R/sfXG.R

In gsDesign: Group Sequential Design

#' Xi and Gallo conditional error spending functions
#'
#' Error spending functions based on Xi and Gallo (2019).
#' The intention of these spending functions is to provide bounds where the
#' conditional error at an efficacy bound is approximately equal to the
#' conditional error rate for crossing the final analysis bound.
#' This is explained in greater detail in
#' \code{vignette("ConditionalErrorSpending")}.
#'
#' @param alpha Real value \eqn{> 0} and no more than 1. Normally,
#'   \code{alpha = 0.025} for one-sided Type I error specification or
#'   \code{alpha = 0.1} for Type II error specification.
#'   However, this could be set to 1 if for descriptive purposes you wish
#'   to see the proportion of spending as a function of the proportion of
#'   sample size/information.
#' @param t A vector of points with increasing values from 0 to 1, inclusive.
#'   Values of the proportion of sample size/information for which the spending
#'   function will be computed.
#' @param param This is the gamma parameter in the Xi and Gallo
#'   spending function paper, distinct for each function.
#'   See the details section for functional forms and range of param
#'   acceptable for each spending function.
#'
#' @return
#' An object of type \code{spendfn}. See spending functions for
#' further details.
#'
#' @details
#' Xi and Gallo use an additive boundary for group sequential designs with
#' connection to conditional error.
#' Three spending functions are defined: \code{sfXG1()}, \code{sfXG2()},
#' and \code{sfXG3()}.
#'
#' Method 1 is defined for \eqn{\gamma \in [0.5, 1)} as
#'
#' \deqn{f(Z_K \ge u_K | Z_k = u_k) = 2 - 2\times \Phi\left(\frac{z_{\alpha/2} - z_\gamma\sqrt{1-t}}{\sqrt{t}} \right).}
#'
#' Method 2 is defined for \eqn{\gamma \in [1 - \Phi(z_{\alpha/2}/2), 1)} as
#'
#' \deqn{f_\gamma(t; \alpha)=2-2\Phi \left(
#' \Phi^{-1}(1-\alpha/2)/ t^{1/2} \right).}{% f(t;
#' alpha)=2-2*Phi(Phi^(-1)(1-alpha/2)/t^(1/2)).}
#'
#' Method 3 is defined as for \eqn{\gamma \in (\alpha/2, 1)} as
#'
#' \deqn{f(t; \alpha)= 2 - 2\times \Phi\left(\frac{z_{\alpha/2} - z_\gamma(1-\sqrt t)}{\sqrt t} \right).}
#'
#' @author Keaven Anderson \email{keaven_anderson@@merck.com}
#'
#' \code{vignette("SpendingFunctionOverview")}, \code{\link{gsDesign}},
#' \code{vignette("gsDesignPackageOverview")}
#'
#' @references
#' Jennison C and Turnbull BW (2000), \emph{Group Sequential
#' Methods with Applications to Clinical Trials}. Boca Raton: Chapman and Hall.
#'
#' Xi D and Gallo P (2019), An additive boundary for group sequential designs
#' with connection to conditional error. \emph{Statistics in Medicine}; 38 (23),
#' 4656--4669.
#'
#' @keywords design
#'
#' @rdname sfXG
#'
#' @aliases sfXG sfXG2 sfXG3
#'
#' @export
#'
#' @examples
#' # Plot conditional error spending spending functions across
#' # a range of values of interest
#' pts <- seq(0, 1.2, 0.01)
#' pal <- palette()
#'
#' plot(
#'   pts,
#'   sfXG1(0.025, pts, 0.5)$spend,
#'   type = "l", col = pal[1],
#'   xlab = "t", ylab = "Spending", main = "Xi-Gallo, Method 1"
#' )
#' lines(pts, sfXG1(0.025, pts, 0.6)$spend, col = pal[2])
#' lines(pts, sfXG1(0.025, pts, 0.75)$spend, col = pal[3])
#' lines(pts, sfXG1(0.025, pts, 0.9)$spend, col = pal[4])
#' legend(
#'   "topleft",
#'   legend = c("gamma=0.5", "gamma=0.6", "gamma=0.75", "gamma=0.9"),
#'   col = pal[1:4],
#'   lty = 1
#' )
#'
#' plot(
#'   pts,
#'   sfXG2(0.025, pts, 0.14)$spend,
#'   type = "l", col = pal[1],
#'   xlab = "t", ylab = "Spending", main = "Xi-Gallo, Method 2"
#' )
#' lines(pts, sfXG2(0.025, pts, 0.25)$spend, col = pal[2])
#' lines(pts, sfXG2(0.025, pts, 0.5)$spend, col = pal[3])
#' lines(pts, sfXG2(0.025, pts, 0.75)$spend, col = pal[4])
#' lines(pts, sfXG2(0.025, pts, 0.9)$spend, col = pal[5])
#' legend(
#'   "topleft",
#'   legend = c("gamma=0.14", "gamma=0.25", "gamma=0.5", "gamma=0.75", "gamma=0.9"),
#'   col = pal[1:5],
#'   lty = 1
#' )
#'
#' plot(
#'   pts,
#'   sfXG3(0.025, pts, 0.013)$spend,
#'   type = "l", col = pal[1],
#'   xlab = "t", ylab = "Spending", main = "Xi-Gallo, Method 3"
#' )
#' lines(pts, sfXG3(0.025, pts, 0.02)$spend, col = pal[2])
#' lines(pts, sfXG3(0.025, pts, 0.05)$spend, col = pal[3])
#' lines(pts, sfXG3(0.025, pts, 0.1)$spend, col = pal[4])
#' lines(pts, sfXG3(0.025, pts, 0.25)$spend, col = pal[5])
#' lines(pts, sfXG3(0.025, pts, 0.5)$spend, col = pal[6])
#' lines(pts, sfXG3(0.025, pts, 0.75)$spend, col = pal[7])
#' lines(pts, sfXG3(0.025, pts, 0.9)$spend, col = pal[8])
#' legend(
#'   "bottomright",
#'   legend = c(
#'     "gamma=0.013", "gamma=0.02", "gamma=0.05", "gamma=0.1",
#'     "gamma=0.25", "gamma=0.5", "gamma=0.75", "gamma=0.9"
#'   ),
#'   col = pal[1:8],
#'   lty = 1
#' )
sfXG1 <- function(alpha, t, param) {
  # Check for scalar parameter in [0.5, 1)
  checkScalar(param, "numeric", c(.5, 1), c(TRUE, FALSE))

