# Species Abundance Distribution: Non-parametric Approaches#

Unlike the parametric approach which seeks to describe an SAD by a given statistical distribution, the non-parametric approach summarizes different aspects of an SAD in a single, univariate measure. These univariate measures are collectively known as biodiversity measures or biodiversity indices.

## The Components of Biodiversity#

The information on biodiversity contained within an SAD can be separated into two components:

Richness: The number of species in the assemblage.

Evenness: How abundance is distributed among species in the assemblage (an assemblage where every species is equally abundant is maximally even).

Accordingly, biodiversity measures can be grouped based on which component they consider:

Measures of richness: For a fully known assemblage, this is simply the number of species. However, we typically only have samples of an assemblage and different measures of richness correct for unseen species in different ways.

Measures of evenness: These measure how abundance is distributed among species in an assemblage. The various measures differ in how they partition the evenness component from the richness component.

Diversity measures: So-called diversity measures consider both richness and evenness together. More species rich communities are more diverse. For a given richness, more even communities are more diverse.

## Effective Number of Species: A Common Scale for Diversity Measures#

There are numerous diversity measures that capture different aspects of biodiversity. For example, two common measures of diversity are the Shannon index {cite} `weaver1963mathematical`

and the Gini-Simpson index [Simpson, 1949].

### Gini-Simpson Index#

For a community containing \(S\) species, the Gini-Simpson index is calculated as

where \(p_i\) is the proportion of species \(i\) in the community.

### Shannon Diversity Index#

For a community with \(S\) species, the Shannon diversity index is calculated as

where \(p_i\) is the proportion of species \(i\) in the community.

Both indices are formulated to capture different aspects of biodiversity. Specifically, the Gini-Simpson index is the probability of interspecific encounter, i.e. the probability that two draws yield separate species when sampling randomly from the community. The Shannon index measures the entropy of the community. The choice of base for the logarithm is arbitrary and simply expresses the entropy in different units. For \(\log_2\), the Shannon index can be interpreted as the number of yes/no questions it would take on average to identify a species drawn randomly from the community. However, the natural log is more often used in ecology. Because different diversity indices measure different aspects of biodiversity, it is difficult to compare the results of studies that use different indices. It is also difficult to interpret the magnitude of biodiversity change in a given assemblage based on these indices. For example, if species richness is halved, many diversity indices will not necessarily decrease by half.

Jost [2006] has shown that Hill numbers provide a unifying framework for comparing biodiversity change using univariate diversity indices. Hill numbers [Hill, 1973] express the diversity of a community as the “effective number of species.” The Hill number, or effective number of species, is the number of equally common species needed to produce the given value of a diversity index [Jost, 2007]. This yields a much more intuitive and biologically meaningful measure of biodiversity for a single community. This also makes comparing the magnitude of biodiversity change between two communities more straightforward. For example, halving the species richness will decrease the effective number of species by half. It is simple to convert the value for a given index to the effective number of species. Or the effective number of species can be calculated directly from the community data as the Hill number.

### Converting an Index to Effective Number of Species#

To convert the value for a given index to the effective number of species, do the following:

Write the index for \(S\) equally common species (\(p_i = \frac{1}{S}\))

Set resulting expression equal to the value of the index.

Solve for \(S\)

### Gini-Simpson Index#

Solving for \(S\) yields

Note that this can also be written as \(S_{GS} = \frac{1}{\sum_i^S = p_i^2}\) which is also known as the inverse Simpson index and is the true diversity of order 2 (see below).

### Shannon Diversity Index#

Solving for S yields

This is known as the Shannon Diversity.

### Hill Numbers#

The general form for calculating the effective number of species from community data is:

For \(q \neq 1\):

\( S_q = (\sum_i^S p_i^q)^{\frac{1}{1 - q}} \)

For \(q = 1\)

\( S_1 = e^{-\sum_i^S p_i \ln(p_i)} \)

where \(q\) is a weighting factor that places different emphasis on rare and common species. These yield the so-called true diversity of order \(q\).

\(q = 0\) is equivalent to presence/absence based measures.

\(0 < q < 1\) puts greater weight on rare species.

\(q = 1\) weights all species equally.

\(q > 1\) puts greater weight on abundant species.

The question of how to weight abundant vs rare species is an ecological one and depends on the question at hand. For instance, if one is interested in functional changes to a community, it may be fair to assume that the most abundant species have the greatest effect on ecological processes. On the other hand, one might be interested in the conservation of rare species in which case giving more weight to rare species will make such changes more apparent. Note, however, that indices with a high weighting of rare species are more difficult to estimate from field data since rare species are more rarely encountered and less well-represented in samples.

## References#

- Hil73
Mark O Hill. Diversity and evenness: a unifying notation and its consequences.

*Ecology*, 54(2):427–432, 1973.- Jos06
Lou Jost. Entropy and diversity.

*Oikos*, 113(2):363–375, 2006.- Jos07
Lou Jost. Partitioning diversity into independent alpha and beta components.

*Ecology*, 88(10):2427–2439, 2007.- Sim49
Edward H Simpson. Measurement of diversity.

*nature*, 163(4148):688–688, 1949.