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## Skew-symmetric association model with layer effect (extension of van der Heijden & Mooijaart, 1995)
hmskewL <- function(tab, nd.symm=NA, layer.effect.skew=c("homogeneous.scores", "heterogeneous", "none"),
layer.effect.symm=c("heterogeneous", "uniform", "regression.type",
"homogeneous.scores", "none"),
diagonal=c("none", "heterogeneous", "homogeneous"),
weighting=c("marginal", "uniform", "none"), se=c("none", "jackknife", "bootstrap"),
nreplicates=100, ncpus=getOption("boot.ncpus"),
family=poisson, weights=NULL, start=NULL, etastart=NULL, tolerance=1e-8, iterMax=5000,
trace=FALSE, verbose=TRUE, ...) {
layer.effect.skew <- match.arg(layer.effect.skew)
layer.effect.symm <- match.arg(layer.effect.symm)
diagonal <- match.arg(diagonal)
weighting <- match.arg(weighting)
se <- match.arg(se)
tab <- as.table(tab)
if(length(dim(tab)) < 3)
stop("tab must have (at least) three dimensions")
if(nrow(tab) != ncol(tab))
stop("tab must be a square table for asymmetry models")
if(!all(rownames(tab) == colnames(tab)))
stop("tab must have identical row and column names for asymmetry models")
if(!is.na(nd.symm) && nd.symm < 0)
stop("nd.symm must be NA, zero or positive")
if(!is.na(nd.symm) && nd.symm/2 > min(nrow(tab), ncol(tab)) - 1)
stop("Number of dimensions of symmetric association cannot exceed 2 * (min(nrow(tab), ncol(tab)) - 1)")
if(is.na(nd.symm) && diagonal != "none")
stop("diagonal parameters are redundant with nd.symm=NA")
if(is.na(nd.symm) && layer.effect.symm == "homogeneous.scores")
stop("layer.effect.symm == \"homogeneous.scores\" is only valid when nd.symm is not NA")
else if(!is.na(nd.symm) && layer.effect.symm == "uniform")
stop("layer.effect.symm == \"uniform\" is only valid when nd.symm is NA")
else if(!is.na(nd.symm) && layer.effect.symm == "regression.type")
stop("layer.effect.symm == \"regression.type\" is only valid when nd.symm is NA")
if(length(dim(tab)) > 3)
tab <- margin.table(tab, 1:3)
tab <- prepareTable(tab, FALSE)
vars <- names(dimnames(tab))
if(diagonal == "heterogeneous")
diagstr <- sprintf("+ %s:Diag(%s, %s) ", vars[3], vars[1], vars[2])
else if(diagonal == "homogeneous")
diagstr <- sprintf("+ Diag(%s, %s) ", vars[1], vars[2])
else
diagstr <- ""
f1 <- sprintf("Freq ~ %s + %s + %s + %s:%s + %s:%s",
vars[1], vars[2], vars[3], vars[1], vars[3], vars[2], vars[3])
if(is.na(nd.symm)) {
if(layer.effect.symm == "homogeneous.scores")
# Handled at the top of the function
stop()
else if(layer.effect.symm == "heterogeneous")
f1.symm <- sprintf("+ %s:Symm(%s, %s)",
vars[3], vars[1], vars[2])
else if(layer.effect.symm == "uniform")
f1.symm <- sprintf("+ Mult(Exp(%s), Symm(%s, %s))",
vars[3], vars[1], vars[2])
else if(layer.effect.symm == "regression.type")
f1.symm <- sprintf("+ Symm(%s, %s) + Mult(%s, Symm(%s, %s))",
vars[1], vars[2], vars[3], vars[1], vars[2])
else
f1.symm <- sprintf("+ Symm(%s, %s)",
vars[1], vars[2])
}
else if(!is.na(nd.symm) && nd.symm > 0) {
if(layer.effect.symm %in% c("uniform", "regression.type")) {
# Handled at the top of the function
stop()
}
else if(layer.effect.symm == "homogeneous.scores") {
f1.symm <- ""
for(i in 1:nd.symm)
f1.symm <- paste(f1.symm, sprintf("+ Mult(%s, MultHomog(%s, %s), inst = %i)",
vars[3], vars[1], vars[2], i))
}
else if(layer.effect.symm == "heterogeneous") {
stop("Symmetric association with heterogeneous layer effect is currently not supported")
