**Figure 6.**
Sensitivity of evenness measures to ultra-rare species for a variety of measures. The initial community consists of two species, each with abundance 100,000. From left to right, vanishingly rare species are added, one at a time. Each vanishingly rare species consists of one individual.

#### 8.2. Partitioning Higher-Order Diversity Measures

I derived evenness and inequality measures by partitioning species richness S (which is

^{0}D of course) into independent diversity and inequality components. This dooms us from the start if we are worried about statistical reliability of our measures, since it is based directly on the difficult-to-estimate value of S. Many authors have trued to avoid this problem by partitioning not

^{0}D but

^{1}D, which can be accurately estimated from small samples [

25].

^{1}D can be partitioned into any higher-order diversity and an independent “inequality” component. The most logical higher-order diversity to use is

^{2}D, so that

This implies that

and its reciprocal is a measure of “evenness”,

This evenness measure was introduced by Hill [

10] as his E

_{1, 2}.

Unfortunately Hill’s evenness factors based on higher-order diversities have a fatal flaw. They can equal unity in two contradictory circumstances. If the assemblage is completely even, then ^{1}D = ^{2}D and the evenness is unity, as it should be. If the assemblage is extremely uneven, so that the diversity profile was very steep around q=0, then the profile would also be nearly horizontal for q ≥ 1. This would cause ^{1}D and ^{2}D to be nearly equal, and the evenness would again be close to unity, even though this assemblage is maximally uneven. These measures are therefore non-monotonic with respect to increasing evenness. For example, the highly uneven assemblage whose species frequencies are [0.999, 0.001] has an “evenness” EF_{1, 2} equal to 0.9941, close to unity. The more even assemblage (according to the Principle of Transfers) with frequencies [0.9, 0.1] has a lower “evenness”; EF_{1, 2} = 0.881.

The concept of a relative measure of evenness, discussed above, also applies to this higher-order “evenness”. Given some observed value for

^{1}D, the minimum possible value for the evenness factor

^{2}D/

^{1}D is 1/

^{1}D, since the minimum possible value of

^{2}D is 1. The maximum possible value is unity for the perfectly even community (since then

^{2}D =

^{1}D = S). Using the standard transformation (x − x

_{min})/(x

_{max} − x

_{min}), we obtain the higher-order relative evenness RE

_{1, 2}:

This modification of Hill’s E

_{1, 2} had been proposed by Alatalo [

26].

Earlier we saw that it was better to form a relative evenness measure out of ln EF

_{0, 1} instead of EF

_{0, 1} itself. We do the same thing here, transforming ln EF

_{1, 2} into a relative measure using (x − x

_{min})/(x

_{max} − x

_{min}):

This has a direct graphical interpretation in terms of the Renyi spectrum, much like the relative logarithmic evenness based on EF

_{0, 1}.

These relative higher-order evennesses RE_{1, 2} and RLE_{1, 2 } are apparently not affected by the problems of the absolute evenness EF_{1, 2}. The relative evenness RE_{1, 2} of [0.999, 0.001] is 0.25, and the relative evenness RE_{1, 2} of [0.9, 0.1] is 0.57. The relative logarithmic evenness RLE_{1, 2} of [0.999, 0.001] is 0.25 while the RLE_{1, 2} of [0.9, 0.1] is 0.61. These measures correctly show that relative evenness of the second community is greater than the relative evenness of the first community. However, to my knowledge, the relation between these measures and the Lorentz partial ordering is not yet known.

#### 8.3. An Estimable Evenness Measure

One way to improve the estimation of the evenness indices derived in this paper would be to improve the estimation of S. There are many reviews of this subject, and many excellent nonparametric estimators, such as the Chao estimators [

27]. These should always be used rather than the observed sample value of S, if the richness of a population is estimated by taking incomplete samples.

However, these nonparametric estimators generally provide only lower bounds for the population value of S [

28]. There is no guarantee that there are not some very rare species dispersed through the ecosystem in densities so low that they will never be detected through normal sampling. This makes the true value of S an unknowable quantity. It is difficult even to quantify the uncertainty in a particular estimate of S without making parametric assumptions. On the other hand, if some species are so rare that they are impossible to detect, then they are also so rare that they make little difference to the day-to-day functioning of the ecosystem. Why not forget about them and satisfy ourselves with characterizing the bulk of the population?

