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#' Coefficient Alpha and Item Statistics
#'
#' This function computes point estimate and confidence interval for the (ordinal)
#' coefficient alpha (aka Cronbach's alpha) along with the corrected item-total
#' correlation and coefficient alpha if item deleted.
#'
#' Ordinal coefficient alpha was introduced by Zumbo, Gadermann and Zeisser (2007)
#' which is obtained by applying the formula for computing coefficient alpha to the
#' polychoric correlation matrix instead of the variance-covariance or product-moment
#' correlation matrix. Note that Chalmers (2018) highlighted that the ordinal
#' coefficient alpha should be interpreted only as a hypothetical estimate of an
#' alternative reliability, whereby a test's ordinal categorical response options
#' have be modified to include an infinite number of ordinal response options and
#' concludes that coefficient alpha should not be reported as a measure of a test's
#' reliability. However, Zumbo and Kroc (2019) argued that Chalmers' critique of
#' ordinal coefficient alpha is unfounded and that ordinal coefficient alpha may
#' be the most appropriate quantifier of reliability when using Likert-type measurement
#' to study a latent continuous random variable.
#' Confidence intervals are computed using the procedure by Feldt, Woodruff and Salih
#' (1987). When computing confidence intervals using pairwise deletion, the average
#' sample size from all pairwise samples is used. Note that there are at least 10
#' other procedures for computing the confidence interval (see Kelley and
#' Pornprasertmanit, 2016), which are implemented in the \code{ci.reliability()}
#' function in the \pkg{MBESSS} package by Ken Kelley (2019).
#'
#' @param ... a matrix, data frame, variance-covariance or correlation
#' matrix. Note that raw data is needed to compute ordinal
#' coefficient alpha, i.e., \code{ordered = TRUE}. Alternatively,
#' an expression indicating the variable names in \code{data}
#' e.g., \code{item.alpha(x1, x2, x3, data = dat)}. Note that
#' the operators \code{.}, \code{+}, \code{-}, \code{~}, \code{:},
#' \code{::}, and \code{!} can also be used to select variables,
#' see 'Details' in the \code{\link{df.subset}} function.
#' @param data a data frame when specifying one or more variables in the
#' argument \code{...}. Note that the argument is \code{NULL}
#' when specifying a matrix, data frame, variance-covariance
#' or correlation matrix for the argument \code{...}.
#' @param exclude a character vector indicating items to be excluded from the
#' analysis.
#' @param std logical: if \code{TRUE}, the standardized coefficient alpha
#' is computed.
#' @param ordered logical: if \code{TRUE}, variables are treated as ordered (ordinal)
#' variables to compute ordinal coefficient alpha.
#' @param na.omit logical: if \code{TRUE}, incomplete cases are removed before
#' conducting the analysis (i.e., listwise deletion); if
#' \code{FALSE} (default), pairwise deletion is used.
#' @param print a character vector indicating which results to show, i.e.
#' \code{"all"} (default), for all results \code{"alpha"} for
#' the coefficient alpha, and \code{"item"} for item statistics.
#' @param digits an integer value indicating the number of decimal places to
#' be used for displaying coefficient alpha and item-total correlations.
#' @param conf.level a numeric value between 0 and 1 indicating the confidence level
#' of the interval.
#' @param as.na a numeric vector indicating user-defined missing values,
#' i.e. these values are converted to \code{NA} before conducting
#' the analysis.
#' @param write a character string for writing the results into a Excel file
#' naming a file with or without file extension '.xlsx', e.g.,
#' \code{"Results.xlsx"} or \code{"Results"}.
#' @param check logical: if \code{TRUE} (default), argument specification
#' is checked.
#' @param output logical: if \code{TRUE} (default), output is shown.
#'
#' @author
#' Takuya Yanagida \email{takuya.yanagida@@univie.ac.at}
#'
#' @seealso
#' \code{\link{item.omega}}, \code{\link{item.cfa}}, \code{\link{item.invar}},
#' \code{\link{item.reverse}}, \code{\link{item.scores}}, \code{\link{write.result}}
#'
#' @references
#' Chalmers, R. P. (2018). On misconceptions and the limited usefulness of ordinal alpha.
