ENS rarefaction and related
approaches
Our approach relies on a family of diversity measures that was first
introduced as “Hurlbert ENS” by Dauby and Hardy (2012). Here, we use
the term “ENS rarefaction” to emphasize that these measures are simply
an effective number of species (ENS) transformation of the
individual-based rarefaction curve (Hurlbert, 1971). Since ENS
rarefaction is one of the lesser-known, but quite powerful, families of
diversity measures, we briefly explain it below and compare it with the
related Hill number framework, and the individual-based rarefaction
framework that it is based on (see Table 1).
Relating the complementary information given by a set of diversity
measures to the diversity components discussed above is challenging
because many metrics are sensitive to more than one component (Chase and
Knight, 2013). Furthermore, diversity metrics often differ in their
numerical ranges and units (i.e. their numerical constraints), and in
the degree to which they are affected by passive sampling effects, which
in statistics is called estimation bias (Gotelli & Chao, 2013). For
example, species richness, which counts all species independent of their
abundance, can attain any integer number, and is strongly affected by
the number of individuals in the sample. In contrast, Simpson’s index,
which gives disproportionately high weight to the dominant species of
the SAD, ranges between 0 and 1 and is almost unaffected by sample size
(i.e., the number of individuals). Although the two metrics hold
complementary information on the SAD and passive sampling effects, their
different numerical constraints and estimation biases make it difficult
to disentangle the two components and compare their effect sizes (Jost,
2006).
The Hill number framework solves the problem of incompatible numerical
constraints by converting diversity index values to effective numbers of
species (Eqn 1). This encompasses all diversity indices that are a
function of the term \(\sum_{i=1}^{S}{p_{i}}^{q}\)(e.g., species
richness for q=0, Shannon index for q=1 and Simpson’s index for q=2),
where the diversity order, q, tunes the weight of species
abundances \(p_{i}\) (Rényi, 1961; Hill, 1973; Jost, 2006). The term ENS
refers to the hypothetical number of species that a perfectly even
sample would have if it produced the same index value as the real
sample. Hence, Hill numbers relieve diversity indices of their numerical
constraints by re-expressing them in units equivalent to that of species
richness (Jost, 2006). However, like most diversity metrics, Hill
numbers retain a downward estimation bias, whose strength diminishes
with increasing values of the diversity order q (Chao et al., 2014).
Therefore, differences in Hill number profiles cannot unambiguously be
attributed to changes in the regional SAD or changes in total abundance.
For example, if 2D (corresponding to Simpson’s index)
is constant along a gradient of interest while 0D
(i.e. species richness) increases, this pattern can be underlain by a
change in the regional SAD (i.e. an increase in the number of rare
species), a passive sampling effect (i.e. an increase in total
abundance) or both.
Individual-based rarefaction (IBR) is a framework that explicitly
addresses passive sampling effects by expressing diversity in terms of
the expected number of species for a standardized number of individuals
(Eqn 2) (Hurlbert, 1971; Gotelli & Colwell, 2001). The resulting
non-linear scaling relationship between the number of individuals
(n ) and expected species richness (i.e. rarefied richness,
Sn) is the IBR curve (Fig 1). Rarefied richness
estimates are unbiased for random samples, which means that they only
respond to changes in the SAD but not to the original number of
individuals present in the sample N. By varying the reference sample
size n, IBR can give more or less influence to species abundances
(Gotelli & Colwell, 2001). However, the value of n also constrains the
numerical range of rarefied richness values. Thus, effect sizes at the
base of the IBR curve (representing mostly common species) are not
directly comparable to those at higher values of n (representing both
common and rare species; Dauby and Hardy, 2012). In other words, if we
find a species richness gradient to be steeper than a corresponding
gradient in rarefied richness, part of the numerical difference has
nothing to do with more individual effects, but is merely the null
expectation from the different numerical constraints of the two metrics.
ENS rarefaction is method that converts the IBR curve into effective
numbers of species with consistent numerical constraints along the curve
(Fig 1). There is no simple closed-form equation for ENS rarefaction but
Dauby and Hardy (2012) showed that numerical approximation of Eqn 3 can
be used to convert any Sn value to its corresponding
effective number (En). Again, ENS refers to the number
of species in a hypothetical, perfectly even community that has the same
rarefied richness as the real community (Dauby and Hardy, 2012). The
base of the resulting “ENS curve” (i.e. E2) is also
the ENS transformation of Hurlbert’s (1971) unbiased probability of
interspecific encounter (SPIE, Olszewski 2004), and is
equal to an asymptotic estimate of the Hill number 2D
(Chao et al., 2014, Dauby and Hardy, 2012). It can be interpreted as the
number of dominant species in the species pool because being at the base
of the curve it gives disproportionately high weight to species with
high relative abundances. As n increases along the ENS curve, rarer and
rarer species influence the diversity estimate until it practically
converges onto the observed total species richness, where all species
are counted regardless of their abundance (i.e. EN).
Increases along the ENS curve are entirely due to the incremental
influence of rare species and do not result from variable numerical
constraints along the curve. Therefore, the ENS transformation makes it
easy to assess relative evenness; random samples from perfectly even
communities (i.e. communities without rare species) produce ENS curves
that are flat horizontal lines (Dauby & Hardy, 2012). In some sense,
ENS rarefaction combines the advantages of Hill numbers and
individual-based rarefaction in a single family of diversity measures.
It has unconstrained values for all values of n and, being a simple
transformation of rarefied richness, its values for a reference sample
size n are only affected by the SAD and not by the actual number of
individuals captured in the sample. Therefore, differences in
En values for a constant n can be unambiguously
attributed to changes in the SAD, while comparisons between different
levels of n reflect a quantification of the more-individual effect.
These properties make ENS rarefaction a useful tool for the
decomposition approach we present here.