#' Rank-based Tests for Multivariate Data in Nonparametric Factorial Designs
#'
#' The rankMANOVA function calculates an ANOVA-type
#' statistic (ATS) with (wild) bootstrap p-values for nonparametric factorial designs with
#' multivariate data.
#'
#' @param formula A model \code{\link{formula}} object. The left hand side
#' contains the response variables and the right hand side contains the factor
#' variables of interest. Data must be
#' provided in wide format.
#' @param data A data.frame containing the variables in
#' \code{formula}.
#' @param iter The number of iterations used for calculating the resampled
#' statistic. The default option is 10,000.
#' @param alpha A number specifying the significance level; the default is 0.05.
#' @param resampling The resampling method to be used, one of "bootstrap"
#' (sample-specific bootstrap approach) and "WildBS" (wild bootstrap approach with
#' Rademacher weights). The default is "WildBS".
#' @param para Logical: should parallel computing be used? Default is FALSE.
#' @param CPU The number of cores used for parallel computing. If omitted, cores are
#' detected via \code{\link[parallel]{detectCores}}.
#' @param seed A random seed for the resampling procedure. If omitted, no
#' reproducible seed is set.
#' @param nested.levels.unique A logical specifying whether the levels of the nested factor(s)
#' are labeled uniquely or not. Default is FALSE, i.e., the levels of the nested
#' factor are the same for each level of the main factor. For an example and more explanations
#' see the GFD package and the corresponding vignette.
#' @param dec Number of decimals the results should be rounded to. Default is 3.
#'
#' @details Implemented is an ANOVA-type test statistic for testing hypotheses formulated in Mann-Whitney-type
#' effects in nonparametric factorial designs. Statistical inference is based on a wild or a sample-specific
#' bootstrap approach. The unweighted treatment effects considered do not depend on sample sizes and allow for
#' transitive ordering. The package thus provides an extension of the univariate \code{\link[rankFD]{rankFD}}
#' package to multivariate data.
#'
#' @return A \code{rankMANOVA} object containing the following components:
#' \item{Descriptive}{Some descriptive statistics of the data for all factor
#' level combinations. Displayed are the number of individuals per factor
#' level combination and the unweighted treatment effects for each dimension.}
#' \item{Test}{The test statistic(s) and p-value(s) based on the
#' chosen bootstrap approach.}
#'
#'@section NOTE: The number of bootstrap iterations has been set to 100 in the examples due
#' to runtime restrictions on CRAN. Usually it is recommended to use at least 1000 iterations.
#'
#' @examples
#' data("marketing")
#' mymar <- marketing[, c("Sex", "Income", "Edu")]
#' mymar2 <- na.omit(mymar)
#' test <- rankMANOVA(cbind(Income, Edu) ~ Sex, data = mymar2, iter=100,
#' resampling = "WildBS", CPU = 1)
#' summary(test)
#'
#'
#' @seealso \code{\link[rankFD]{rankFD}}
#'
#' @references
#' Dobler, D., Friedrich, S., and Pauly, M. (2017). Nonparametric MANOVA in Mann-Whitney effects.
#'
#' @importFrom graphics axis legend par plot title abline points
#' @importFrom stats ecdf formula model.frame pchisq pf qt terms var cov rbinom quantile
#' @importFrom methods hasArg
#' @importFrom parallel makeCluster parSapply detectCores
#'
#' @export
rankMANOVA <- function(formula, data,
iter = 10000, alpha = 0.05,
para = FALSE, CPU, dec = 3,
seed, resampling = "bootstrap", nested.levels.unique = FALSE){
if (!(resampling %in% c("bootstrap", "WildBS"))){
stop("Resampling must be one of 'bootstrap' and 'WildBS'!")
}
output <- list()
if(para){
test1 <- hasArg(CPU)
if(!test1){
CPU <- parallel::detectCores()
}
}
test2 <- hasArg(seed)
if(!test2){
seed <- 0
}
input_list <- list(formula = formula, data = data,
iter = iter, alpha = alpha, resampling = resampling, dec = dec,
seed = seed)
dat <- model.frame(formula, data)
nr_hypo <- attr(terms(formula), "factors")
perm_names <- t(attr(terms(formula), "factors")[-1, ])
fac_names <- colnames(nr_hypo)
outcome_names <- rownames(nr_hypo)[1] # names of outcome variables
# extract names of outcome variables
if (grepl("cbind", outcome_names)){
split1 <- strsplit(outcome_names, "(", fixed = TRUE)[[1]][-1]
split2 <- strsplit(split1, ")", fixed = TRUE)[[1]]
split3 <- strsplit(split2, ",")[[1]]
} else {
split3 <- outcome_names
}
EF <- rownames(nr_hypo)[-1] # names of influencing factors
nf <- length(EF)
names(dat) <- c("response", EF)
#no. dimensions
d <- ncol(dat$response)
if (!is.numeric(d)){
d <- 1
}
fl <- NA
for (aa in 1:nf) {
fl[aa] <- nlevels(as.factor(as.character(dat[, (aa + 1)])))
}
levels <- list()
for (jj in 1:nf) {
levels[[jj]] <- levels(as.factor(as.character(dat[, (jj + 1)])))
}
lev_names <- expand.grid(levels)
# number of hypotheses
tmp <- 0
for (i in 1:nf) {
tmp <- c(tmp, choose(nf, i))
nh <- sum(tmp)
}
# correct formula?
if (length(fac_names) != nf && length(fac_names) != nh){
stop("Something is wrong with the formula. Please specify all or no interactions in crossed designs.")