#' \emph{Educational and Psychological Measurement, 78}, 1056-1071.
#' https://doi.org/10.1177/0013164417727036
#'
#' Cronbach, L.J. (1951). Coefficient alpha and the internal structure of tests.
#' \emph{Psychometrika, 16}, 297-334. https://doi.org/10.1007/BF02310555
#'
#' Cronbach, L.J. (2004). My current thoughts on coefficient alpha and successor
#' procedures. \emph{Educational and Psychological Measurement, 64}, 391-418.
#' https://doi.org/10.1177/0013164404266386
#'
#' Feldt, L. S., Woodruff, D. J., & Salih, F. A. (1987). Statistical inference for
#' coefficient alpha. \emph{Applied Psychological Measurement}, 11 93-103.
#' https://doi.org/10.1177/014662168701100107
#'
#' Kelley, K., & Pornprasertmanit, S. (2016). Confidence intervals for population
#' reliability coefficients: Evaluation of methods, recommendations, and software
#' for composite measures. \emph{Psychological Methods, 21}, 69-92.
#' https://doi.org/10.1037/a0040086.
#'
#' Ken Kelley (2019). \emph{MBESS: The MBESS R Package}. R package version 4.6.0.
#' https://CRAN.R-project.org/package=MBESS
#'
#' Zumbo, B. D., & Kroc, E. (2019). A measurement is a choice and Stevens' scales
#' of measurement do not help make it: A response to Chalmers. \emph{Educational
#' and Psychological Measurement, 79}, 1184-1197.
#' https://doi.org/10.1177/0013164419844305
#'
#' Zumbo, B. D., Gadermann, A. M., & Zeisser, C. (2007). Ordinal versions of coefficients
#' alpha and theta for Likert rating scales. \emph{Journal of Modern Applied Statistical
#' Methods, 6}, 21-29. https://doi.org/10.22237/jmasm/1177992180
#'
#' @return
#' Returns an object of class \code{misty.object}, which is a list with following
#' entries:
#' \tabular{ll}{
#' \code{call} \tab function call \cr
#' \code{type} \tab type of analysis \cr
#' \code{data} \tab data frame used for the current analysis \cr
#' \code{args} \tab specification of function arguments \cr
#' \code{result} \tab list with result tables \cr
#' }
#'
#' @export
#'
#' @examples
#' dat <- data.frame(item1 = c(4, 2, 3, 4, 1, 2, 4, 2),
#' item2 = c(4, 3, 3, 3, 2, 2, 4, 1),
#' item3 = c(3, 2, 4, 2, 1, 3, 4, 1),
#' item4 = c(4, 1, 2, 3, 2, 3, 4, 2))
#'
#' # Example 1a: Compute unstandardized coefficient alpha and item statistics
#' item.alpha(dat)
#'
#' # Example 1b: Alternative specification using the 'data' argument
#' item.alpha(., data = dat)
#'
#' # Example 2: Compute standardized coefficient alpha and item statistics
#' item.alpha(dat, std = TRUE)
#'
#' # Example 3: Compute unstandardized coefficient alpha
#' item.alpha(dat, print = "alpha")
#'
#' # Example 4: Compute item statistics
#' item.alpha(dat, print = "item")
#'
#' # Example 5: Compute unstandardized coefficient alpha and item statistics while excluding item3
#' item.alpha(dat, exclude = "item3")
#'
#' # Example 6: Compute variance-covariance matrix
#' dat.cov <- cov(dat)
#' # Compute unstandardized coefficient alpha based on the variance-covariance matrix
#' item.alpha(dat.cov)
#'
#' # Compute correlation matrix
#' dat.cor <- cor(dat)
#' # Example 7: Compute standardized coefficient alpha based on the correlation matrix
#' item.alpha(dat.cor)
#'
#' # Example 8: Compute ordinal coefficient alpha
#' item.alpha(dat, ordered = TRUE)
#'
#' \dontrun{
#' # Example 9a: Write Results into a text file
#' result <- item.alpha(dat, write = "Alpha.xlsx")
#'
#' # Example 9b: Write Results into a Excel file
#' result <- item.alpha(dat, write = "Alpha.xlsx")
#'
#' result <- item.alpha(dat, output = FALSE)
#' write.result(result, "Alpha.xlsx")
#' }
item.alpha <- function(..., data = NULL, exclude = NULL, std = FALSE, ordered = FALSE,
na.omit = FALSE, print = c("all", "alpha", "item"), digits = 2,
conf.level = 0.95, as.na = NULL, write = NULL, append = TRUE,
check = TRUE, output = TRUE) {
#_____________________________________________________________________________
