CDAM.post.R
`CADM.post` <-
function(Dmat, nmat, n, nperm=99, make.sym=TRUE, weights=NULL, mult="holm", mantel=FALSE, silent=FALSE)
{
### Function to carry out a posteriori tests of the contribution of individual
### matrices to the congruence of a group of distance matrices.
###
### copyleft - Pierre Legendre, December 2008
###
### Reference -
### Legendre, P. and F.-J. Lapointe. 2004. Assessing congruence among distance
### matrices: single malt Scotch whiskies revisited. Australian and New Zealand
### Journal of Statistics 46: 615-629.
###
### Parameters of the function --
###
### Dmat = A text file listing the distance matrices one after the other, with
### or without blank lines.
### Each matrix is in the form of a square distance matrix with 0's
### on the diagonal.
###
### nmat = number of distance matrices in file Dmat.
###
### n = number of objects in each distance matrix. All matrices have same n.
###
### nperm = number of permutations for the tests.
###
### make.sym = TRUE: turn asymmetric matrices into symmetric matrices by
### averaging the two triangular portions.
### = FALSE: analyse asymmetric matrices as they are.
###
### weights = a vector of positive weights for the distance matrices.
### Example: weights = c(1,2,3)
### = NULL (default): all matrices have same weight in calculation of W.
###
### mult = method for correcting P-values due to multiple testing. The methods
### are "holm" (default), "sidak", and "bonferroni". The Bonferroni
### correction is overly conservative; it is not recommended. It is
### included to allow comparisons with the other methods.
###
### mantel = TRUE: Mantel statistics are computed from ranked distances,
### as well as permutational P-values.
### = FALSE (default): Mantel statistics and tests are not computed.
###
### silent = TRUE: informative messages will not be printed, except stopping
### messages. Option useful for simulation work.
### = FALSE: informative messages will be printed.
###
################################################################################
mult <- match.arg(mult, c("sidak", "holm", "bonferroni"))
if(nmat < 2)
stop("Analysis requested for a single D matrix: CADM is useless")
a <- system.time({
## Check the input file
if(ncol(Dmat) != n)
stop("Error in the value of 'n' or in the D matrices themselves")
nmat2 <- nrow(Dmat)/n
if(nmat2 < nmat) # OK if 'nmat' < number of matrices in the input file
stop("Number of input D matrices = ",nmat2,"; this value is < nmat")
nd <- n*(n-1)/2
if(is.null(weights)) {
w <- rep(1,nmat)
} else {
if(length(weights) != nmat)
stop("Incorrect number of values in vector 'weights'")
if(length(which(weights < 0)) > 0)
stop("Negative weights are not permitted")
w <- weights*nmat/sum(weights)
if(!silent) cat("Normalized weights =",w,'\n')
}
## Are asymmetric D matrices present?
asy <- rep(FALSE, nmat)
asymm <- FALSE
end <- 0
for(k in 1:nmat) {
begin <- end+1
end <- end+n
D.temp <- Dmat[begin:end,]
if(sum(abs(diag(as.matrix(D.temp)))) > 0)
stop("Diagonal not 0: matrix #",k," is not a distance matrix")
vec1 <- as.vector(as.dist(D.temp))
vec2 <- as.vector(as.dist(t(D.temp)))
if(sum(abs((vec1-vec2))) > 0) {
if(!silent) cat("Matrix #",k," is asymmetric",'\n')
asy[k] <- TRUE
asymm <- TRUE
}
}
D1 <- as.list(1:nmat)
if(asymm) {
if(make.sym) {
if(!silent) cat("\nAsymmetric matrices were transformed to be symmetric",'\n')
} else {
nd <- nd*2
if(!silent) cat("\nAnalysis carried out on asymmetric matrices",'\n')
D2 <- as.list(1:nmat)
}
} else {
if(!silent) cat("Analysis of symmetric matrices",'\n')
}
Y <- rep(NA,nd)
## String out the distance matrices (vec) and assemble them as columns into matrix 'Y'
## Construct also matrices of ranked distances D1[[k]] and D2[[k]] for permutation test
end <- 0
for(k in 1:nmat) {
begin <- end+1
end <- end+n
D.temp <- as.matrix(Dmat[begin:end,])
vec <- as.vector(as.dist(D.temp))
if(asymm) {
if(!make.sym) {
## Analysis carried out on asymmetric matrices:
## The ranks are computed on the whole matrix except the diagonal values.
## The two halves are stored as symmetric matrices in D1[[k]] and D2[[k]]
vec <- c(vec, as.vector(as.dist(t(D.temp))))
diag(D.temp) <- NA
D.temp2 <- rank(D.temp)
diag(D.temp2) <- 0
# cat("nrow =",nrow(D.temp2)," ncol =",ncol(D.temp2),'\n')
# cat("Matrix ",k," min =",min(D.temp2)," max =",max(D.temp2),'\n')
# cat("Matrix ",k," max values #",which(D.temp2 == max(D.temp2)),'\n')
D1[[k]] <- as.matrix(as.dist(D.temp2))
D2[[k]] <- as.matrix(as.dist(t(D.temp2)))
} else {
## Asymmetric matrices transformed to be symmetric, stored in D1[[k]]
vec <- (vec + as.vector(as.dist(t(D.temp)))) / 2
D.temp2 <- (D.temp + t(D.temp)) / 2
D.temp2 <- as.dist(D.temp2)
D.temp2[] <- rank(D.temp2)
D.temp2 <- as.matrix(D.temp2)
D1[[k]] <- D.temp2
}
} else {
## Symmetric matrices are stored in D1[[k]]
D.temp2 <- as.dist(D.temp)
D.temp2[] <- rank(D.temp2)
D1[[k]] <- as.matrix(D.temp2)
}
Y <- cbind(Y, vec)
}
Y <- as.matrix(Y[,-1])
colnames(Y) <- colnames(Y,do.NULL = FALSE, prefix = "Dmat.")
