# Effective Numbers for Alpha, Beta, and Gamma Diversity Indices

### Description

The function `dz`

calculates the 'effective number' of traits or species in terms of multiplicative alpha, beta, or gamma diversity for any order of q (Hill 1973; Jost 2006, 2007). As the order q increases, the diversity calculated by `dz`

becomes increasingly sensitive to differential abundance or amount; rare or low-abundance traits/species become downweighted in importance.

### Usage

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### Arguments

`abundances` |
A matrix of community or phenotypic data where columns are individual species or traits and rows are replicates (e.g., sites, individuals, groups). Matrix elements are abundances or amounts (e.g., concentrations, counts). |

`lev` |
Partitioning level of diversity. Can be "alpha," "beta," or "gamma". |

`q` |
Order of diversity. Default is the exponential of Shannon's entropy where q = 1. |

### Details

`dz`

is a direct modification of the `d`

function from the `vegetarian`

package (Charney & Record 2012). The function was optimized to work with `hierDiversity`

to return the desired multiplicative diversity partition of interest for a given order q.

The diversity orders most familiar to ecologists are q = 0 (richness), q = 1 (Shannon entropy), and q = 2 (Simpson's probability of identity). As q increases, the diversity measure becomes less sensitive to rare subcomponents of low amount, while components of high abundance are increasingly upweighted (Jost 2006, 2007).

### Value

`D.VALUE` |
A scalar that gives the effective number of alpha, beta, or gamma diversity for the desired order of q. |

### Note

If `abundances`

is a class other than a matrix or vector, `dz`

will return an error.

### Author(s)

Zachary Marion zmarion@vols.utk.edu, James Fordyce jfordyce@utk.edu, and Benjamin Fitzpatrick benfitz@utk.edu

### References

Charney, N. and S. Record. (2012). *vegetarian: Jost Diversity Measures for Community Data.* R package version 1.2.

Hill, M.O. (1973). Diversity and evenness: a unifying notation and its consequences. *Ecology*, 54(2):427–432.

Jost, L. (2006). Entropy and diversity. *Oikos*, 113(2):363–375.

Jost, L. (2007). Partitioning diversity into independent alpha and beta components. *Ecology*, 88(10):2427–2439.

### See Also

`hierDiversity`

, d

### Examples

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