R/des_nearopt.R

In mjg211/OptGS: Optimal and Near-optimal Group-sequential Designs for Clinical Trials with Continuous Outcomes

#' Design a near-optimal group-sequential clinical trial for a normally
#' distributed primary outcome
#'
#' \code{des_nearopt()} determines near-optimal group-sequential clinical trial
#' designs assuming the primary outcome variable is normally distributed, using
#' the approach proposed in Wason (2015).
#'
#' @param J A \code{\link{numeric}} indicating the chosen value for
#' \ifelse{html}{\out{<i>J</i>}}{\eqn{J}}, the maximal allowed number of stages.
#' Must be an integer greater than or equal to 2. Defaults to \code{2}.
#' @param alpha A \code{\link{numeric}} indicating the chosen value for
#' \ifelse{html}{\out{<i>&alpha;</i>}}{\eqn{\alpha}}, the desired type-I
#' error-rate. Must be strictly between 0 and 1. Defaults to \code{0.05}.
#' @param beta A \code{\link{numeric}} indicating the chosen value for
#' \ifelse{html}{\out{<i>&beta;</i>}}{\eqn{\beta}}, the desired type-II
#' error-rate. Must be strictly between 0 and 1. Defaults to \code{0.2}.
#' @param delta A \code{\link{numeric}} indicating the chosen value for
#' \ifelse{html}{\out{<i>&delta;</i>}}{\eqn{\delta}}, the treatment effect to
#' power the trial for. Must be strictly positive. Defaults to \code{0.2}.
#' @param sigma0 A \code{\link{numeric}} indicating the chosen value for
#' \ifelse{html}{\out{<i>&sigma;</i><sub>0</sub>}}{\eqn{\sigma_0}}, the
#' standard deviation of the responses in the control arm. Must be strictly
#' positive. Defaults to \code{1}.
#' @param sigma1 A \code{\link{numeric}} indicating the chosen value for
#' \ifelse{html}{\out{<i>&sigma;</i><sub>1</sub>}}{\eqn{\sigma_1}}, the
#' standard deviation of the responses in the experimental arm. Must be strictly
#' positive. Defaults to \code{sigma0}.
#' @param ratio A \code{\link{numeric}} indicating the chosen value for
#' \ifelse{html}{\out{<i>r</i>}}{\eqn{r}}, the allocation ratio to the
#' experimental arm relative to the control arm. Must be strictly positive.
#' Defaults to \code{1}.
#' @param Delta A \code{\link{numeric}} \code{\link{vector}} of length 1 or 2
#' indicating the initial values for the boundary shape parameters to use in the
#' search procedure. All elements must be less than or equal to 0.5. Defaults to
#' \code{0}.
#' @param w A \code{\link{numeric}} \code{\link{vector}} of length 4 indicating
#' the weights to use in the optimality criteria. All elements must be greater
#' than or equal to 0, and at least one of the first 3 elements must be strictly
#' positive. See Wason (2015). Defaults to \code{c(1, 0, 0, 0)}.
#' @param quantile_sub A \code{\link{logical}} variable indicating whether
#' quantile substitution should be applied to the identified stopping
#' boundaries. Defaults to \code{FALSE}.
#' @param integer_n A \code{\link{logical}} variable indicating whether the
#' computed values for \ifelse{html}{\out{<i>n</i><sub>0</sub>}}{\eqn{n_0}} and
#' \ifelse{html}{\out{<i>n</i><sub>1</sub>}}{\eqn{n_1}}, the group sizes in the
#' control and experimental arms, should be forced to be whole numbers. Defaults
#' to \code{TRUE}.
#' @param method A \code{\link{character}} string indicating the optimisation
#' routine to use to identify the optimised boundary shape parameters. Must be
#' either \code{"Nelder-Mead"} or \code{"L-BFGS-B"}. Typically,
#' \code{"Nelder-Mead"} provides better convergence to the optimal solution, but
#' is slower than \code{"L-BFGS-B"}. Defaults to \code{"Nelder-Mead"}.
#' @param summary A \code{\link{logical}} variable indicating whether a summary
#' of the function's progress should be printed to the console. Defaults to
#' \code{FALSE}.
#' @return A \code{\link{list}} with additional class \code{"OptGS_des"}. It
#' will contain each of the input variables (subject to internal modification),
#' relating them to the outputs of the various group-sequential design functions
#' in \code{\link{OptGS}}, along with additional elements including:
#' \itemize{
#' \item \code{CovZ}: A \code{\link{numeric}} \code{\link{matrix}} giving
#' \ifelse{html}{\out{Cov(<b><i>Z</i></b>)}}{\eqn{Cov(\bold{Z})}}, the
#' covariance between the standardised test statistics for the identified
#' design.
#' \item \code{e}: A \code{\link{numeric}} \code{\link{vector}} giving
#' \ifelse{html}{\out{<b><i>e</i></b>}}{\eqn{\bold{e}}}, the efficacy stopping
#' boundaries for the identified design.
#' \item \code{f}: A \code{\link{numeric}} \code{\link{vector}} giving
#' \ifelse{html}{\out{<b><i>f</i></b>}}{\eqn{\bold{f}}}, the futility stopping
#' boundaries for the identified design.
#' \item \code{I}: A \code{\link{numeric}} \code{\link{vector}} giving
#' \ifelse{html}{\out{<b><i>I</i></b>}}{\eqn{\bold{I}}}, the vector of
#' information levels for the identified design.
#' \item \code{n}: A \code{\link{numeric}} \code{\link{vector}} giving
#' \ifelse{html}{\out{<b><i>n</i></b>}}{\eqn{\bold{n}}}, the vector of
#' stage-wise sample sizes for the identified design.
#' \item \code{n_fixed}: A \code{\link{numeric}} giving the sample size required
#' by a corresponding fixed-sample design.
#' \item \code{n0}: A \code{\link{numeric}} giving
#' \ifelse{html}{\out{<i>n</i><sub>0</sub>}}{\eqn{n_0}}, the group size in the
#' control arm for the identified design.
#' \item \code{n1}: A \code{\link{numeric}} giving
#' \ifelse{html}{\out{<i>n</i><sub>1</sub>}}{\eqn{n_1}}, the group size in the
#' experimental arm for the identified design.
#' \item \code{name}: A \code{\link{character}} string giving a name for the
#' identified design.
#' \item \code{opchar}: A \code{\link[tibble]{tibble}} giving the operating
#' characteristics of the identified design when
#' \ifelse{html}{\out{<i>&tau;</i> = 0}}{\eqn{\tau = 0}},
#' \ifelse{html}{\out{<i>&tau;</i> = <i>&delta;</i>}}{\eqn{\tau = \delta}}, and
#' \ifelse{html}{\out{<i>&tau;</i> =
#' argmax<sub>&theta;</sub><i>ESS</i>(<i>&theta;</i>)}}{
#' \eqn{\tau = argmax_{\theta}ESS(\theta)}}.
#' }
#' @examples
#' # The near-optimal group-sequential design for the default parameters
#' des     <- des_nearopt()
#' # A three-stage near-optimal design
#' des_3   <- des_nearopt(J = 3)
#' # Optimal under the alternative hypothesis
#' des_alt <- des_nearopt(w = c(0, 1, 0, 0))
#' @seealso \code{\link{build}}, \code{\link{des_nearopt}},
#' \code{\link{des_opt}}, \code{\link{est}}, \code{\link{opchar}},
#' \code{\link{sim}}, \code{\link{plot.OptGS_des}},
#' \code{\link{print.OptGS_des}}, \code{\link{summary.OptGS_des}}
#' @export
des_nearopt <- function(J = 2, alpha = 0.05, beta = 0.2, delta = 0.2,
                        sigma0 = 1, sigma1 = sigma0, ratio = 1, Delta = c(0, 0),
                        w = c(1, 0, 0, 0), quantile_sub = FALSE,
                        integer_n = TRUE, method = "Nelder-Mead",
                        summary = FALSE) {

