# R/nSeScree.r In nFactors: Parallel Analysis and Other Non Graphical Solutions to the Cattell Scree Test

#### Documented in nSeScree

#' Standard Error Scree and Coefficient of Determination Procedures to
#' Determine the Number of Components/Factors
#'
#' This function computes the \emph{seScree} (\eqn{S_{Y \bullet X}}) indices
#' (Zoski and Jurs, 1996) and the coefficient of determination indices of
#' Nelson (2005) \eqn{R^2} for determining the number of components/factors to
#' retain.
#'
#' The Zoski and Jurs \eqn{S_{Y \bullet X}} index is the standard error of the
#' estimate (predicted) eigenvalues by the regression from the \eqn{(k+1,
#' \ldots, p)} subsequent ranks of the eigenvalues. The standard error is
#' computed as:
#'
#' \hat{\lambda}_k)^2} {p-2} } } \cr
#'
#' A value of \eqn{1/p} is choosen as the criteria to determine the number of
#' components or factors to retain, \emph{p} corresponding to the number of
#' variables.
#'
#' The Nelson \eqn{R^2} index is simply the multiple regresion coefficient of
#' determination for the \eqn{k+1, \ldots, p} eigenvalues.  Note that Nelson
#' didn't give formal prescriptions for the criteria for this index. He only
#' suggested that a value of 0.75 or more must be considered. More is to be
#' done to explore adequate values.
#'
#' @param x numeric: eigenvalues.
#' @param cor logical: if \code{TRUE} computes eigenvalues from a correlation
#' matrix, else from a covariance matrix
#' @param model character: \code{"components"} or \code{"factors"}
#' @param details logical: if \code{TRUE} also returns details about the
#' computation for each eigenvalue.
#' @param r2limen numeric: criterion value retained for the coefficient of
#' determination indices.
#' @param ...  variable: additionnal parameters to give to the
#' \code{eigenComputes} and \code{cor} or \code{cov} functions
#' @return \item{nFactors}{ numeric: number of components/factors retained by
#' the seScree procedure. } \item{details}{ numeric: matrix of the details for
#' each index.}
#' @author Gilles Raiche \cr Centre sur les Applications des Modeles de
#' Reponses aux Items (CAMRI) \cr Universite du Quebec a Montreal\cr
#' \email{raiche.gilles@@uqam.ca}
#'
#' @references
#' Nasser, F. (2002). The performance of regression-based
#' variations of the visual scree for determining the number of common factors.
#' \emph{Educational and Psychological Measurement, 62(3)}, 397-419.
#'
#' Nelson, L. R. (2005). Some observations on the scree test, and on
#' coefficient alpha. \emph{Thai Journal of Educational Research and
#' Measurement, 3(1)}, 1-17.
#'
#' Raiche, G., Walls, T. A., Magis, D., Riopel, M. and Blais, J.-G. (2013). Non-graphical solutions
#' for Cattell's scree test. Methodology, 9(1), 23-29.
#'
#' Zoski, K. and Jurs, S. (1993). Using multiple regression to determine the
#' number of factors to retain in factor analysis. \emph{Multiple Linear
#' Regression Viewpoints, 20}(1), 5-9.
#'
#' Zoski, K. and Jurs, S. (1996). An objective counterpart to the visuel scree
#' test for factor analysis: the standard error scree. \emph{Educational and
#' Psychological Measurement, 56}(3), 443-451.
#' @export
# #' @importFrom stats sd lm
#' @keywords multivariate
#' @examples
#'
#' ## SIMPLE EXAMPLE OF SESCREE AND R2 ANALYSIS
#'
#'  data(dFactors)
#'  eig      <- dFactors$Raiche$eigenvalues
#'
#'  results  <- nSeScree(eig)
#'  results
#'
#'  plotuScree(eig, main=paste(results$nFactors[1], " or ", results$nFactors[2],
#'                             " factors retained by the sescree and R2 procedures",
#'                             sep=""))
#'
nSeScree <-
function(x, cor=TRUE, model="components", details=TRUE, r2limen=0.75, ...) {
x               <- eigenComputes(x, cor=cor, model=model, ...)
detail          <- NULL
n               <- length(x)
criteria        <- 1/n
seScreeCriteria <- R2Criteria <- 0
if (n < 3) stop("The number of variables must be at least 3.")
i               <- 1
seScree         <- R2 <- numeric(n-3)
while ((i) <= (n-2)) {
xa              <- c(i:n)
ya              <- x[i:n]
ma              <- lm(ya ~ xa)
seScree[i]      <- sd(ya)*sqrt((1-summary(ma)$r.squared) * ((length(ya)-1)/(length(ya)-2))) # Howell(2008, p. 253) seScreeCriteria <- seScreeCriteria + as.numeric(seScree[i] > criteria) R2[i] <- summary(ma)$r.squared
R2Criteria      <- R2Criteria + as.numeric(R2[i] < r2limen)
i               <- i + 1
}
if (details == TRUE) detail  <- data.frame(v=(1:(n-2)),values=x[1:(n-2)], seScree, R2)
seScree <- seScreeCriteria
R2      <- R2Criteria
res     <- list(detail=detail, nFactors=c(se=seScree, R2=R2))
class(res) <- c("nFactors","list")
return(res)
}


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nFactors documentation built on Oct. 10, 2022, 5:07 p.m.