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#' Two-Sample F-Test for Variance
#'
#' Given two univariate samples \eqn{x} and \eqn{y}, it tests
#' \deqn{H_0 : \sigma_x^2 \left\lbrace =,\geq,\leq \right\rbrace \sigma_y^2\quad vs\quad H_1 : \sigma_x^2 \left\lbrace \neq,<,>\right\rbrace \sigma_y^2}.
#'
#' @param x a length-\eqn{n} data vector.
#' @param y a length-\eqn{m} data vector.
#' @param alternative specifying the alternative hypothesis.
#'
#' @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
#' x = rnorm(10)
#' y = rnorm(10)
#' var2.F(x, y, alternative="g") ## Ha : var(x) >= var(y)
#' var2.F(x, y, alternative="l") ## Ha : var(x) <= var(y)
#' var2.F(x, y, alternative="t") ## Ha : var(x) =/= var(y)
#'
#' \donttest{
#' ## empirical Type 1 error
#' niter = 1000
#' counter = rep(0,niter) # record p-values
#' for (i in 1:niter){
#' x = rnorm(57) # sample x from N(0,1)
#' y = rnorm(89) # sample y from N(0,1)
#'
#' counter[i] = ifelse(var2.F(x,y)$p.value < 0.05, 1, 0)
#' }
#'
#' ## print the result
#' cat(paste("\n* Example for 'var2.F'\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{snedecor_statistical_1996}{SHT}
#'
#' @concept variance
#' @export
var2.F <- function(x, y, alternative=c("two.sided","less","greater")){
##############################################################
# PREPROCESSING
check_1d(x) # univariate vector of 1st class
check_1d(y) # univariate vector of 2nd class
if (missing(alternative)){
alternative = "two.sided"
} else {
if (pracma::strcmp(alternative,"g")){
alternative = "greater"
} else if (pracma::strcmp(alternative,"t")){
alternative = "two.sided"
} else if (pracma::strcmp(alternative,"l")){
alternative = "less"
}
alternative = match.arg(alternative)
}
##############################################################
# COMPUTATION : PRELIMINARY
n = length(x)
m = length(y)
varx = stats::var(x)
vary = stats::var(y)
thestat = (varx/vary)
##############################################################
# COMPUTATION : HYPOTHESIS and DETERMINATION
if (pracma::strcmp(alternative,"two.sided")){
tmpval = stats::pf(thestat,(n-1),(m-1))
pvalue = 2*min(tmpval, 1-tmpval)
Ha = "two true variances are different."
} else if (pracma::strcmp(alternative,"less")){
pvalue = stats::pf(thestat,(n-1),(m-1),lower.tail = TRUE)
Ha = "true variance of x is smaller than true variance of y."
} else if (pracma::strcmp(alternative,"greater")){
pvalue = stats::pf(thestat,(n-1),(m-1),lower.tail = FALSE)
Ha = "true variance of x is greater than true variance of y."
}
##############################################################
# REPORT
hname = "Two-Sample F Test for Variance."
DNAME = paste(deparse(substitute(x))," and ",deparse(substitute(y)),sep="") # borrowed from HDtest
names(thestat) = "statistic"
res = list(statistic=thestat, p.value=pvalue, alternative = Ha, method=hname, data.name = DNAME)
class(res) = "htest"
return(res)
}
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