calc_PIE returns the probability of interspecific encounter (PIE)
which is also known as Simpson's evenness index and Gini-Simpson index. For
PIE will be converted to an asymptotic effective number of species (S_PIE).
can either be a: 1) mob_in object, 2) community matrix-like object in which rows represent plots and columns represent species, or 3) a vector which contains the abundance of each species.
boolean that determines if the effective number of species should be returned or the raw PIE value. Defaults to FALSE
The formula of Hurlbert (1971) is used to calculate PIE:
PIE = N /(N - 1) * (1 - p_i^2)
where N is the total number of individuals and p_i is the relative abundance of species i. This formulation uses sampling without replacement and it is sometimes referred to as the bias corrected formulation of PIE.
ENS = TRUE, S_PIE will be returned which represents the species richness of
a hypothetical community with equally-abundant species and infinitely many individuals
corresponding to the observed value of PIE. It is computed as
S_PIE = 1 /(1 - PIE), which is equal to the
asymptotic estimator for Hill numbers of diversity order 2 provided by Chao et al (2014).
Note that S_PIE is undefined for communities with exactly one individual per species.
The code in this function borrows heavily from the function vegan::diversity() but computes a different quantity. The function vegan::diversity() computes PIE when sampling with replacement is assumed. The difference between the two formulations will decrease as N becomes large. Jari Oksanen and Bob O'Hara are the original authors of the function vegan::diversity().
Dan McGlinn, Thore Engel
Hurlbert, S. H. (1971) The nonconcept of species diversity: a critique and alternative parameters. Ecology 52, 577<e2><80><93>586.
Chao, A., Gotelli, N. J., Hsieh, T. C., Sander, E. L., Ma, K. H., Colwell, R. K., & Ellison, A. M. (2014). Rarefaction and extrapolation with Hill numbers: A framework for sampling and estimation in species diversity studies. Ecological Monographs 84(1), 45<e2><80><93>67.
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