}
# mixture of nested and crossed designs is not possible
if (length(fac_names) != nf && 2 %in% nr_hypo) {
stop("A model involving both nested and crossed factors is
not implemented!")
}
if (nf == 1) {
# one-way layout
nest <- FALSE
dat2 <- dat[order(dat[, 2]), ]
fac.groups <- dat2[, 2]
hypo_matrices <- list((diag(fl) - matrix(1 / fl, ncol = fl, nrow = fl)) %x% diag(d))
#------------------------ end one-way layout -------------------------------------------------#
} else {
dat2 <- dat[do.call(order, dat[, 2:(nf + 1)]), ]
fac.groups <- do.call(list, dat2[, 2:(nf+1)])
}
Y <- split(dat2, fac.groups)
n <- sapply(Y, nrow)
nested <- grepl(":", formula)
nested2 <- grepl("%in%", formula)
if (sum(nested) > 0 || sum(nested2) > 0) {
# nested
nest <- TRUE
# if nested factor is named uniquely
if (nested.levels.unique){
# delete factorcombinations which don't exist
n <- n[n != 0]
# create correct level combinations
blev <- list()
lev_names <- list()
for (ii in 1:length(levels[[1]])) {
blev[[ii]] <- droplevels(as.factor(as.character(dat[, 3][dat[, 2] == levels[[1]][ii]])))
lev_names[[ii]] <- rep(levels[[1]][ii], length(blev[[ii]]))
}
if (nf == 2) {
lev_names <- as.factor(unlist(lev_names))
blev <- as.factor(unlist(blev))
lev_names <- cbind.data.frame(lev_names, blev)
} else {
lev_names <- lapply(lev_names, rep,
length(levels[[3]]) / length(levels[[2]]))
lev_names <- lapply(lev_names, sort)
lev_names <- as.factor(unlist(lev_names))
blev <- lapply(blev, rep, length(levels[[3]]) / length(levels[[2]]))
blev <- lapply(blev, sort)
blev <- as.factor(unlist(blev))
lev_names <- cbind.data.frame(lev_names, blev, as.factor(levels[[3]]))
}
# correct for wrong counting of nested factors
if (nf == 2) {
fl[2] <- fl[2] / fl[1]
} else if (nf == 3) {
fl[3] <- fl[3] / fl[2]
fl[2] <- fl[2] / fl[1]
}
}
hypo_matrices <- HN_MANOVA(fl, d)
} else {
# crossed
nest <- FALSE
## adapting formula argument, if interaction term missing
if (nrow(perm_names) != nh) {
#stop("For crossed designs, an interaction term must be specified in the formula.")
form2 <- as.formula(paste(outcome_names, "~", paste(fac_names, collapse = "*")))
perm_names2 <- t(attr(terms(form2), "factors")[-1, ])
fac_names2 <- attr(terms(form2), "term.labels")
hyps <- HC_MANOVA(fl, perm_names2, fac_names2, d, nh)
hypo_matrices <- hyps[[1]]
fac_names2 <- hyps[[2]]
# choose only relevant entries of the hypo matrices
indices <- grep(":", fac_names2, invert = T)
hypo_matrices <- lapply(indices, function(x) hypo_matrices[[x]])
} else if(nf !=1){
hyps <- HC_MANOVA(fl, perm_names, fac_names, d, nh)
hypo_matrices <- hyps[[1]]
fac_names <- hyps[[2]]
}
}
# correcting for "empty" combinations (if no interaction specified)
n.groups <- prod(fl)
if(nf != 1 & length(Y) != n.groups){
index <- NULL
for(i in 1:length(Y)){
if(nrow(Y[[i]]) == 0){
index <- c(index, i)
}
}
Y <- Y[-index]
}
Y2 <- lapply(Y, function(x) x$response)
if (d==1){
Y2 <- lapply(Y2, function(x) as.matrix(x))
}
# ---------------------- error detection ------------------------------------
# only 3-way nested designs are possible
if (sum(nested) > 0 && nf >= 4) {
stop("Four- and higher way nested designs are
not implemented!")
}
# no factor combinations with less than 2 observations
if (0 %in% n || 1 %in% n) {
stop("There is at least one factor-level combination
with less than 2 observations!")
}
#--------------------------------------------------------------------------#
statistic_out <- matrix(NA, nrow = length(hypo_matrices), ncol = 2) # Test statistic, p-value
rownames(statistic_out) <- fac_names
colnames(statistic_out) <- c("Test statistic", "p-value")
# calculate results
for (i in 1:length(hypo_matrices)) {
results <- rankbs(Y2, n, hypo_matrices[[i]], d, iter, alpha, para, CPU, seed, resampling)
statistic_out[i, ] <- round(results$statistic, dec)
}
p_out <- round(results$p, dec)
descriptive <- cbind(unique(lev_names), n, p_out)
colnames(descriptive) <- c(EF, "n", split3)
# other information needed, eg, for post-hoc tests-------------------#
other <- list(dim=d, nf = nf, fl = fl, outcomes = split3, fac_names = fac_names,
Y2 = Y2, quant = results$quant)
# Output ------------------------------------------------------
output$input <- input_list
output$Descriptive <- descriptive
output$Test <- statistic_out
output$other <- other
class(output) <- "rankMANOVA"
return(output)
}
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