#' @title Bootstrapped t-test
#' @description
#' `r lifecycle::badge('stable')`
#'
#' A function for a bootstrap method for t-tests.
#' @inheritParams simple_htest
#' @inheritParams boot_t_TOST
#' @return A list with class `"htest"` containing the following components:
#'
#' - "statistic": the value of the t-statistic.
#' - "parameter": the degrees of freedom for the t-statistic.
#' - "p.value": the p-value for the test.
#' - "conf.int": a confidence interval for the mean appropriate to the specified alternative hypothesis.
#' - "estimate": the estimated mean or difference in means depending on whether it was a one-sample test or a two-sample test.
#' - "null.value": the specified hypothesized value of the mean or mean difference. May be 2 values.
#' - "stderr": the standard error of the mean (difference), used as denominator in the t-statistic formula.
#' - "alternative": a character string describing the alternative hypothesis.
#' - "method": a character string indicating what type of t-test was performed.
#' - "data.name": a character string giving the name(s) of the data.
#'
#' @details The implemented test(s) corresponds to the proposal of Chapter 16 of Efron and Tibshirani (1994).
#'
#' For two-sample tests, the test is of \eqn{\bar x - \bar y} (mean of x minus mean of y).
#' For paired samples, the test is of the difference scores (z),
#' wherein \eqn{z = x - y}, and the test is of \eqn{\bar z} (mean of the difference scores).
#' For one-sample tests, the test is of \eqn{\bar x } (mean of x).
#'
#' For details on the calculations in this function see `vignette("robustTOST")`.
#'
#' @examples
#' # example code
#'
#' boot_t_test(extra ~ group, data = sleep)
#' @section References:
#'
#' Efron, B., & Tibshirani, R. J. (1994). An introduction to the bootstrap. CRC press.
#' @family Robust tests
#' @importFrom stats var quantile
#' @name boot_t_test
#' @export boot_t_test
boot_t_test <- function(x, ...){
UseMethod("boot_t_test")
}
#' @rdname boot_t_test
#' @method boot_t_test default
#' @export
boot_t_test.default <- function(x,
y = NULL,
var.equal = FALSE,
paired = FALSE,
alternative = c("two.sided",
"less",
"greater",
"equivalence",
"minimal.effect"),
mu = 0,
alpha = 0.05,
boot_ci = c("stud","basic","perc"),
R = 1999, ...){
alternative = match.arg(alternative)
boot_ci = match.arg(boot_ci)
if(!missing(alpha) && (length(alpha) != 1 || !is.finite(alpha) ||
alpha < 0 || alpha > 1)) {
stop("'alpha' must be a single number between 0 and 1")
}
if (!is.null(y)) {
dname <- paste(deparse(substitute(x)), "and",
deparse(substitute(y)))
}
else {
dname <- deparse(substitute(x))
}
null_test = simple_htest(x = x,
y = y,
test = "t.test",
var.equal = var.equal,
paired = paired,
alternative = alternative,
mu = mu,
alpha = 0.05)
mu = null_test$null.value
m_vec <- rep(NA, times=length(R)) # mean difference vector
m_se_vec <- rep(NA, times=length(R)) # mean difference vector
if(alternative %in% c("equivalence","minimal.effect")){
conf.level = 1-alpha*2
} else {
conf.level = 1-alpha
}
if(!is.null(y)){
dname <- paste(deparse(substitute(x)), "and", deparse(substitute(y)))
if (paired) {
i1 <- y
i2 <- x
data <- data.frame(i1 = i1, i2 = i2)
data <- na.omit(data)
y <- data$i1
x <- data$i2
}
yok <- !is.na(y)
xok <- !is.na(x)
y <- y[yok]
}else{
