Computes Simpson-type diversity indices, including the Gini-Simpson index, the Hunter-Gaston index (bias-corrected for sample size), and the Deltas-corrected Gini-Simpson index (bias-corrected for number of categories).
Usage
simpson(x, method = c("gini", "hunter", "deltas"), na.rm = FALSE)Arguments
- x
a factor or character vector of observations, or a vector of non-negative counts. Note that a numeric vector is always read as counts, never as observations; wrap it in
factor()to have it tabulated instead.- method
character string specifying the index to compute:
"gini","hunter", or"deltas"- na.rm
logical. If
TRUE, missing values are removed before computation. IfFALSEandxcontainsNA, the result will beNA_real_.
Details
The Gini-Simpson index is defined as \(1 - \sum p_i^2\), where \(p_i\) are the relative frequencies of categories.
The Hunter-Gaston index is defined as \(1 - \sum n_i (n_i - 1) / (N (N - 1))\), where \(n_i\) are counts and \(N\) is the total sample size.
The Deltas correction is defined as \((1 - \sum p_i^2) * k / (k - 1)\), where \(k\) is the number of observed (non-empty) categories. The factor is the reciprocal of \((k-1)/k\), the largest value the Gini-Simpson index can attain with \(k\) categories, so the corrected index reaches 1 for a uniform distribution over the observed categories.
All indices represent the probability that two randomly selected observations belong to different categories.
The Hunter-Gaston index corrects for finite sample size, while the Deltas correction adjusts for a small number of observed categories. Note that the finite-sample correction \(N/(N-1)\) applied to the Gini-Simpson index reproduces the Hunter-Gaston index exactly; the two are the same adjustment with \(N\) and \(k\) in the correction factor.
A sample concentrated in a single category is perfectly homogeneous rather
than undefined, so "gini" and "hunter" return 0 for it. Only
the Deltas correction requires \(k \ge 2\), since \(k - 1\) appears in
its denominator.
When x is numeric, it is treated as a vector of counts. Non-integer
values produce a warning; the Hunter-Gaston index requires integer counts.
References
Sachs, L. (1997). Angewandte Statistik. Springer.
Hunter, P. R., & Gaston, M. A. (1988). Numerical index of the discriminatory ability of typing systems. Journal of Clinical Microbiology, 26(11), 2465-2466. https://doi.org/10.1128/jcm.26.11.2465-2466.1988
Deltas, G. (2003). The small-sample bias of the Gini coefficient: Results and implications for empirical research. Review of Economics and Statistics, 85(1), 226-234. https://doi.org/10.1162/rest.2003.85.1.226
See also
Other diversity.concentration:
entropy(),
herfindahl()
Examples
x <- c("A", "A", "B", "C", "C", "C")
simpson(x, method = "gini")
#> [1] 0.6111111
simpson(x, method = "hunter")
#> [1] 0.7333333
simpson(x, method = "deltas")
#> [1] 0.9166667
# Using counts directly
counts <- c(A = 2, B = 1, C = 3)
simpson(counts, method = "hunter")
#> [1] 0.7333333
# a numeric vector of observations must be tabulated first, otherwise
# its values are read as counts
simpson(factor(c(1, 1, 2, 2, 3)), method = "gini")
#> [1] 0.64
# With missing values
x <- c("A", "A", NA, "B")
simpson(x, method = "gini", na.rm = TRUE)
#> [1] 0.4444444