f1.symm <- sprintf("+ instances(MultHomog(%s:%s, %s:%s), %i)",
vars[3], vars[1], vars[3], vars[2], nd.symm)
}
else {
f1.symm <- sprintf("+ instances(MultHomog(%s, %s), %i)",
vars[1], vars[2], nd.symm)
}
}
else {
f1.symm <- ""
}
base <- NULL
if(layer.effect.symm %in% c("heterogeneous", "regression.type"))
eliminate <- eval(parse(text=sprintf("quote(Symm(%s, %s))", vars[1], vars[2])))
else
eliminate <- eval(parse(text=sprintf("quote(%s:%s)", vars[1], vars[3])))
# gnm can give incorrect results with contrasts other than treatment
contr <- getOption("contrasts")
on.exit(options(contrasts=contr))
options(contrasts=c("contr.treatment", "contr.treatment"))
if(identical(start, NA)) {
cat("Running base model to find starting values...\n")
args <- list(formula=as.formula(paste(f1, diagstr)), data=tab,
family=family, weights=weights,
eliminate=eval(parse(text=sprintf("quote(%s:%s)", vars[1], vars[3]))),
tolerance=1e-6, iterMax=iterMax)
args <- c(args, list(...))
base <- do.call("gnm", args)
args$method <- "coefNames"
args$formula <- as.formula(paste(f1, diagstr, f1.symm))
args$eliminate <- eliminate
npar <- length(do.call("gnm", args))
# Use base model without symmetric interaction as this seems to give better results
# when quasi-symmetry is specified.
# Using NA for all linear parameters usually works better than their values in the base model
if(layer.effect.skew == "homogeneous.scores")
start <- c(rep(NA, npar), rep(NA, dim(tab)[3]),
residEVDL(base, 1, layer.effect.skew, skew=TRUE))
else
start <- c(rep(NA, npar),
residEVDL(base, 1, layer.effect.skew, skew=TRUE))
if(is.null(etastart))
etastart <- as.numeric(predict(base))
cat("Running real model...\n")
}
if(layer.effect.skew == "heterogeneous")
f2.skew <- sprintf("+ HMSkew(%s:%s, %s:%s)",
vars[1], vars[3], vars[2], vars[3])
else if(layer.effect.skew == "homogeneous.scores")
f2.skew <- sprintf("+ Mult(%s, HMSkew(%s, %s))",
vars[3], vars[1], vars[2])
else
f2.skew <- sprintf("+ HMSkew(%s, %s)",
vars[1], vars[2])
args <- list(formula=as.formula(paste(f1, diagstr, f1.symm, f2.skew)), data=tab,
family=family, start=start, etastart=etastart, eliminate=eliminate,
tolerance=tolerance, iterMax=iterMax, verbose=verbose, trace=trace)
model <- do.call("gnm", c(args, list(...)))
if(is.null(model))
return(NULL)
class(model) <- c("hmskewL", "assocmod", class(model))
model$call.gnm <- model$call
model$call <- match.call()
if(is.na(nd.symm) || nd.symm == 0) {
assoc1 <- NULL
}
else if(layer.effect.symm == "none") {
model$assoc <- assoc.rc.symm(model, weighting=weighting)
assoc1 <- assoc.rc.symm
}
else {
model$assoc <- assoc.rcL.symm(model, weighting=weighting)
assoc1 <- assoc.rcL.symm
}
if(!is.null(model$assoc))
class(model$assoc) <- c("assoc.rcL", "assoc.symm", "assoc")
assoc2 <- NULL
if(layer.effect.skew == "none") {
model$assoc.hmskew <- assoc.hmskew(model, weighting=weighting)
if(is.null(assoc1))
assoc1 <- assoc.hmskew
else
assoc2 <- assoc.hmskew
}
else {
model$assoc.hmskew <- assoc.hmskewL(model, weighting=weighting)
if(is.null(assoc1))
assoc1 <- assoc.hmskewL
else
assoc2 <- assoc.hmskewL
}
class(model$assoc.hmskew) <- c("assoc.hmskewL", "assoc.symm", "assoc")
if(se %in% c("jackknife", "bootstrap")) {
jb <- jackboot(se, ncpus, nreplicates, tab, model, assoc1, assoc2,
weighting, NULL, NULL, family, weights,
verbose, trace, start, etastart, ...)