#
# Initial Check --------------------------------------------------------------
# Check if input '...' is missing
if (isTRUE(missing(...))) { stop("Please specify the argument '...'.", call. = FALSE) }
# Check if input '...' is NULL
if (isTRUE(is.null(substitute(...)))) { stop("Input specified for the argument '...' is NULL.", call. = FALSE) }
# Check if input 'data' is data frame
if (isTRUE(!is.null(data) && !is.data.frame(data))) { stop("Please specify a data frame for the argument 'data'.", call. = FALSE) }
#_____________________________________________________________________________
#
# Data -----------------------------------------------------------------------
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Data using the argument 'data' ####
if (isTRUE(!is.null(data))) {
# Variable names
var.names <- .var.names(..., data = data, check.chr = "a matrix, data frame, variance-covariance or correlation matrix")
# Extract variables
x <- data[, var.names]
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Data without using the argument 'data' ####
} else {
# Extract data
x <- eval(..., enclos = parent.frame())
}
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Raw data or cor/cov matrix ####
if (isTRUE(nrow(x) == ncol(x))) {
if (isTRUE(isSymmetric(x))) {
sym <- TRUE
x.raw <- FALSE
} else {
sym <- FALSE
x.raw <- TRUE
}
# Diagonal is all 1?
if (isTRUE(sym)) {
std <- ifelse(all(diag(x) == 1L), TRUE, FALSE)
}
} else {
x.raw <- TRUE
}
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Ordered ####
if (isTRUE(ordered)) {
# Check if raw data is availeble
if (!isTRUE(x.raw)) {
stop("Please submit raw data to the argument 'x' to compute ordinal coefficient alpha.", call. = FALSE)
}
# Compute polychoric correlation matrix
x <- misty::cor.matrix(x, method = "poly", output = FALSE)$result$cor
x.raw <- FALSE
std <- TRUE
}
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## As data frame ####
x <- as.data.frame(x, stringsAsFactors = FALSE)
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Numeric Variables ####
# Non-numeric variables
non.num <- !vapply(x, is.numeric, FUN.VALUE = logical(1L))
if (isTRUE(any(non.num))) {
x <- x[, -which(non.num), drop = FALSE]
# Variables left
if (isTRUE(ncol(x) == 0L)) { stop("No variables left for analysis after excluding non-numeric variables.", call. = FALSE) }
warning(paste0("Non-numeric variables were excluded from the analysis: ", paste(names(which(non.num)), collapse = ", ")), call. = FALSE)
}
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Exclude items and specify user-defined NA ####
# Raw data
if (isTRUE(x.raw)) {
if (isTRUE(!is.null(exclude))) {
# Check input 'exclude'
check.ex <- !exclude %in% colnames(x)
if (isTRUE(any(check.ex))) { stop(paste0("Items to be excluded from the analysis were not found in 'x': ", paste(exclude[check.ex], collapse = ", ")), call. = FALSE) }
x <- x[, which(!colnames(x) %in% exclude)]
# One item left
if (isTRUE(is.null(dim(x)))) {
stop("At least two items after excluding items are needed to compute coefficient alpha.", call. = FALSE)
}
}
# Convert user-missing values into NA
if (isTRUE(!is.null(as.na))) { x <- .as.na(x, na = as.na) }
# Covariance or correlation matrix
} else {
if (isTRUE(!is.null(exclude))) {
x <- x[, which(!colnames(x) %in% exclude)]
x <- x[which(!rownames(x) %in% exclude), ]
# One item left
if (isTRUE(is.null(dim(x)))) {
stop("At least two items after excluding items are needed to compute coefficient alpha.", call. = FALSE)
}
}
}
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Method for handling missing data ####
# Listwise deletion
if (isTRUE(na.omit)) {
x <- na.omit(x)
}
#_____________________________________________________________________________
#
# Input Check ----------------------------------------------------------------
# Check input 'check'
if (isTRUE(!is.logical(check))) { stop("Please specify TRUE or FALSE for the argument 'check'.", call. = FALSE) }
