Multiple Double Principal Coordinate Analysis
Description
The DPCoA analysis (see dpcoa
) has been developed by Pavoine et al. (2004).
It has been used in genetics for describing interpopulation nucleotide
diversity. However, this procedure can only be used with one locus. In order to measure
and describe nucleotide diversity with more than one locus, we developed three versions of
multiple DPCoA by using three ordination methods: multiple coinertia analysis, STATIS, and
multiple factorial analysis.
The multiple DPCoA allows the impact of various loci in the
measurement and description of diversity to be quantified and described. This method is general enough to handle a large variety
of data sets. It complements existing methods such as the analysis of molecular variance or other
analyses based on linkage disequilibrium measures, and is very useful to study the impact of various
loci on the measurement of diversity.
Usage
1 2 3 4 5 6 7 8 9 10  mdpcoa(msamples, mdistances = NULL, method =
c("mcoa", "statis", "mfa"),
option = c("inertia", "lambda1", "uniform", "internal"),
scannf = TRUE, nf = 3, full = TRUE,
nfsep = NULL, tol = 1e07)
kplotX.mdpcoa(object, xax = 1, yax = 2, mfrow = NULL,
which.tab = 1:length(object$nX), includepop = FALSE,
clab = 0.7, cpoi = 0.7, unique.scale = FALSE,
csub = 2, possub = "bottomright")
prep.mdpcoa(dnaobj, pop, model, ...)

Arguments
msamples 
A list of data frames with the populations as columns, alleles as rows and abundances as entries. All the tables should have equal numbers of columns (populations). Each table corresponds to a locus; 
mdistances 
A list of objects of class 'dist', corresponding to the distances among alleles. The order of the loci should be the same in msamples as in mdistances; 
method 
One of the three possibilities: "mcoa", "statis", or "mfa". If a vector is given, only its first value is considered; 
option 
One of the four possibilities for normalizing the population coordinates over the loci: "inertia", "lambda1", "uniform", or "internal". These options are used with MCoA and MFA only; 
scannf 
a logical value indicating whether the eigenvalues bar plots should be displayed; 
nf 
if scannf is FALSE, an integer indicating the number of kept axes for the multiple analysis; 
full 
a logical value indicating whether all the axes should be kept in the separated analyses (one analysis, DPCoA, per locus); 
nfsep 
if full is FALSE, a vector indicating the number of kept axes for each of the separated analyses; 
tol 
a tolerance threshold for null eigenvalues (a value less than tol times the first one is considered as null); 
object 
an object of class 'mdpcoa'; 
xax 
the number of the xaxis; 
yax 
the number of the yaxis; 
mfrow 
a vector of the form 'c(nr,nc)', otherwise computed by as special own function 'n2mfrow'; 
which.tab 
a numeric vector containing the numbers of the loci to analyse; 
includepop 
a logical indicating if the populations must be displayed. In that case, the alleles are displayed by points and the populations by labels; 
clab 
a character size for the labels; 
cpoi 
a character size for plotting the points, used with 'par("cex")'*cpoint. If zero, no points are drawn; 
unique.scale 
if TRUE, all the arrays of figures have the same scale; 
csub 
a character size for the labels of the arrays of figures used with 'par("cex")*csub'; 
possub 
a string of characters indicating the subtitle position ("topleft", "topright", "bottomleft", "bottomright"); 
dnaobj 
a list of dna sequences that can be obtained with the function 
pop 
a factor that gives the name of the population to which each sequence belongs; 
model 
a vector giving the model to be applied for the calculations of the distances for each locus. One model should be attributed to each locus, given that the loci are in alphabetical order. The models can take the following values: "raw", "JC69", "K80" (the default), "F81", "K81", "F84", "BH87", "T92", "TN93", "GG95", "logdet", or "paralin". See the help documentation for the function "dist.dna" of ape for a describtion of the models. 
... 

Details
An object obtained by the function mdpcoa has two classes. The first one is "mdpcoa" and the second is either "mcoa", or "statis", or "mfa", depending on the method chosen. Consequently, other functions already available in ade4 for displaying graphical results can be used: With MCoA,  plot.mcoa: this function displays (1) the differences among the populations according to each locus and the compromise, (2) the projection of the principal axes of the individual analyses onto the synthetic variables, (3) the projection of the principal axes of the individual analyses onto the coinertia axes, (4) the squared vectorial covariance among the coinertia scores and the synthetic variables;  kplot.mcoa: this function divides previous displays (figures 1, 2, or 3 described in plot.mcoa) by giving one plot per locus.
With STATIS,  plot.statis: this function displays (1) the scores of each locus according to the two first eigenvectors of the matrix Rv, (2) the scatter diagram of the differences among populations according to the compromise, (3) the weight attributed to each locus in abscissa and the vectorial covariance among each individual analysis with the notations in the main text of the paper) and the compromise analysis in ordinates, (4) the covariance between the principal component inertia axes of each locus and the axes of the compromise space;  kplot.statis: this function displays for each locus the projection of the principal axes onto the compromise space.
With MFA,  plot.mfa: this function displays (1) the differences among the populations according to each locus and the compromise, (2) the projection of the principal axes of the individual analyses onto the compromise, (3) the covariance between the principal component inertia axes of each locus and the axes of the compromise space, (4) for each axis of the compromise, the amount of inertia conserved by the projection of the individual analyses onto the common space.  kplot.mfa: this function displays for each locus the projection of the principal axes and populations onto the compromise space.
Value
The functions provide the following results:
dist.ktab 
returns an object of class 
Author(s)
Sandrine Pavoine pavoine@mnhn.fr
References
Pavoine, S. and Bailly, X. (2007) New analysis for consistency among markers in the study of genetic diversity:
development and application to the description of bacterial diversity. BMC Evolutionary Biology, 7, e156.
Pavoine, S., Dufour, A.B. and Chessel, D. (2004) From dissimilarities among species to dissimilarities among communities: a double principal coordinate analysis. Journal of Theoretical Biology, 228, 523–537.
See Also
dpcoa
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24  # The functions used below require the package ape
data(rhizobium)
if (requireNamespace("ape", quiet = TRUE)) {
dat < prep.mdpcoa(rhizobium[[1]], rhizobium[[2]],
model = c("F84", "F84", "F84", "F81"),
pairwise.deletion = TRUE)
sam < dat$sam
dis < dat$dis
# The distances should be Euclidean.
# Several transformations exist to render a distance object Euclidean
# (see functions cailliez, lingoes and quasieuclid in the ade4 package).
# Here we use the quasieuclid function.
dis < lapply(dis, quasieuclid)
mdpcoa1 < mdpcoa(sam, dis, scannf = FALSE, nf = 2)
# Reference analysis
plot(mdpcoa1)
# Differences between the loci
kplot(mdpcoa1)
# Alleles projected on the population maps.
kplotX.mdpcoa(mdpcoa1)
}