dname <- deparse(substitute(x))
#if (paired) {
# stop("'y' is missing for paired test")
#}
xok <- !is.na(x)
yok <- NULL
}
x <- x[xok]
if(paired && !is.null(y)){
x <- x - y
y <- NULL
}
nx <- length(x)
mx <- mean(x)
vx <- var(x)
if (is.null(y)) {
if (nx < 2)
stop("not enough 'x' observations")
df <- nx - 1
stderr <- sqrt(vx/nx)
if (stderr < 10 * .Machine$double.eps * abs(mx)){
stop("data are essentially constant")
}
tstat <- (mx - mu)/stderr
method <- if (paired) "Bootstrapped Paired t-test" else "Bootstrapped One Sample t-test"
#estimate <- setNames(mx, if (paired) "mean of the differences" else "mean of x")
#x.cent <- x - mx # remove to have an untransformed matrix
X <- matrix(sample(x, size = nx*R, replace = TRUE), nrow = R)
MX <- rowMeans(X - mx)
VX <- rowSums((X - MX) ^ 2) / (nx - 1)
MZ2 = NA
VZ2 = NA
for(i in 1:R){
zi = X[i,]
MZ2[i] = mean(zi - mx)
VZ2[i] <- sum((zi - MZ2[i]) ^ 2) / (nx - 1) #rowSums((X - MX) ^ 2) / (nx - 1)
}
STDERR <- sqrt(VX/nx)
TSTAT <- (MX)/STDERR
#TSTAT_low <- (MX-low_eqbound)/STDERR
#TSTAT_high <- (MX-high_eqbound)/STDERR
EFF <- MX+mx
for(i in 1:nrow(X)){
dat = X[i,]
m_vec[i] <- mean(dat, na.rm=TRUE) # mean difference vector
m_se_vec[i] <- sd(dat, na.rm = TRUE)/sqrt(length(na.omit(dat)))
}
}
if(!is.null(y) && !paired){
ny <- length(y)
if(nx < 1 || (!var.equal && nx < 2))
stop("not enough 'x' observations")
if(ny < 1 || (!var.equal && ny < 2))
stop("not enough 'y' observations")
if(var.equal && nx + ny < 3)
stop("not enough observations")
my <- mean(y)
vy <- var(y)
method <- paste("Bootstrapped", paste(if (!var.equal) "Welch", "Two Sample t-test"))
estimate <- c(mx, my)
names(estimate) <- c("mean of x", "mean of y")
if(var.equal){
df <- nx + ny - 2
v <- 0
if (nx > 1){
v <- v + (nx - 1) * vx
}
if (ny > 1){
v <- v + (ny - 1) * vy
}
v <- v/df
stderr <- sqrt(v * (1/nx + 1/ny))
z <- c(x, y)
mz <- mean(z)
#Z <- matrix(sample(z, size = (nx+ny)*R, replace = TRUE), nrow = R)
X <- matrix(sample(x, size = nx*R, replace = TRUE), nrow = R)
Y <- matrix(sample(y, size = ny*R, replace = TRUE), nrow = R)
MX <- rowMeans(X - mx + mz)
MY <- rowMeans(Y - my + mz)
V <- (rowSums((X-MX)^2) + rowSums((Y-MY)^2))/df
STDERR <- sqrt(V*(1/nx + 1/ny))
EFF <- (MX+mx) - (MY+my)
#d_vec <- rep(NA, times=length(R))
for(i in 1:nrow(X)){
#dat = Z[i,]
dat_x = X[i,]#dat[1:nx]
dat_y = Y[i,]#dat[(nx+1):(nx+ny)]
m_vec[i] <- mean(dat_x, na.rm=TRUE) - mean(dat_y,na.rm=TRUE) # mean difference vector
m_se_vec[i] <- sqrt(sd(dat_x, na.rm=TRUE)^2/length(na.omit(dat_x)) + sd(dat_y, na.rm=TRUE)^2/length(na.omit(dat_y)))
}
}else{
stderrx <- sqrt(vx/nx)
stderry <- sqrt(vy/ny)
stderr <- sqrt(stderrx^2 + stderry^2)
df <- stderr^4/(stderrx^4/(nx - 1) + stderry^4/(ny - 1))
z <- c(x, y)
mz <- mean(z)
x.cent <- x - mx + mz
y.cent <- y - my + mz
X <- matrix(sample(x, size = nx*R, replace = TRUE), nrow = R)
Y <- matrix(sample(y, size = ny*R, replace = TRUE), nrow = R)
MX <- rowMeans(X - mx + mz)
MY <- rowMeans(Y - my + mz)
VX <- rowSums((X-MX)^2)/(nx-1)
VY <- rowSums((Y-MY)^2)/(ny-1)
STDERR <- sqrt(VX/nx + VY/ny)
EFF <- (MX+mx) - (MY+my)
for(i in 1:nrow(X)){
#dat = Z[i,]
dat_x = X[i,]#dat[1:nx]
dat_y = Y[i,]#dat[(nx+1):(nx+ny)]
m_vec[i] <- mean(dat_x, na.rm=TRUE) - mean(dat_y,na.rm=TRUE) # mean difference vector
