R/mean2_2004KY.R

In kisungyou/SHT: Statistical Hypothesis Testing Toolbox

#' Two-sample Test for Multivariate Means by Krishnamoorthy and Yu (2004)
#' 
#' Given two multivariate data \eqn{X} and \eqn{Y} of same dimension, it tests
#' \deqn{H_0 : \mu_x = \mu_y\quad vs\quad H_1 : \mu_x \neq \mu_y}
#' using the procedure by Krishnamoorthy and Yu (2004), which is a modified 
#' version of Nel and Van der Merwe (1986).
#' 
#' @param X an \eqn{(n_x \times p)} data matrix of 1st sample.
#' @param Y an \eqn{(n_y \times p)} data matrix of 2nd sample.
#' 
#' @return a (list) object of \code{S3} class \code{htest} containing: \describe{
#' \item{statistic}{a test statistic.}
#' \item{p.value}{\eqn{p}-value under \eqn{H_0}.}
#' \item{alternative}{alternative hypothesis.}
#' \item{method}{name of the test.}
#' \item{data.name}{name(s) of provided sample data.}
#' }
#' 
#' @examples 
#' ## CRAN-purpose small example
#' smallX = matrix(rnorm(10*3),ncol=3)
#' smallY = matrix(rnorm(10*3),ncol=3)
#' mean2.2004KY(smallX, smallY) # run the test
#' 
#' \dontrun{
#' ## empirical Type 1 error 
#' niter   = 1000
#' counter = rep(0,niter)  # record p-values
#' for (i in 1:niter){
#'   X = matrix(rnorm(50*5), ncol=10)
#'   Y = matrix(rnorm(50*5), ncol=10)
#'   
#'   counter[i] = ifelse(mean2.2004KY(X,Y)$p.value < 0.05, 1, 0)
#' }
#' 
#' ## print the result
#' cat(paste("\n* Example for 'mean2.2004KY'\n","*\n",
#' "* number of rejections   : ", sum(counter),"\n",
#' "* total number of trials : ", niter,"\n",
#' "* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))
#' }
#' 
#' @references 
#' \insertRef{krishnamoorthy_modified_2004}{SHT}
#' 
#' @concept mean_multivariate
#' @export
mean2.2004KY <- function(X, Y){
  # First two parts are commonly available for 
  #   mean2.1965Yao
  #   mean2.1980Johansen
  #   mean2.1986NVM
  #   mean2.2004KY
  ##############################################################
  # PREPROCESSING
  check_nd(X)
  check_nd(Y)
  if (ncol(X)!=ncol(Y)){
    stop("* mean2.2004KY : two samples X and Y should be of same dimension.")
  }
  p = ncol(X)
  
  ##############################################################
  # PARAMETERS AND PRELIMINARY COMPUTATIONS
  N1 = nrow(X); n1 = N1-1
  N2 = nrow(Y); n2 = N2-1
  
  x1bar = as.vector(colMeans(X)) # means
  x2bar = as.vector(colMeans(Y))
  xbardiff = (x1bar-x2bar)
  
  S1 = cov(X)/N1 # sample tilde' covariances
  S2 = cov(Y)/N2
  SS  = (S1+S2)
  
  # S1inv = pracma::pinv(S1) # inverse of covariances
  # S2inv = pracma::pinv(S2)
  SSinv = pracma::pinv(SS)
  
  T2 = aux_quadform(SSinv, xbardiff) # Hotelling's T statistic
  
  ##############################################################
  # SPECIFICS
  SS1inv = S1%*%SSinv # S_1 S^{-1}
  SS2inv = S2%*%SSinv # S_2 S^{-1}
  
  v.top = p*(p+1)
  v.bot = (1/n1)*(aux_trace(SS1inv%*%SS1inv) + (aux_trace(SS1inv)^2)) + (1/n2)*(aux_trace(SS2inv%*%SS2inv) + (aux_trace(SS2inv)^2))
  v     = (v.top/v.bot)
  
  thestat = T2
  T2adj   = T2*(v-p+1)/(v*p)
  pvalue  = pf(T2adj, p, (v-p+1), lower.tail = FALSE)
  
  ##############################################################
  # REPORT
  hname   = "Two-sample Test for Multivariate Means by Krishnamoorthy and Yu (2004)"
  Ha      = "true means are different."
  
  
  DNAME = paste(deparse(substitute(X))," and ",deparse(substitute(Y)),sep="") # borrowed from HDtest
  names(thestat) = "T2"
  res   = list(statistic=thestat, p.value=pvalue, alternative = Ha, method=hname, data.name = DNAME)
  class(res) = "htest"
  return(res)
}