Description Usage Arguments Details Value Note Author(s) References Examples

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

1 | ```
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 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`

.

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,
http://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.

1 2 3 4 5 6 7 | ```
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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