if (isTRUE(check)) {
# Matrix or data frame for the argument 'x'?
if (isTRUE(!is.matrix(x) && !is.data.frame(x))) { stop("Please specify a matrix, a data frame, a variance-covariance or correlation matrix for the argument 'x'.", call. = FALSE) }
# Check input 'x': One item
if (isTRUE(ncol(x) == 1L)) { stop("Please specify at least two items to compute coefficient alpha.", call. = FALSE) }
# Check input 'x': Zero variance
if (isTRUE(nrow(x) != ncol(x))) {
x.check <- vapply(as.data.frame(x, stringsAsFactors = FALSE), function(y) length(na.omit(unique(y))) == 1L, FUN.VALUE = logical(1L))
if (isTRUE(any(x.check))) {
stop(paste0("Following variables in the matrix or data frame specified in 'x' have zero variance: ", paste(names(which(x.check)), collapse = ", ")), call. = FALSE)
}
}
# Check input 'std'
if (isTRUE(!is.logical(std))) { stop("Please specify TRUE or FALSE for the argument 'std'.", call. = FALSE) }
# Check input 'ordered'
if (isTRUE(!is.logical(ordered))) { stop("Please specify TRUE or FALSE for the argument 'ordered'.", call. = FALSE) }
# Check input 'na.omit'
if (isTRUE(!is.logical(na.omit))) { stop("Please specify TRUE or FALSE for the argument 'na.omit'.", call. = FALSE) }
# Check input 'print'
if (isTRUE(!all(print %in% c("all", "alpha", "item")))) { stop("Character strings in the argument 'print' do not all match with \"all\", \"alpha\", or \"item\".", call. = FALSE) }
# Check input 'conf.level'
if (isTRUE(conf.level >= 1L || conf.level <= 0L)) { stop("Please specifiy a numeric value between 0 and 1 for the argument 'conf.level'.", call. = FALSE) }
# Check input 'digits'
if (isTRUE(digits %% 1L != 0L || digits < 0L)) { stop("Specify a positive integer number for the argument 'digits'.", call. = FALSE) }
# Check input 'append'
if (isTRUE(!is.logical(append))) { stop("Please specify TRUE or FALSE for the argument 'append'.", call. = FALSE) }
# Check input 'output'
if (isTRUE(!is.logical(output))) { stop("Please specify TRUE or FALSE for the argument 'output'.", call. = FALSE) }
}
#_____________________________________________________________________________
#
# Arguments ------------------------------------------------------------------
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Print coefficient alpha and/or item statistic ####
if (isTRUE(all(c("all", "alpha", "item") %in% print))) { print <- c("alpha", "item") }
if (isTRUE(length(print) == 1L && "all" %in% print)) { print <- c("alpha", "item") }
#_____________________________________________________________________________
#
# Main Function --------------------------------------------------------------
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Correlation or variance-covariance matrix ####
if (isTRUE(x.raw)) {
if (isTRUE(std)) {
mat.sigma <- cor(x, use = "pairwise.complete.obs", method = "pearson")
} else {
mat.sigma <- cov(x, use = "pairwise.complete.obs", method = "pearson")
}
} else {
mat.sigma <- x
}
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Coefficient Alpha ####
# Define Coefficient alpha function
alpha.function <- function(mat.sigma, p) {
return((p / (p - 1)) * (1L - sum(diag(as.matrix(mat.sigma))) / sum(as.matrix(mat.sigma))))
}
p <- ncol(mat.sigma)
alpha.mat.sigma <- alpha.function(mat.sigma, p)
if (isTRUE(x.raw)) {
alpha.x <- data.frame(n = nrow(x), items = ncol(mat.sigma), alpha = alpha.mat.sigma,