m_se_vec[i] <- sqrt(sd(dat_x, na.rm=TRUE)^2/length(na.omit(dat_x)) + sd(dat_y, na.rm=TRUE)^2/length(na.omit(dat_y)))
}
}
if (stderr < 10 * .Machine$double.eps * max(abs(mx), abs(my))){
stop("data are essentially constant")
}
tstat <- (mx - my - mu)/stderr
# Remember tstat[which.max( abs(tstat) )]
#TSTAT <- (MX - MY)/STDERR
TSTAT <- (MX-MY)/STDERR
#TSTAT_low <- (MX-low_eqbound)/STDERR
#TSTAT_high <- (MX-high_eqbound)/STDERR
}
if(is.null(y)){
diff = mx
}else{
diff = mx-my
}
if(alternative %in% c("equivalence", "minimal.effect")){
tstat_l = (diff-min(mu))/stderr
tstat_u = (diff-max(mu))/stderr
}
if (alternative == "less") {
boot.pval <- mean(TSTAT < tstat)
boot.cint <- switch(boot_ci,
"stud" = stud(m_vec,
boots_se = m_se_vec,
t0 = diff,
se0 = null_test$stderr,
alpha = alpha*2),
"basic" = basic(m_vec, t0 = diff, alpha*2),
"perc" = perc(m_vec, alpha*2))
}
if(alternative == "greater") {
boot.pval <- mean(TSTAT > tstat)
boot.cint <- switch(boot_ci,
"stud" = stud(m_vec,
boots_se = m_se_vec,
t0 = diff,
se0 = null_test$stderr,
alpha*2),
"basic" = basic(m_vec, t0 = diff, alpha*2),
"perc" = perc(m_vec, alpha*2))
}
if(alternative == "two.sided"){
boot.pval <- 2*min(mean(TSTAT <= tstat), mean(TSTAT > tstat))
boot.cint <- switch(boot_ci,
"stud" = stud(m_vec,
boots_se = m_se_vec,
t0 = diff,
se0 = null_test$stderr,
alpha),
"basic" = basic(m_vec, t0 = diff, alpha),
"perc" = perc(m_vec, alpha))
}
if(alternative == "equivalence") {
p_l = mean(TSTAT > tstat_l)
p_u = mean(TSTAT < tstat_u)
boot.pval <- max(p_l, p_u)
boot.cint <- switch(boot_ci,
"stud" = stud(m_vec,
boots_se = m_se_vec,
t0 = diff,
se0 = null_test$stderr,
alpha*2),
"basic" = basic(m_vec, t0 = diff, alpha*2),
"perc" = perc(m_vec, alpha*2))
}
if(alternative == "minimal.effect") {
p_l = mean(TSTAT < tstat_l)
p_u = mean(TSTAT > tstat_u)
boot.pval <- min(p_l,p_u)
boot.cint <- switch(boot_ci,
"stud" = stud(m_vec,
boots_se = m_se_vec,
t0 = diff,
se0 = null_test$stderr,
alpha*2),
"basic" = basic(m_vec, t0 = diff, alpha*2),
"perc" = perc(m_vec, alpha*2))
}
boot.se = sd(m_vec, na.rm = TRUE)
attr(boot.cint, "conf.level") <- conf.level
rval = list(
p.value = boot.pval,
stderr = boot.se,
conf.int = boot.cint,
estimate = null_test$estimate,
null.value = null_test$null.value,
alternative = alternative,
method = method,
boot = m_vec,
data.name = null_test$data.name,
call = match.call()
)
class(rval) = "htest"
return(rval)
}
#' @rdname boot_t_test
#' @method boot_t_test formula
#' @export
#'
boot_t_test.formula <- function (formula, data, subset, na.action, ...){
if(missing(formula)
|| (length(formula) != 3L)
|| (length(attr(terms(formula[-2L]), "term.labels")) != 1L))
stop("'formula' missing or incorrect")
m <- match.call(expand.dots = FALSE)
if(is.matrix(eval(m$data, parent.frame())))
m$data <- as.data.frame(data)
## need stats:: for non-standard evaluation
m[[1L]] <- quote(stats::model.frame)
m$... <- NULL
mf <- eval(m, parent.frame())
DNAME <- paste(names(mf), collapse = " by ")
names(mf) <- NULL
response <- attr(attr(mf, "terms"), "response")
g <- factor(mf[[-response]])
if(nlevels(g) != 2L)
stop("grouping factor must have exactly 2 levels")
DATA <- setNames(split(mf[[response]], g), c("x", "y"))
y <- do.call("boot_t_test", c(DATA, list(...)))
y$data.name <- DNAME
y
}
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