Morisita index of intraspecific aggregation
Calculates the Morisita index of dispersion, standardized index values, and the so called clumpedness and uniform indices.
dispindmorisita(x, unique.rm = FALSE, crit = 0.05, na.rm = FALSE)
x |
community data matrix, with sites (samples) as rows and species as columns. |
unique.rm |
logical, if |
crit |
two-sided p-value used to calculate critical Chi-squared values. |
na.rm |
logical.
Should missing values (including |
The Morisita index of dispersion is defined as (Morisita 1959, 1962):
Imor = n * (sum(xi^2) - sum(xi)) / (sum(xi)^2 - sum(xi))
where xi is the count of individuals in sample i, and n is the number of samples (i = 1, 2, …, n). Imor has values from 0 to n. In uniform (hyperdispersed) patterns its value falls between 0 and 1, in clumped patterns it falls between 1 and n. For increasing sample sizes (i.e. joining neighbouring quadrats), Imor goes to n as the quadrat size approaches clump size. For random patterns, Imor = 1 and counts in the samples follow Poisson frequency distribution.
The deviation from random expectation (null hypothesis)
can be tested using critical values of the Chi-squared
distribution with n-1 degrees of freedom.
Confidence intervals around 1 can be calculated by the clumped
Mclu and uniform Muni indices (Hairston et al. 1971, Krebs
1999) (Chi2Lower and Chi2Upper refers to e.g. 0.025 and 0.975 quantile
values of the Chi-squared distribution with n-1 degrees of
freedom, respectively, for crit = 0.05):
Mclu = (Chi2Lower - n + sum(xi)) / (sum(xi) - 1)
Muni = (Chi2Upper - n + sum(xi)) / (sum(xi) - 1)
Smith-Gill (1975) proposed scaling of Morisita index from [0, n] interval into [-1, 1], and setting up -0.5 and 0.5 values as confidence limits around random distribution with rescaled value 0. To rescale the Morisita index, one of the following four equations apply to calculate the standardized index Imst:
(a) Imor >= Mclu > 1: Imst = 0.5 + 0.5 (Imor - Mclu) / (n - Mclu),
(b) Mclu > Imor >= 1: Imst = 0.5 (Imor - 1) / (Mclu - 1),
(c) 1 > Imor > Muni: Imst = -0.5 (Imor - 1) / (Muni - 1),
(d) 1 > Muni > Imor: Imst = -0.5 + 0.5 (Imor - Muni) / Muni.
Returns a data frame with as many rows as the number of columns
in the input data, and with four columns. Columns are: imor the
unstandardized Morisita index, mclu the clumpedness index,
muni the uniform index, imst the standardized Morisita
index, pchisq the Chi-squared based probability for the null
hypothesis of random expectation.
A common error found in several papers is that when standardizing
as in the case (b), the denominator is given as Muni - 1. This
results in a hiatus in the [0, 0.5] interval of the standardized
index. The root of this typo is the book of Krebs (1999), see the Errata
for the book (Page 217,
https://www.zoology.ubc.ca/~krebs/downloads/errors_2nd_printing.pdf).
Péter Sólymos, solymos@ualberta.ca
Morisita, M. 1959. Measuring of the dispersion of individuals and analysis of the distributional patterns. Mem. Fac. Sci. Kyushu Univ. Ser. E 2, 215–235.
Morisita, M. 1962. Id-index, a measure of dispersion of individuals. Res. Popul. Ecol. 4, 1–7.
Smith-Gill, S. J. 1975. Cytophysiological basis of disruptive pigmentary patterns in the leopard frog, Rana pipiens. II. Wild type and mutant cell specific patterns. J. Morphol. 146, 35–54.
Hairston, N. G., Hill, R. and Ritte, U. 1971. The interpretation of aggregation patterns. In: Patil, G. P., Pileou, E. C. and Waters, W. E. eds. Statistical Ecology 1: Spatial Patterns and Statistical Distributions. Penn. State Univ. Press, University Park.
Krebs, C. J. 1999. Ecological Methodology. 2nd ed. Benjamin Cummings Publishers.
data(dune) x <- dispindmorisita(dune) x y <- dispindmorisita(dune, unique.rm = TRUE) y dim(x) ## with unique species dim(y) ## unique species removed
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