One approach, also used in estimating S, is to standardize on a particular sample size N, and estimate the mean evenness of a sample of that size. This would be done by repeatedly rarefying a larger sample down to the standard size. However, sample sizes that are sufficiently large to characterize a low-diversity community will often not be large enough to characterize a high-diversity community.

Furthermore, sampling to a fixed size does not preserve the important theoretical properties of a diversity measure like S. Diversities follow the replication principle, so if we pool two equally large populations with richness S, and with identical species frequencies but no shared species, the pooled population will have richness 2∙S, twice the richness of either of the original populations. However, the richness of a sample of fixed size taken from the pooled population will not be twice the richness of a sample from one of the original populations. Sampling strategies should preserve, as much as possible, the mathematical properties of the measure being estimated, and sampling at a fixed, standardized sample size does not do this. Instead of sampling at a fixed size, we need an adaptive approach to choosing the sample size.

The concept of “sample coverage” was introduced by Good [

29] and Good and Toulmin [

30] and underlies many nonparametric estimation techniques [

28]. The sample coverage is the proportion of the population belonging to sampled species. For example, suppose the true population frequencies of the species in an ecosystem are {0.5, 0.3, 0.18. 0.02}. Suppose we make a sample of size N and we find Species 1, 2, and 3, but not Species 4. The coverage of our sample is 0.98, because the species in our sample make up 98 % of the population. The species we have not sampled will represent individuals that make up about 2% of the population, and these can be ignored.

The sample coverage serves as an adaptive “stopping rule” for choosing sample size [

28,

31]. The mean relative evenness of a sample that gives, say, 95% coverage is a well-defined number that can be estimated with precision. The possible presence of nearly undetectable ultra-rare species simply has no effect on this number. The number will measure the relative evenness not of the population but of a standardized percentage of the population.

It may seem that in order to estimate this number, we would need to know the complete species list and the true population frequencies of each species, so that we would know when our sample reached 95% coverage. However, Good [

29] discovered a simple way to estimate sample coverage without knowing anything about the population. The sample coverage C is approximately equal to

where f

_{1} is the number of singleton species in the sample (the number of species represented by exactly one individual in the sample). This estimate is most accurate when f

_{1} is large. If we wanted to estimate the richness of a community at 95% coverage, we would keep sampling until C = 0.95, and then measure the evenness of the sample. More accurate would be to make a sample that exceeds 95% coverage, and repeatedly rarefy it down to 95% coverage, averaging the evenness of each rarefied sample.

The richness at fixed coverage, unlike the richness at fixed sample size, will approximately obey the replication principle. Suppose a community has relative abundances [0.4. 0.4, 0.1, 0.05, 0.025, 0.025]. If this is sampled at 95% coverage, on the average the observed richness will be 4 (the four most common species make up 95% of the population). If a replicate community is added to this one, the relative abundances of the new community will be [0.2. 0.2, 0.2, 0.2, 0.05, 0.05, 0.025, 0.025, 0.0125, 0.0125, 0.0125, 0.0125]. Now the eight most common species make up 95% of the population, so the most likely observed richness at 95% coverage will be about eight, double the observed richness of the original community.

Richness estimated in this way will depend very much on the choice of coverage chosen. The best way to facilitate comparison with the results of others is to make a rarefaction curve based on coverage values instead of the usual sample sizes (Anne Chao, pers. com.). These rarefaction curves for different communities may intersect, just like rarefaction curves based on sample sizes.

The richness and diversity at a given coverage can be used in the formulas for evenness and relative logarithmic evenness. The resulting measures should not be considered as estimates of true population values of the parent measures, but as valid descriptive measures in their own right. They will approximately share the theoretical properties of their parent measures. While the true population values of the parent measures are virtually unknowable without a complete census, their values for a sample with coverage X can be reliably estimated and meaningfully compared across communities, resolving the problem of sensitivity to S inherent in these measures.