Computes Simpson-type diversity indices: the Gini-Simpson index, the Hunter-Gaston index (its unbiased, finite-sample version) and the index of qualitative variation (the Gini-Simpson index rescaled to a maximum of 1).
Usage
simpson(x, metric = c("gini", "hunter", "iqv"), levels = NULL, 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. A data frame with a single row of numeric columns (e.g. one site of a community matrix) is read as counts.- metric
character string specifying the index to compute:
"gini","hunter", or"iqv".- levels
the possible categories for
metric = "iqv": either their number (a single number) or the categories themselves (a vector, e.g.levels(x)). IfNULL(default), the number of observed, non-empty categories is used. Ignored, with a warning, for the other methods.- na.rm
logical. If
TRUE, missing values are removed before computation. IfFALSEandxcontainsNA, the result will beNA_real_.
Value
a numeric scalar between 0 and 1. NA_real_ (with a warning)
when the index is undefined: no observations, \(N < 2\) for
"hunter", \(K < 2\) for "iqv"; NA_real_ without a warning for
missing values with na.rm = FALSE. Invalid input (negative, infinite
or, for "hunter", non-integer counts; invalid levels) is an
error.
Details
The Gini-Simpson index is defined as \(1 - \sum p_i^2\), where \(p_i\) are the relative frequencies of the categories.
The Hunter-Gaston index is defined as \(1 - \sum n_i (n_i - 1) / (N (N - 1))\), where \(n_i\) are the counts and \(N\) is the total sample size. It equals the Gini-Simpson index times \(N / (N - 1)\). The complement, \(\sum n_i (n_i - 1) / (N (N - 1))\), is Simpson's unbiased \(\lambda\), the probability that two observations are of the same category.
The index of qualitative variation (IQV) is defined as \((1 - \sum p_i^2) \, K / (K - 1)\), where \(K\) is the number of categories. Since \((K - 1)/K\) is the largest value the Gini-Simpson index can attain with \(K\) categories, the IQV reaches 1 exactly for a uniform distribution over all \(K\) categories.
"gini" and "hunter" are the probability that two randomly selected
observations belong to different categories, drawn with and without
replacement respectively. "iqv" is not a probability but the
Gini-Simpson index relative to its maximum.
Number of categories in the IQV. \(K\) should be the number of
categories that were possible, not only those observed. With the
default, a sample spread evenly over 3 of 5 possible categories gets an
IQV of 1, although it is far from the maximum diversity the coding scheme
allows. Supply levels whenever the set of categories is known; for a
factor, levels = levels(x) uses all levels, including empty ones.
A sample concentrated in a single category is perfectly homogeneous rather
than undefined, so "gini" returns 0 for it, and so does "hunter" as
long as \(N \ge 2\). The IQV requires \(K \ge 2\), since
\(K - 1\) appears in its denominator; with levels given, a single
occupied category gives an IQV of 0.
When x is numeric, it is treated as a vector of counts. Relative
frequencies are fine for "gini" and "iqv"; the Hunter-Gaston index
counts pairs of observations and therefore requires integer counts
(non-integer counts can push it above 1).
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
Mueller, J. H., & Schuessler, K. F. (1961). Statistical Reasoning in Sociology. Houghton Mifflin.
Agresti, A., & Agresti, B. F. (1978). Statistical analysis of qualitative variation. Sociological Methodology, 9, 204-237. https://doi.org/10.2307/270810
See also
Other diversity.concentration:
entropy(),
herfindahl()
Examples
x <- c("A", "A", "B", "C", "C", "C")
simpson(x, metric = "gini")
#> [1] 0.6111111
simpson(x, metric = "hunter")
#> [1] 0.7333333
simpson(x, metric = "iqv")
#> [1] 0.9166667
# the same sample, if five categories were possible
simpson(x, metric = "iqv", levels = 5)
#> [1] 0.7638889
simpson(x, metric = "iqv", levels = c("A", "B", "C", "D", "E"))
#> [1] 0.7638889
# Using counts directly
counts <- c(A = 2, B = 1, C = 3)
simpson(counts, metric = "hunter")
#> [1] 0.7333333
# Hunter-Gaston = Gini-Simpson * N / (N - 1)
simpson(counts, metric = "gini") * 6 / 5
#> [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)), metric = "gini")
#> [1] 0.64
# With missing values
x <- c("A", "A", NA, "B")
simpson(x, metric = "gini", na.rm = TRUE)
#> [1] 0.4444444