if(!is.na(nd.symm) && nd.symm > 0) {
model$assoc$covtype <- se
model$assoc$covmat <- jb$covmat1
model$assoc$adj.covmats <- jb$adj.covmats1
model$assoc$jack.results <- jb$jack.results1
model$assoc$boot.results <- jb$boot.results1
model$assoc.hmskew$covtype <- se
model$assoc.hmskew$covmat <- jb$covmat2
model$assoc.hmskew$adj.covmats <- jb$adj.covmats2
model$assoc.hmskew$jack.results <- jb$jack.results2
model$assoc.hmskew$boot.results <- jb$boot.results2
}
else {
model$assoc.hmskew$covtype <- se
model$assoc.hmskew$covmat <- jb$covmat
model$assoc.hmskew$adj.covmats <- jb$adj.covmats
model$assoc.hmskew$jack.results <- jb$jack.results
model$assoc.hmskew$boot.results <- jb$boot.results
}
}
else {
if(!is.na(nd.symm) && nd.symm > 0) {
model$assoc$covtype <- se
model$assoc$covmat <- numeric(0)
model$assoc$adj.covmats <- numeric(0)
model$assoc$boot.results <- numeric(0)
model$assoc$jack.results <- numeric(0)
}
model$assoc.hmskew$covtype <- se
model$assoc.hmskew$covmat <- numeric(0)
model$assoc.hmskew$adj.covmats <- numeric(0)
model$assoc.hmskew$boot.results <- numeric(0)
model$assoc.hmskew$jack.results <- numeric(0)
}
model
}
assoc.hmskewL <- function(model, weighting=c("marginal", "uniform", "none"), ...) {
if(!inherits(model, "gnm"))
stop("model must be a gnm object")
tab <- prepareTable(model$data, FALSE)
vars <- names(dimnames(tab))
nr <- nrow(tab)
nc <- ncol(tab)
nl <- dim(tab)[3]
# Weight with marginal frequencies, cf. Clogg & Shihadeh (1994), p. 83-84, and Becker & Clogg (1989), p. 144.
weighting <- match.arg(weighting)
if(weighting == "marginal")
p <- prop.table(apply(tab, 1, sum, na.rm=TRUE) + apply(tab, 2, sum, na.rm=TRUE))
else if(weighting == "uniform")
p <- rep(1/nr, nr)
else
p <- rep(1, nr)
homogeneous <- TRUE
mu1 <- parameters(model)[pickCoef(model, sprintf("HMSkew.*\\)([.:]\\Q%s\\E\\|\\Q%s\\E)?(\\Q%s\\E)(:(\\Q%s\\E))?$",
vars[1], vars[2],
paste(c(rownames(tab), colnames(tab)), collapse="\\E|\\Q"),
paste(dimnames(tab)[[3]], collapse="\\E|\\Q")))]
mu2 <- parameters(model)[pickCoef(model, sprintf("HMSkew.*\\)([.:]\\Q%s\\E\\|\\Q%s\\E)?(\\Q%s\\E)(:(\\Q%s\\E))?\\.1$",
vars[1], vars[2],
paste(c(rownames(tab), colnames(tab)), collapse="\\E|\\Q"),
paste(dimnames(tab)[[3]], collapse="\\E|\\Q")))]
phi <- parameters(model)[pickCoef(model, sprintf("Mult\\(.*HMSkew\\(.*[.:]\\Q%s\\E(\\Q%s\\E)$",
vars[3],
paste(dimnames(tab)[[3]], collapse="\\E|\\Q")))]
if(length(phi) == nl &&
length(mu1) == length(mu2) &&
length(mu1) == nr) {
homogeneous <- TRUE
layer <- matrix(phi, nl, 2)
sc <- array(NA, dim=c(nr, 2, 1))
sc[,1,1] <- mu1
sc[,2,1] <- mu2
}
else if(length(phi) == 0 &&
length(mu1) == length(mu2) &&
length(mu1) == nr * nl) {
homogeneous <- FALSE
layer <- matrix(1, nl, 2)
sc <- array(NA, dim=c(nr, 2, nl))
sc[,1,] <- t(matrix(mu1, nl, nr))
sc[,2,] <- t(matrix(mu2, nl, nr))
}
else {
stop("No dimensions found. Are you sure this is a van der Heijden & Mooijaart skewness model with layer effect?")
