven communities (Feature 8) and that they are asymmetric with regard to minor and abundant species (Feature 13). £' has the advantage of good mid-range behaviour (Feature 11); it is let down in comparison with £, -„ by failing to achieve a minimum of zero with small numbers of species (Feature 10). and in having a Molinari shape different from our ideal 80 OIKOS 76:1 (1996) Table 2. Summary of the results of testing evenness indices. Key: ,7 = good, J = poor, ■ = fail. Requirement Feature ■Hr: r Index /' EH £1 Eyo £-lni) O Ľ £" NHC 1 2 3 4 _ _ 3 v, ^ \ v. v. L □ ■ v 2 v/ \A r/ v, V f v ■ V, / vŕ v. v y, v, . v, v, í V v' V v v i v v, v. í v, v, 3 2 7 V v v \ v; j, j, 10 II 12 13 14 7 v/ v/ v. v, v, -J. g ^ V v^ n/ ■ V, ■• ^ v6 Ľ V, V V', \/, D V, V VÍ V v/ & ■ J Ľ D D ■J, t J. J O C V V,. □ v/ S u u v, r (Feature 12). Both of the latter features could be considered desirable by other judgement, and if so then E' can be recommended. f,,, has the advantage that it can reach the minimum of zero with any particular number of species (Feature 10), and that it has an excellent Molinari shape (Feature 12). EyD also has a particular advantage in its relation to diversity. Informally, diversity can be decomposed into richness and evenness. Usually, no such formal equation can be made. However, E,,„ is formally related to diversity index \jD (a form of the well-known Simpson index of 'dominance') by the simple relation: l/D = El/DxS Any evenness index can be multiplied by S to obtain an index of species diversity (Bulla 1994); the difference is ;'::al in '."::.' case of £", ,, a standard index of diversity is recovered. The other two indices that are independent of S (Requirement 1), EQ and Evar, are both symmetric to degrees of abundance, and are the only choices if this feature is required. Symmetry of response to abundance is a controversial question. Alatalo (1981) suggested symmetry was an undesirable feature, because the minor species in a community are likely to be inaccurately estimated, or even missed altogether. However, this depends on the type of sampling; with plant biomass sampling the minor species can sometimes be estimated more accurately than the abundant ones. Moreover, there seems no theoretical justification for asymmetry, minor species contain as much information on evenness as abundant ones do (Pielou 1975). Eg has a perfect scorecard (Table 2) except for its response to Molinarľs intuitive scale of evenness (Feature 12). In that test it achieves values appreciably-above zero when the distribution is still quite uneven (Fig. 2), and also falls away from 1.0 very rapidly when the distribution is only slightly uneven. The latter prob- lem can be cured by taking the square root of the index, but then the former problem is exacerbated. In contrast, Emr has an excellent response to Molinarľs test data. It is the only index with no severe problems (Table 2), but it has several minor disadvantages: it does not reach a low value until species abundances are very uneven (Feature 8), it has too high a value for distributions that seem to us intermediate (Feature 11), and it has a higher evenness than we would expect for highly skewed distributions (Feature 14). Alatalo (1981) commented that "'there is no single way to measure evenness". This is true. Our recommendations are: If symmetry between minor and abundant species is not required: If a minimum of zero with any number of species is essential, or a good Molinari shape is required: Elr> If good mid-range behaviour is desired: E' If symmetry between minor and abundant species is required: If Molinari shape is not important: Eg If the Molinari shape is important: E,.ar Our overall recommendation, for general use. is £,,,„. References Alatalo, R. V. 1981. 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