Coefficients of Biogeographical Dispersal Direction

This function computes coefficients of dispersal direction between geographically connected areas, as defined by Legendre and Legendre (1984), and also described in Legendre and Legendre (2012, section 13.3.4).

Usage

bgdispersal(mat, PAonly = FALSE, abc = FALSE)

Arguments

mat: Data frame or matrix containing a community composition data table (species presence-absence or abundance data).
PAonly: FALSE if the four types of coefficients, DD1 to DD4, are requested; TRUE if DD1 and DD2 only are sought (see Details).
abc: If TRUE, return tables a, b and c used in DD1 and DD2.

Details

The signs of the DD coefficients indicate the direction of dispersal, provided that the asymmetry is significant. A positive sign indicates dispersal from the first (row in DD tables) to the second region (column); a negative sign indicates the opposite. A McNemar test of asymmetry is computed from the presence-absence data to test the hypothesis of a significant asymmetry between the two areas under comparison.

In the input data table, the rows are sites or areas, the columns are taxa. Most often, the taxa are species, but the coefficients can be computed from genera or families as well. DD1 and DD2 only are computed for presence-absence data. The four types of coefficients are computed for quantitative data, which are converted to presence-absence for the computation of DD1 and DD2. PAonly = FALSE indicates that the four types of coefficients are requested. PAonly = TRUE if DD1 and DD2 only are sought.

Value

Function bgdispersal returns a list containing the following matrices:

DD1: \(DD1_{j,k} = (a(b - c))/((a + b + c)^2)\)
DD2: \(DD2_{j,k} = (2 a (b - c))/((2a + b + c) (a + b + c))\) where \(a\), \(b\), and \(c\) have the same meaning as in the computation of binary similarity coefficients.
DD3: \(DD3_{j,k} = {W(A-B) / (A+B-W)^2} \)
DD4: \(DD4_{j,k} = 2W(A-B) / ((A+B)(A+B-W))\) where W = sum(pmin(vector1, vector2)), A = sum(vector1), B = sum(vector2)
McNemar: McNemar chi-square statistic of asymmetry (Sokal and Rohlf 1995): \(2(b \log(b) + c \log(c) - (b+c) \log((b+c)/2)) / q\), where \(q = 1 + 1/(2(b+c))\) (Williams correction for continuity)
prob.McNemar: probabilities associated with McNemar statistics, chi-square test. H0: no asymmetry in \((b-c)\).

References

Legendre, P. and V. Legendre. 1984. Postglacial dispersal of freshwater fishes in the Québec peninsula. Can. J. Fish. Aquat. Sci. 41: 1781-1802.

Legendre, P. and L. Legendre. 2012. Numerical ecology, 3rd English edition. Elsevier Science BV, Amsterdam.

Sokal, R. R. and F. J. Rohlf. 1995. Biometry. The principles and practice of statistics in biological research. 3rd edn. W. H. Freeman, New York.

Author

Pierre Legendre, Departement de Sciences Biologiques, Universite de Montreal

Note

The function uses a more powerful alternative for the McNemar test than the classical formula. The classical formula was constructed in the spirit of Pearson's Chi-square, but the formula in this function was constructed in the spirit of Wilks Chi-square or the \(G\) statistic. Function mcnemar.test uses the classical formula. The new formula was introduced in vegan version 1.10-11, and the older implementations of bgdispersal used the classical formula.

Examples

mat <- matrix(c(32,15,14,10,70,30,100,4,10,30,25,0,18,0,40,
  0,0,20,0,0,0,0,4,0,30,20,0,0,0,0,25,74,42,1,45,89,5,16,16,20),
  4, 10, byrow=TRUE)
bgdispersal(mat)
#> $DD1
#>       [,1]  [,2] [,3]  [,4]
#> [1,]  0.00  0.24 0.21  0.00
#> [2,] -0.24  0.00 0.08 -0.24
#> [3,] -0.21 -0.08 0.00 -0.21
#> [4,]  0.00  0.24 0.21  0.00
#> 
#> $DD2
#>            [,1]       [,2]      [,3]       [,4]
#> [1,]  0.0000000  0.3428571 0.3230769  0.0000000
#> [2,] -0.3428571  0.0000000 0.1142857 -0.3428571
#> [3,] -0.3230769 -0.1142857 0.0000000 -0.3230769
#> [4,]  0.0000000  0.3428571 0.3230769  0.0000000
#> 
#> $DD3
#>             [,1]       [,2]      [,3]        [,4]
#> [1,]  0.00000000  0.1567922 0.1420408 -0.01325831
#> [2,] -0.15679216  0.0000000 0.1101196 -0.20049485
#> [3,] -0.14204082 -0.1101196 0.0000000 -0.13586560
#> [4,]  0.01325831  0.2004949 0.1358656  0.00000000
#> 
#> $DD4
#>             [,1]       [,2]      [,3]        [,4]
#> [1,]  0.00000000  0.2513176 0.2425087 -0.01960102
#> [2,] -0.25131757  0.0000000 0.1725441 -0.30993929
#> [3,] -0.24250871 -0.1725441 0.0000000 -0.23381521
#> [4,]  0.01960102  0.3099393 0.2338152  0.00000000
#> 
#> $McNemar
#>      [,1]     [,2]      [,3]     [,4]
#> [1,]   NA 7.677938 9.0571232 0.000000
#> [2,]   NA       NA 0.2912555 7.677938
#> [3,]   NA       NA        NA 9.057123
#> [4,]   NA       NA        NA       NA
#> 
#> $prob.McNemar
#>      [,1]        [,2]        [,3]        [,4]
#> [1,]   NA 0.005590001 0.002616734 1.000000000
#> [2,]   NA          NA 0.589417103 0.005590001
#> [3,]   NA          NA          NA 0.002616734
#> [4,]   NA          NA          NA          NA
#>