adegenet (version 2.1.10)

dist.genpop: Genetic distances between populations

Description

This function computes measures of genetic distances between populations using a genpop object.
Currently, five distances are available, some of which are euclidian (see details).

A non-euclidian distance can be transformed into an Euclidean one using cailliez in order to perform a Principal Coordinate Analysis dudi.pco (both functions in ade4).

The function dist.genpop is based on former dist.genet function of ade4 package.

Usage

dist.genpop(x, method = 1, diag = FALSE, upper = FALSE)

Value

returns a distance matrix of class dist between the rows of the data frame

Arguments

x

a list of class genpop

method

an integer between 1 and 5. See details

diag

a logical value indicating whether the diagonal of the distance matrix should be printed by print.dist

upper

a logical value indicating whether the upper triangle of the distance matrix should be printed by print.dist

Author

Thibaut Jombart
Former dist.genet code by Daniel Chessel
and documentation by Anne B. Dufour

Details

Let A a table containing allelic frequencies with t populations (rows) and m alleles (columns).
Let \(\nu\) the number of loci. The locus j gets m(j) alleles. \(m=\sum_{j=1}^{\nu} m(j)\)

For the row i and the modality k of the variable j, notice the value \(a_{ij}^k\) (\(1 \leq i \leq t\), \(1 \leq j \leq \nu\), \(1 \leq k \leq m(j)\)) the value of the initial table.

\(a_{ij}^+=\sum_{k=1}^{m(j)}a_{ij}^k\) and \(p_{ij}^k=\frac{a_{ij}^k}{a_{ij}^+}\)

Let P the table of general term \(p_{ij}^k\)
\(p_{ij}^+=\sum_{k=1}^{m(j)}p_{ij}^k=1\), \(p_{i+}^+=\sum_{j=1}^{\nu}p_{ij}^+=\nu\), \(p_{++}^+=\sum_{j=1}^{\nu}p_{i+}^+=t\nu\)

The option method computes the distance matrices between populations using the frequencies \(p_{ij}^k\).

1. Nei's distance (not Euclidean):
\(D_1(a,b)=- \ln(\frac{\sum_{k=1}^{\nu} \sum_{j=1}^{m(k)} p_{aj}^k p_{bj}^k}{\sqrt{\sum_{k=1}^{\nu} \sum_{j=1}^{m(k)} {(p_{aj}^k) }^2}\sqrt{\sum_{k=1}^{\nu} \sum_{j=1}^{m(k)} {(p_{bj}^k)}^2}})\)

2. Angular distance or Edwards' distance (Euclidean):
\(D_2(a,b)=\sqrt{1-\frac{1}{\nu} \sum_{k=1}^{\nu} \sum_{j=1}^{m(k)} \sqrt{p_{aj}^k p_{bj}^k}}\)

3. Coancestrality coefficient or Reynolds' distance (Eucledian):
\(D_3(a,b)=\sqrt{\frac{\sum_{k=1}^{\nu} \sum_{j=1}^{m(k)}{(p_{aj}^k - p_{bj}^k)}^2}{2 \sum_{k=1}^{\nu} (1- \sum_{j=1}^{m(k)}p_{aj}^k p_{bj}^k)}}\)

4. Classical Euclidean distance or Rogers' distance (Eucledian):
\(D_4(a,b)=\frac{1}{\nu} \sum_{k=1}^{\nu} \sqrt{\frac{1}{2} \sum_{j=1}^{m(k)}{(p_{aj}^k - p_{bj}^k)}^2}\)

5. Absolute genetics distance or Provesti 's distance (not Euclidean):
\(D_5(a,b)=\frac{1}{2{\nu}} \sum_{k=1}^{\nu} \sum_{j=1}^{m(k)} |p_{aj}^k - p_{bj}^k|\)

References

To complete informations about distances:

Distance 1:
Nei, M. (1972) Genetic distances between populations. American Naturalist, 106, 283--292.
Nei M. (1978) Estimation of average heterozygosity and genetic distance from a small number of individuals. Genetics, 23, 341--369.
Avise, J. C. (1994) Molecular markers, natural history and evolution. Chapman & Hall, London.

Distance 2:
Edwards, A.W.F. (1971) Distance between populations on the basis of gene frequencies. Biometrics, 27, 873--881.
Cavalli-Sforza L.L. and Edwards A.W.F. (1967) Phylogenetic analysis: models and estimation procedures. Evolution, 32, 550--570.
Hartl, D.L. and Clark, A.G. (1989) Principles of population genetics. Sinauer Associates, Sunderland, Massachussetts (p. 303).

Distance 3:
Reynolds, J. B., B. S. Weir, and C. C. Cockerham. (1983) Estimation of the coancestry coefficient: basis for a short-term genetic distance. Genetics, 105, 767--779.

Distance 4:
Rogers, J.S. (1972) Measures of genetic similarity and genetic distances. Studies in Genetics, Univ. Texas Publ., 7213, 145--153.
Avise, J. C. (1994) Molecular markers, natural history and evolution. Chapman & Hall, London.

Distance 5:
Prevosti A. (1974) La distancia genetica entre poblaciones. Miscellanea Alcobe, 68, 109--118.
Prevosti A., Ocaña J. and Alonso G. (1975) Distances between populations of Drosophila subobscura, based on chromosome arrangements frequencies. Theoretical and Applied Genetics, 45, 231--241.

For more information on dissimilarity indexes:
Gower J. and Legendre P. (1986) Metric and Euclidean properties of dissimilarity coefficients. Journal of Classification, 3, 5--48

Legendre P. and Legendre L. (1998) Numerical Ecology, Elsevier Science B.V. 20, pp274--288.

See Also

cailliez,dudi.pco

Examples

if (FALSE) {
data(microsatt)
obj <- as.genpop(microsatt$tab)

listDist <- lapply(1:5, function(i) cailliez(dist.genpop(obj,met=i)))
for(i in 1:5) {attr(listDist[[i]],"Labels") <- popNames(obj)}
listPco <- lapply(listDist, dudi.pco,scannf=FALSE)

par(mfrow=c(2,3))
for(i in 1:5) {scatter(listPco[[i]],sub=paste("Dist:", i))}

}