}
if(length(pickCoef(model, "Diag\\(")) > nr)
dg <- matrix(pickCoef(model, "Diag\\(", value=TRUE), nl, nr)
else if(length(pickCoef(model, "Diag\\(")) > 0)
dg <- matrix(pickCoef(model, "Diag\\(", value=TRUE), 1, nr)
else
dg <- numeric(0)
## Normalize, cf. Clogg & Shihadeh (1994), eq. 5.3 et 5.4 (p. 83)
# Center
sc <- sweep(sc, 2:3, margin.table(sweep(sc, 1, p/sum(p), "*"), 2:3), "-")
for(l in 1:dim(sc)[3]) {
# Technique proposed in van der Heijden & Mooijaart (1995), Appendix C
# Weighted SVD taken from Goodman (1991), Appendix 4
lambda <- sc[,2,l] %o% sc[,1,l] - sc[,1,l] %o% sc[,2,l]
lambda0 <- sqrt(p %o% p) * lambda # Eq. A.4.3
sv <- svd(lambda0)
sc[,1:2,l] <- diag(1/sqrt(p)) %*% sv$u[,1:2] # Eq. A.4.7
phi <- sv$d[1]
# Since both dimensions share the same singular value, we cannot distinguish them
# and their order is random. The sign of the association reconstructed from scores
# must be the same as the original one: we use one cell, since all signs are the same.
# This operation affects the results, while the next one is merely a display convention.
sc[,1,l] <- sc[,1,l] * sign((sc[1,2,l] * sc[2,1,l] - sc[1,1,l] * sc[2,2,l])/lambda[1,2])
# Use the convention that we want the null score to be for the first category
# on the second dimension, and a positive score on the first dimension
# (like original article does with *last* category).
# The SVD always sets to 0 the score of the first category on one dimension.
# These two operations do not change the actual association.
if(which.min(abs(sc[1,,l])) == 1)
sc[,,l] <- sc[,,l] %*% matrix(c(0, 1, -1, 0), 2, 2)
if(sc[1,1,l] < 0)
sc[,,l] <- -sc[,,l]
# Integrate phi to layer coefficients
if(homogeneous)
layer <- layer * phi
else
layer[l,] <- layer[l,] * phi
}
# By convention, keep layer coefficients positive for the first layer category
if(homogeneous) {
if(layer[1,1] < 0) {
layer <- -layer
sc[, 2,] <- -sc[, 2,]
}
}
# For heterogeneous scores, always keep phi positive
else {
sc <- sweep(sc, 3:2, sign(layer), "*")
}
# The reference category is not really at 0, and this makes the display ugly
sc[abs(sc) < sqrt(.Machine$double.eps)] <- 0
## Prepare objects
rownames(sc) <- rownames(tab)
colnames(sc) <- colnames(layer) <- paste("Dim", 1:2, sep="")
rownames(layer) <- dimnames(tab)[[3]]
if(!homogeneous)
dimnames(sc)[[3]] <- dimnames(tab)[[3]]
else
dimnames(sc)[[3]] <- "All levels"
if(length(dg) > 0) {
# Diag() sorts coefficients alphabetically!
dg[,order(rownames(tab))] <- dg
colnames(dg) <- if(all(rownames(tab) == colnames(tab))) rownames(tab)
else paste(rownames(tab), colnames(tab), sep=":")
if(nrow(dg) > 1)
rownames(dg) <- dimnames(tab)[[3]]
else
rownames(dg) <- "All levels"
}
row.weights <- apply(tab, c(1, 3), sum, na.rm=TRUE)
col.weights <- apply(tab, c(2, 3), sum, na.rm=TRUE)
obj <- list(phi = layer, row = sc, col = sc, diagonal = dg,
weighting = weighting, row.weights = row.weights, col.weights = col.weights,
vars = vars)
class(obj) <- c("assoc.hmskewL", "assoc.symm", "assoc")
obj
}
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