stringsAsFactors = FALSE)
} else {
alpha.x <- data.frame(items = ncol(mat.sigma), alpha = alpha.mat.sigma,
stringsAsFactors = FALSE)
}
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Confidence interval ####
if (isTRUE(x.raw)) {
if (isTRUE(any(is.na(x)) && !isTRUE(na.omit))) {
df1 <- mean(apply(combn(ncol(x), 2L), 2L, function(y) nrow(na.omit(cbind(x[, y[1L]], x[, y[2L]]))))) - 1L
} else {
df1 <- nrow(na.omit(x)) - 1L
}
df2 <- (ncol(x) - 1L) * df1
alpha.low <- 1L - (1L - alpha.mat.sigma) * qf(1L - (1L - conf.level) / 2L, df1, df2)
alpha.upp <- 1L - (1L - alpha.mat.sigma) * qf((1L - conf.level) / 2L, df1, df2)
alpha.x <- data.frame(alpha.x, low = alpha.low, upp = alpha.upp, stringsAsFactors = FALSE)
}
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Corrected item-total correlation and alpha if item deleted ####
if (isTRUE(x.raw)) {
itemstat <- matrix(rep(NA, times = ncol(x)*2L), ncol = 2L,
dimnames = list(NULL, c("it.cor", "alpha")))
for (i in seq_len(ncol(x))) {
var <- colnames(x)[i]
itemstat[i, 1L] <- ifelse(ncol(x) > 2L, cor(x[, i], rowMeans(x[, -grep(var, colnames(x))], na.rm = TRUE),
use = "pairwise.complete.obs"), NA)
if (isTRUE(std)) {
itemstat[i, 2L] <- ifelse(ncol(x) > 2L, alpha.function(cor(x[, -grep(var, colnames(x))],
use = "pairwise.complete.obs", method = "pearson"), p = (ncol(x) - 1L)), NA)
} else {
itemstat[i, 2L] <- ifelse(ncol(x) > 2L, alpha.function(cov(x[, -grep(var, colnames(x))],
use = "pairwise.complete.obs", method = "pearson"), p = (ncol(x) - 1L)), NA)
}
}
#...................
### Descriptive statistics ####
itemstat <- data.frame(var = colnames(x),
misty::descript(x, output = FALSE)$result[, c("n", "nNA", "pNA", "m", "sd", "min", "max")],
itemstat,
stringsAsFactors = FALSE)
} else {
itemstat <- NULL
}
#_____________________________________________________________________________
#
# Return object --------------------------------------------------------------
object <- list(call = match.call(),
type = "item.alpha",
data = x,
args = list(exclude = exclude, std = std, ordered = ordered, na.omit = na.omit,
print = print, digits = digits, conf.level = conf.level, as.na = as.na,
write = write, append = append, check = check, output = output),
result = list(alpha = alpha.x, itemstat = itemstat))
class(object) <- "misty.object"
#_____________________________________________________________________________
#
# Write Results --------------------------------------------------------------
if (isTRUE(!is.null(write))) {
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Text file ####
if (isTRUE(grepl("\\.txt", write))) {
# Send R output to textfile
sink(file = write, append = ifelse(isTRUE(file.exists(write)), append, FALSE), type = "output", split = FALSE)
if (isTRUE(append && file.exists(write))) { write("", file = write, append = TRUE) }
# Print object
print(object, check = FALSE)
# Close file connection
sink()
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Excel file ####
} else {
misty::write.result(object, file = write)
}
}
#_____________________________________________________________________________
#
# output ---------------------------------------------------------------------
if (isTRUE(output)) { print(object, check = FALSE) }
return(invisible(object))
}
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