Morisita index of intraspecific aggregation

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 (\(i = 1, 2, \ldots, n\)). \(Imor\) 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),

(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.

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.

Author

Péter Sólymos, solymos@ualberta.ca

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

Examples

data(dune)
x <- dispindmorisita(dune)
x
#>                imor      mclu         muni        imst       pchisq
#> Achimill  2.1666667  1.923488  0.327101099  0.50672636 9.157890e-03
#> Agrostol  1.8085106  1.294730  0.785245032  0.51373357 1.142619e-05
#> Airaprae  8.0000000  4.463082 -1.523370880  0.61382303 3.571702e-04
#> Alopgeni  2.5396825  1.395781  0.711614757  0.53074307 3.024441e-08
#> Anthodor  2.6666667  1.692616  0.495325824  0.52660266 5.897217e-05
#> Bellpere  2.0512821  2.154361  0.158876373  0.45535255 3.451547e-02
#> Bromhord  3.2380952  1.989452  0.279036892  0.53466422 1.170437e-04
#> Chenalbu        NaN       Inf         -Inf         NaN          NaN
#> Cirsarve 20.0000000 14.852327 -9.093483518  1.00000000 5.934709e-03
#> Comapalu  6.6666667  5.617442 -2.364494506  0.53647558 1.055552e-02
#> Eleopalu  3.7333333  1.577180  0.579438187  0.55851854 2.958285e-10
#> Elymrepe  2.7692308  1.554093  0.596260659  0.53293787 1.180195e-06
#> Empenigr 20.0000000 14.852327 -9.093483518  1.00000000 5.934709e-03
#> Hyporadi  6.6666667  2.731541 -0.261685440  0.61393969 7.832274e-07
#> Juncarti  3.1372549  1.814843  0.406265675  0.53635966 2.066336e-05
#> Juncbufo  4.1025641  2.154361  0.158876373  0.55458486 1.503205e-05
#> Lolipere  1.5849970  1.243023  0.822921342  0.50911591 5.873839e-05
#> Planlanc  2.4615385  1.554093  0.596260659  0.52459747 1.921730e-05
#> Poaprat   1.1702128  1.294730  0.785245032  0.28876015 1.046531e-01
#> Poatriv   1.4644137  1.223425  0.837201879  0.50641728 2.747301e-04
#> Ranuflam  2.4175824  2.065564  0.223578191  0.50981405 7.010483e-03
#> Rumeacet  3.9215686  1.814843  0.406265675  0.55792432 1.530085e-07
#> Sagiproc  2.4210526  1.729070  0.468764025  0.51893672 4.956394e-04
#> Salirepe  5.8181818  2.385233 -0.009348352  0.59744520 2.687397e-07
#> Scorautu  0.9643606  1.261365  0.809556915 -0.09356972 5.823404e-01
#> Trifprat  6.6666667  2.731541 -0.261685440  0.61393969 7.832274e-07
#> Trifrepe  1.2210916  1.301138  0.780576445  0.36709402 6.335449e-02
#> Vicilath  3.3333333  5.617442 -2.364494506  0.25266513 1.301890e-01
#> Bracruta  1.1904762  1.288590  0.789719093  0.33001160 8.071762e-02
#> Callcusp  5.3333333  2.539147 -0.121498169  0.58001287 7.982634e-06
y <- dispindmorisita(dune, unique.rm = TRUE)
y
#>               imor     mclu         muni        imst       pchisq
#> Achimill 2.1666667 1.923488  0.327101099  0.50672636 9.157890e-03
#> Agrostol 1.8085106 1.294730  0.785245032  0.51373357 1.142619e-05
#> Airaprae 8.0000000 4.463082 -1.523370880  0.61382303 3.571702e-04
#> Alopgeni 2.5396825 1.395781  0.711614757  0.53074307 3.024441e-08
#> Anthodor 2.6666667 1.692616  0.495325824  0.52660266 5.897217e-05
#> Bellpere 2.0512821 2.154361  0.158876373  0.45535255 3.451547e-02
#> Bromhord 3.2380952 1.989452  0.279036892  0.53466422 1.170437e-04
#> Comapalu 6.6666667 5.617442 -2.364494506  0.53647558 1.055552e-02
#> Eleopalu 3.7333333 1.577180  0.579438187  0.55851854 2.958285e-10
#> Elymrepe 2.7692308 1.554093  0.596260659  0.53293787 1.180195e-06
#> Hyporadi 6.6666667 2.731541 -0.261685440  0.61393969 7.832274e-07
#> Juncarti 3.1372549 1.814843  0.406265675  0.53635966 2.066336e-05
#> Juncbufo 4.1025641 2.154361  0.158876373  0.55458486 1.503205e-05
#> Lolipere 1.5849970 1.243023  0.822921342  0.50911591 5.873839e-05
#> Planlanc 2.4615385 1.554093  0.596260659  0.52459747 1.921730e-05
#> Poaprat  1.1702128 1.294730  0.785245032  0.28876015 1.046531e-01
#> Poatriv  1.4644137 1.223425  0.837201879  0.50641728 2.747301e-04
#> Ranuflam 2.4175824 2.065564  0.223578191  0.50981405 7.010483e-03
#> Rumeacet 3.9215686 1.814843  0.406265675  0.55792432 1.530085e-07
#> Sagiproc 2.4210526 1.729070  0.468764025  0.51893672 4.956394e-04
#> Salirepe 5.8181818 2.385233 -0.009348352  0.59744520 2.687397e-07
#> Scorautu 0.9643606 1.261365  0.809556915 -0.09356972 5.823404e-01
#> Trifprat 6.6666667 2.731541 -0.261685440  0.61393969 7.832274e-07
#> Trifrepe 1.2210916 1.301138  0.780576445  0.36709402 6.335449e-02
#> Vicilath 3.3333333 5.617442 -2.364494506  0.25266513 1.301890e-01
#> Bracruta 1.1904762 1.288590  0.789719093  0.33001160 8.071762e-02
#> Callcusp 5.3333333 2.539147 -0.121498169  0.58001287 7.982634e-06
dim(x) ## with unique species
#> [1] 30  5
dim(y) ## unique species removed
#> [1] 27  5