#' Two-sample Student's t-test for Univariate Means
#'
#' Given two univariate samples \eqn{x} and \eqn{y}, it tests
#' \deqn{H_0 : \mu_x^2 \left\lbrace =,\geq,\leq \right\rbrace \mu_y^2\quad vs\quad H_1 : \mu_x^2 \left\lbrace \neq,<,>\right\rbrace \mu_y^2}
#' using the procedure by Student (1908) and Welch (1947).
#'
#' @param x a length-\eqn{n} data vector.
#' @param y a length-\eqn{m} data vector.
#' @param alternative specifying the alternative hypothesis.
#' @param paired a logical; whether consider two samples as paired.
#' @param var.equal a logical; if \code{FALSE}, use Welch's correction.
#'
#' @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
#' \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(mean2.ttest(x,y)$p.value < 0.05, 1, 0)
#' }
#'
#' ## print the result
#' cat(paste("\n* Example for 'mean2.ttest'\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{student_probable_1908}{SHT}
#'
#' \insertRef{student_probable_1908a}{SHT}
#'
#' \insertRef{welch_generalization_1947}{SHT}
#'
#' @concept mean_univariate
#' @export
mean2.ttest <- function(x, y, alternative=c("two.sided","less","greater"), paired=FALSE, var.equal=FALSE){
##############################################################
# 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 (alternative=="g"){
alternative = "greater"
} else if (alternative=="t"){
alternative = "two.sided"
} else if (alternative=="l"){
alternative = "less"
}
alternative = match.arg(alternative)
}
##############################################################
# CASE 1 : PAIRED/DEPENDENT T-TEST
if (paired==TRUE){
res = mean2.Student.paired(x,y,alternative)
} else {
##############################################################
# CASE 2 : INDEPENDENT T-TEST
# 2-1. Common Parameters and Computations
n1 = length(x)
n2 = length(y)
s1 = sd(x)
s2 = sd(y)
xbar1 = mean(x)
xbar2 = mean(y)
# 2-2. branching
if (var.equal==TRUE){
hname = "Student's t-test for Independent Samples with Equal Variance Assumption"
df = (n1+n2-2)
sp = sqrt(((n1-1)*(s1^2) + (n2-1)*(s2^2))/(n1+n2-2))
t = ((xbar1-xbar2)/(sp*sqrt((1/n1)+(1/n2))))
} else if (var.equal==FALSE){
hname = "Student's t-test for Independent Samples with Unequal Variance Assumption"
df1 = ((((s1^2)/n1) + ((s2^2)/n2))^2)
df2 = (((((s1^2)/n1)^2)/(n1-1)) + ((((s2^2)/n2)^2)/(n2-1)))
df = (df1/df2)
sp = sqrt(((s1^2)/n1) + ((s2^2)/n2))
t = ((xbar1-xbar2)/sp)
}
# 2-3. hypothesis and determination
if (alternative=="two.sided"){
pvalue = 2*pt(abs(t),df,lower.tail = FALSE)
Ha = "two true means are different."
} else if (alternative=="less"){
pvalue = pt(t,df,lower.tail = TRUE)
Ha = "true mean of x is smaller than true mean of y."
} else if (alternative=="greater"){
pvalue = pt(t,df,lower.tail = FALSE)
Ha = "true mean of x is greater than true mean of y."
}
# if (pvalue < alpha){
# conclusion = "Reject Null Hypothesis"
# } else {
# conclusion = "Not Reject Null Hypothesis"
# }
thestat = t
DNAME = paste(deparse(substitute(x))," and ",deparse(substitute(y)),sep="") # borrowed from HDtest
names(thestat) = "t"
res = list(statistic=thestat, p.value=pvalue, alternative = Ha, method=hname, data.name = DNAME)
class(res) = "htest"
}
##############################################################
# REPORT
return(res)
}
# paired case -------------------------------------------------------------
#' @keywords internal
#' @noRd
mean2.Student.paired <- function(x, y, paired.alternative){
if (length(x)!=length(y)){
stop("* mean2.Student : for paired t-test, length of x and y should be identical.")
}
diff = (x-y)
tmpout = mean1.ttest(diff,mu0=0,alternative=paired.alternative)
hname = "Two-sample Student's t-test for Paired/Dependent Data."
if (paired.alternative=="two.sided"){
Ha = "true mean difference is not equal to 0."
} else if (paired.alternative=="less"){
Ha = "true mean difference is less than 0."
} else if (paired.alternative=="greater"){
Ha = "true mean difference is greater than 0."
}
thestat = tmpout$statistic
DNAME = paste(deparse(substitute(x))," and ",deparse(substitute(y)),sep="") # borrowed from HDtest
names(thestat) = "t"
res = list(statistic=thestat, p.value=tmpout$p.value, alternative = Ha, method=hname, data.name = DNAME)
class(res) = "htest"
return(res)
}
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