Shannon or Shannon--Weaver (or Shannon--Wiener) index is defined as
\(H' = -\sum_i p_i \log_{b} p_i\), where
\(p_i\) is the proportional abundance of species \(i\) and \(b\)
is the base of the logarithm. It is most popular to use natural
logarithms, but some argue for base \(b = 2\) (which makes sense,
but no real difference).
Both variants of Simpson's index are based on \(D = \sum p_i^2\). Choice simpson
returns \(1-D\) and
invsimpson
returns \(1/D\).
simpson.unb
finds unbiased Simpson indices for discrete
samples (Hurlbert 1971, eq. 5). These are less sensitive to sample
size than the basic Simpson indices. The unbiased indices can be only
calculated for data of integer counts.
The diversity
function can find the total (or gamma) diversity
of pooled communities with argument groups
. The average alpha
diversity can be found as the mean of diversities by the same groups,
and their difference or ratio is an estimate of beta diversity (see
Examples). The pooling can be based either on the observed
abundancies, or all communities can be equalized to unit total before
pooling; see Jost (2007) for discussion. Functions
adipart
and multipart
provide canned
alternatives for estimating alpha, beta and gamma diversities in
hierarchical settings.
fisher.alpha
estimates the \(\alpha\) parameter of
Fisher's logarithmic series (see fisherfit
).
The estimation is possible only for genuine
counts of individuals.
None of these diversity indices is usable for empty sampling units
without any species, but some of the indices can give a numeric
value. Filtering out these cases is left for the user.
Function specnumber
finds the number of species. With
MARGIN = 2
, it finds frequencies of species. If groups
is given, finds the total number of species in each group (see
example on finding one kind of beta diversity with this option).
Better stories can be told about Simpson's index than about
Shannon's index, and still grander narratives about
rarefaction (Hurlbert 1971). However, these indices are all very
closely related (Hill 1973), and there is no reason to despise one
more than others (but if you are a graduate student, don't drag me in,
but obey your Professor's orders). In particular, the exponent of the
Shannon index is linearly related to inverse Simpson (Hill 1973)
although the former may be more sensitive to rare species. Moreover,
inverse Simpson is asymptotically equal to rarefied species richness
in sample of two individuals, and Fisher's \(\alpha\) is very
similar to inverse Simpson.