## Begin calculations: compute reference value of S
## Transform the distances to ranks, by column
Rmat <- apply(Y,2,rank)
## Compute the S = Sum-of-Squares of the row-marginal sums of ranks (eq. 1a)
## The ranks are weighted during the sum by the vector of matrix weights 'w'
sumRanks <- as.vector(Rmat%*%w)
S <- (nd-1)*var(sumRanks)
## Begin a posteriori tests of individual matrices
## Statistics displayed for each matrix: "Mantel.mean" and "W.per.matrix"
## Calculate the mean of the Mantel correlations on ranks for each matrix
Mantel.cor <- cor(Rmat)
diag(Mantel.cor) <- 0
spear.mean <- as.vector(Mantel.cor%*%w)/(nmat-1)
## Calculate Kendall's W for each variable
## W.var <- ((nmat-1)*spear.mean+1)/nmat
## P-value for each matrix: test of S, permuting values in matrix[[k]] only
## as in program CADM.f (2004)
## Initialize the counters
counter <- rep(1,nmat)
## Test each matrix 'k' in turn
for(k in 1:nmat) {
## Create a new Rmat table where the permuted column has been removed
Rmat.mod <- Rmat[,-k]
## Permutation loop: string out permuted matrix 'k' only
for(j in 1:nperm) {
order <- sample(n)
if(asymm & !make.sym) {
## For asymmetric matrices: permute the values within each triangular
## portion, stored as square matrices in D1[[]] and D2[[]]
vec <- as.vector(as.dist(D1[[k]][order,order]))
vec <- c(vec, as.vector(as.dist(D2[[k]][order,order])))
} else {
vec <- as.vector(as.dist(D1[[k]][order,order]))
}
Rmat.perm <- cbind(Rmat.mod, vec)
S.perm <- (nd-1)*var(as.vector(Rmat.perm%*%w))
if(S.perm >= S) counter[k] <- counter[k]+1
}
}
## Calculate P-values
counter <- counter/(nperm+1)
## Correction to P-values for multiple testing
if(mult == "sidak") {
vec.corr = NA
for(i in 1:p) vec.corr = c(vec.corr, (1-(1-counter[i])^p))
vec.corr <- vec.corr[-1]
}
if(mult == "holm") vec.corr <- p.adjust(counter, method="holm")
if(mult == "bonferroni") vec.corr <- p.adjust(counter, method="bonferroni")
## Create a data frame containing the results
# table <- rbind(spear.mean, W.var, counter, vec.corr)
# rownames(table) <- c("Mantel.mean", "W.per.matrix", "Prob", "Corrected prob")
table <- rbind(spear.mean, counter, vec.corr)
rownames(table) <- c("Mantel.mean", "Prob", "Corrected.prob")
colnames(table) <- colnames(table,do.NULL = FALSE, prefix = "Dmat.")
## Mantel tests
if(mantel) {
diag(Mantel.cor) <- 1
rownames(Mantel.cor) <- colnames(table)
colnames(Mantel.cor) <- colnames(table)
Mantel.prob <- matrix(1,nmat,nmat)
rownames(Mantel.prob) <- colnames(table)
colnames(Mantel.prob) <- colnames(table)
for(j in 1:nperm) { # Each matrix is permuted independently
# There is no need to permute the last matrix
Rmat.perm <- rep(NA,nd)
##
if(asymm & !make.sym) {
## For asymmetric matrices: permute the values within each triangular
## portion, stored as square matrices in D1[[]] and D2[[]]
for(k in 1:(nmat-1)) {
order <- sample(n)
vec <- as.vector(as.dist(D1[[k]][order,order]))
vec <- c(vec, as.vector(as.dist(D2[[k]][order,order])))
Rmat.perm <- cbind(Rmat.perm, vec)
}
vec <- as.vector(as.dist(D1[[nmat]]))
vec <- c(vec, as.vector(as.dist(D2[[nmat]])))
Rmat.perm <- cbind(Rmat.perm, vec)
} else {
for(k in 1:(nmat-1)) {
order <- sample(n)
vec <- as.vector(as.dist(D1[[k]][order,order]))
Rmat.perm <- cbind(Rmat.perm, vec)
}
vec <- as.vector(as.dist(D1[[nmat]]))
Rmat.perm <- cbind(Rmat.perm, vec)
}
# Remove the first column of Rmat.perm containing NA
Rmat.perm <- as.matrix(Rmat.perm[,-1])
# Compute Mantel correlations on ranks under permutation
Mantel.cor.perm <- cor(Rmat.perm)
for(j2 in 1:(nmat-1)) { # Compute prob in the upper tail
for(j1 in (j2+1):nmat) {
if(Mantel.cor.perm[j1,j2] >= Mantel.cor[j1,j2]) Mantel.prob[j1,j2] <- Mantel.prob[j1,j2]+1
}
}
}
Mantel.prob <- as.matrix(as.dist(Mantel.prob/(nperm+1)))
diag(Mantel.prob) <- 1
}
})
a[3] <- sprintf("%2f",a[3])
if(!silent) cat("Time to compute a posteriori tests (per matrix) =",a[3]," sec",'\n')
out <- list(A_posteriori_tests=table, Correction.type=mult)
if(mantel) {
out$Mantel.cor <- Mantel.cor
out$Mantel.prob <- Mantel.prob
}
out$nperm <- nperm
class(out) <- "CADM.post"
out
}