## Indices of beta Diversity

### Description

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).

### Usage

```betadiver(x, index = NA, order = FALSE, help = FALSE, ...)
## S3 method for class 'betadiver':
plot(x, ...)
## S3 method for class 'betadiver':
scores(x, triangular = TRUE, ...)
```

### Arguments

 `x` Community data matrix, or the `betadiver` result for `plot` and `scores` functions. `index` The index of beta diversity as defined in Koleff et al. (2003), Table 1. You can use either the subscript of β or the number of the index. See argument `help` below. `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 `FALSE`, returns a 3-column matrix of raw counts. `...` Other arguments to functions.

### Details

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 SÃ¸rensen 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` which uses different notation, 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`, `adonis` 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 `index = 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.

### Value

With `index = NA`, the function returns an object of class `"betadisper"` with elements `a`, `b`, and `c`. If `index` 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, `adonis` 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.

### Warning

Some indices return similarities instead of dissimilarities.

Jari Oksanen

### References

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` for an alternative to implement all these functions, `vegdist` for some canned alternatives, and `betadisper`, `adonis`, `mantel` for analysing beta diversity objects.

### Examples

```## Raw data and plotting
data(sipoo)