The function estimates any of the 24 indices of beta diversity reviewed by Koleff et al. (2003). Alternatively, it finds the co-occurrence frequencies for triangular plots (Koleff et al. 2003).

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`x` |
Community data matrix, or the |

`method` |
The index of beta diversity as defined in Koleff et al.
(2003), Table 1. You can use either the subscript of |

`order` |
Order sites by increasing number of species. This will influence the configuration in the triangular plot and non-symmetric indices. |

`help` |
Show the numbers, subscript names and the defining equations of the indices and exit. |

`triangular` |
Return scores suitable for triangular plotting of
proportions. If |

`...` |
Other arguments to functions. |

The most commonly used index of beta diversity is *β_w =
S/α - 1*, where *S* is the total number of species, and
*α* is the average number of species per site (Whittaker
1960). A drawback of this model is that *S* increases with
sample size, but the expectation of *α* remains constant,
and so the beta diversity increases with sample size. A solution to
this problem is to study the beta diversity of pairs of sites. If we
denote the number of species shared between two sites as *a* and
the numbers of unique species (not shared) as *b* and *c*,
then *S = a + b + c* and *α = (2 a + b + c)/2* so that
*β_w = (b+c)/(2 a + b + c)*. This is the Sørensen
dissimilarity as defined in vegan function
`vegdist`

with argument `binary = TRUE`

. Many
other indices are dissimilarity indices as well.

Function `betadiver`

finds all indices reviewed by Koleff et
al. (2003). All these indices could be found with function
`designdist`

, but the current function provides a
conventional shortcut. The function only finds the indices. The proper
analysis must be done with functions such as `betadisper`

,
`adonis2`

or `mantel`

.

The indices are directly taken from Table 1 of Koleff et al. (2003),
and they can be selected either by the index number or the subscript
name used by Koleff et al. The numbers, names and defining equations
can be seen using `betadiver(help = TRUE)`

. In all cases where
there are two alternative forms, the one with the term *-1* is
used. There are several duplicate indices, and the number of distinct
alternatives is much lower than 24 formally provided. The formulations
used in functions differ occasionally from those in Koleff et
al. (2003), but they are still mathematically equivalent. With
`method = NA`

, no index is calculated, but instead an object of
class `betadiver`

is returned. This is a list of elements
`a`

, `b`

and `c`

. Function `plot`

can be used to
display the proportions of these elements in triangular plot as
suggested by Koleff et al. (2003), and `scores`

extracts the
triangular coordinates or the raw scores. Function `plot`

returns
invisibly the triangular coordinates as an `"ordiplot"`

object.

With `method = NA`

, the function returns an object of
class `"betadisper"`

with elements `a`

, `b`

, and
`c`

. If `method`

is specified, the function returns a
`"dist"`

object which can be used in any function analysing
dissimilarities. For beta diversity, particularly useful functions
are `betadisper`

to study the betadiversity in groups,
`adonis2`

for any model, and `mantel`

to
compare beta diversities to other dissimilarities or distances
(including geographical distances). Although `betadiver`

returns a `"dist"`

object, some indices are similarities and
cannot be used as such in place of dissimilarities, but that is a
severe user error. Functions 10 (`"j"`

) and 11 (`"sor"`

)
are two such similarity indices.

Some indices return similarities instead of dissimilarities.

Jari Oksanen

Baselga, A. (2010) Partitioning the turnover and nestedness
components of beta diversity. *Global Ecology and Biogeography*
19, 134–143.

Koleff, P., Gaston, K.J. and Lennon, J.J. (2003) Measuring beta
diversity for presence-absence data. *Journal of Animal
Ecology* 72, 367–382.

Whittaker, R.H. (1960) Vegetation of Siskiyou mountains, Oregon and
California. *Ecological Monographs* 30, 279–338.

`designdist`

can be used to implement all these
functions, and also allows using notation with `alpha`

and
`gamma`

diversities. `vegdist`

has some canned
alternatives. Functions `betadisper`

,
`adonis2`

and `mantel`

can be used for
analysing beta diversity objects. The returned dissimilarities can
be used in any distance-based methods, such as
`metaMDS`

, `capscale`

and
`dbrda`

. Functions `nestedbetasor`

and
`nestedbetajac`

implement decomposition beta diversity
measures (Sørensen and Jaccard) into turnover and
nestedness components following Baselga (2010).

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