  # For values of t > 1, compute value as if t = 1
  t <- pmin(t, 1)

  # Compute spending
  y <- 2 - 2 * pnorm((qnorm(1 - alpha / 2) -
    qnorm(1 - param) * sqrt(1 - t)) / sqrt(t))

  # Assemble return value and return
  x <- list(
    name = "Xi-Gallo, method 1", param = param,
    parname = "gamma", sf = sfXG1,
    spend = y,
    bound = NULL,
    prob = NULL
  )
  class(x) <- "spendfn"
  x
}

#' @export
#'
#' @rdname sfXG
sfXG2 <- function(alpha, t, param) {
  # Check for scalar parameter in appropriate range
  checkScalar(param, "numeric", c(1 - pnorm(qnorm(1 - alpha / 2)), 1), c(TRUE, FALSE))

  # For values of t > 1, compute value as if t = 1
  t <- pmin(t, 1)

  # Compute spending
  y <- 2 - 2 * pnorm((qnorm(1 - alpha / 2) -
    qnorm(1 - param) * (1 - t)) / sqrt(t))

  # Assemble return value and return
  x <- list(
    name = "Xi-Gallo, method 2", param = param,
    parname = "gamma", sf = sfXG2,
    spend = y,
    bound = NULL,
    prob = NULL
  )
  class(x) <- "spendfn"
  x
}

#' @export
#'
#' @rdname sfXG
sfXG3 <- function(alpha, t, param) {
  # Check for scalar parameter in appropriate range
  checkScalar(param, "numeric", c(alpha / 2, 1), c(FALSE, FALSE))

  # For values of t > 1, compute value as if t = 1
  t <- pmin(t, 1)

  # Compute spending
  y <- 2 - 2 * pnorm((qnorm(1 - alpha / 2) -
    qnorm(1 - param) * (1 - sqrt(t))) / sqrt(t))

  # Assemble return value and return
  x <- list(
    name = "Xi-Gallo, method 1", param = param,
    parname = "gamma", sf = sfXG3,
    spend = y,
    bound = NULL,
    prob = NULL
  )
  class(x) <- "spendfn"
  x
}