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 $1/D$.
Function rarefy gives the expected species richness in random
subsamples of size sample from the community. The size of
sample should be smaller than total community size, but the
function will silently work for larger sample as well and
return non-rarefied species richness (and standard error = 0). If
sample is a vector, rarefaction of all observations is
performed for each sample size separately. Rarefaction can be
performed only with genuine counts of individuals. The function
rarefy is based on Hurlbert's (1971) formulation, and the
standard errors on Heck et al. (1975).
Function rrarefy generates one randomly rarefied community data
frame or vector of given sample size. The sample can be
a vector giving the sample sizes for each row, and its values must be
less or equal to observed number of individuals. The random
rarefaction is made without replacement so that the variance of
rarefied communities is rather related to rarefaction proportion than
to to the size of the sample.
Function drarefy returns probabilities that species occur in a
rarefied community of size sample. The sample can be a
vector giving the sample sizes for each row.
Function rarecurve draws a rarefaction curve for each row of
the input data. The rarefaction curves are evaluated using the
interval of step sample sizes, always including 1 and total
sample size. If sample is specified, a vertical line is
drawn at sample with horizontal lines for the rarefied
species richnesses.
fisher.alpha estimates the $\alpha$ parameter of
Fisher's logarithmic series (see fisherfit).
The estimation is possible only for genuine
counts of individuals. The function can optionally return standard
errors of $\alpha$. These should be regarded only as rough
indicators of the accuracy: the confidence limits of $\alpha$ are
strongly non-symmetric and the standard errors cannot be used in
Normal inference.
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.