ade4 (version 1.7-5)

dist.genet: Genetic distances from gene frequencies

Description

This program computes any one of five measures of genetic distance from a set of gene frequencies in different populations with several loci.

Usage

dist.genet(genet, method = 1, diag = FALSE, upper = FALSE)

Arguments

genet

a list of class genet

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

Value

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

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: $D_1(a,b)=- \ln(\frac{\sum_{k=1}^{\nu} \sum_{j=1}^{m(k)} p_{aj}^k p_{bj}^k}{\sqrt{\sum_{k=1}^{\nu} \sum_{j=1}^{m(k)} {(p_{aj}^k) }^2}\sqrt{\sum_{k=1}^{\nu} \sum_{j=1}^{m(k)} {(p_{bj}^k)}^2}})$

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

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

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

5. Absolute genetics distance or Provesti 's distance: $D_5(a,b)=\frac{1}{2{\nu}} \sum_{k=1}^{\nu} \sum_{j=1}^{m(k)} |p_{aj}^k - p_{bj}^k|$

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 genética entre poblaciones. Miscellanea Alcobé, 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.

To find some useful explanations: Sanchez-Mazas A. (2003) Cours de Génétique Moléculaire des Populations. Cours VIII Distances génétiques - Représentation des populations. http://anthro.unige.ch/GMDP/Alicia/GMDP_dist.htm

Examples

data(casitas)
casi.genet <- char2genet(casitas,
    as.factor(rep(c("dome", "cast", "musc", "casi"), c(24,11,9,30))))
ldist <- lapply(1:5, function(method) dist.genet(casi.genet,method))
ldist
unlist(lapply(ldist, is.euclid))
kdist(ldist)