R: Morisita index of intraspecific aggregation

dispindmorisita {vegan} | R Documentation

## Morisita index of intraspecific aggregation

### Description

Calculates the Morisita index of dispersion, standardized index values, and the so called clumpedness and uniform indices.

### Usage

    dispindmorisita(x, unique.rm = FALSE, crit = 0.05, na.rm = FALSE)

### Arguments

`x` | community data matrix, with sites (samples) as rows and species as columns.
`unique.rm` | logical, if `TRUE`, unique species (occurring in only one sample) are removed from the result.
`crit` | two-sided p-value used to calculate critical Chi-squared values.
`na.rm` | logical. Should missing values (including `NaN`) be omitted from the calculations?

### Details

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 criticalvalues 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`.

### Value

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.

### Note

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, http://www.zoology.ubc.ca/~krebs/downloads/errors_2nd_printing.pdf).

### Author(s)

Péter Sólymos, solymos@ualberta.ca

### References

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.

### Examples

    data(dune)
    x <- dispindmorisita(dune)
    x
    y <- dispindmorisita(dune, unique.rm = TRUE)
    y
    dim(x) ## with unique species
    dim(y) ## unique species removed

[Package vegan version 2.4-3 Index]