  ##### Check input variables ##################################################

  J             <- check_integer_range(J, "J", c(1, Inf), 1L)
  check_real_range_strict( alpha,  "alpha", c(0,   1), 1)
  check_real_range_strict(  beta,   "beta", c(0,   1), 1)
  check_real_range_strict( delta,  "delta", c(0, Inf), 1)
  check_real_range_strict(sigma0, "sigma0", c(0, Inf), 1)
  check_real_range_strict(sigma1, "sigma1", c(0, Inf), 1)
  check_real_range_strict( ratio,  "ratio", c(0, Inf), 1)
  Delta         <- check_Delta(Delta, "nearopt")
  w             <- check_w(w)
  check_logical(quantile_sub, "quantile_sub")
  check_logical(   integer_n,    "integer_n")
  check_belong(method, "method", c("Nelder-Mead", "L-BFGS-B"), 1)
  check_logical(     summary,      "summary")

  ##### Print summary ##########################################################

  if (summary) {
    #summary_des_nearopt(J, alpha, beta, delta, sigma0, sigma1, ratio, Delta, w,
    #                    quantile_sub, integer_n)
    message("")
  }

  ##### Perform main computations ##############################################

  seq_J         <- 1:J
  CovZ          <- covariance(sqrt(seq_J))
  sqrt_I_fac    <- sqrt(b_fac <- seq_J/J)
  seq_j         <- lapply(seq_J, function(j) { 1:j })
  seq_jm1       <- lapply(seq_J, function(j) { seq_len(j - 1) })
  if (method == "Nelder-Mead") {
    Delta       <- stats::optim(par        = Delta,
                                fn         = eval_Delta_pf,
                                J          = J,
                                alpha      = alpha,
                                beta       = beta,
                                delta      = delta,
                                sigma0     = sigma0,
                                sigma1     = sigma1,
                                ratio      = ratio,
                                w          = w,
                                CovZ       = CovZ,
                                b_fac      = b_fac,
                                sqrt_I_fac = sqrt_I_fac,
                                n_fac      =
                                  (sigma0^2 + sigma1^2/ratio)*(1 + ratio),
                                seq_j      = seq_j,
                                seq_jm1    = seq_jm1)$par
  } else {
    Delta       <- stats::optim(par        = Delta,
                                fn         = eval_Delta_pf,
                                method     = "L-BFGS-B",
                                lower      = c(-5, -5),
                                upper      = c(0.5, 0.5),
                                J          = J,
                                alpha      = alpha,
                                beta       = beta,
                                delta      = delta,
                                sigma0     = sigma0,
                                sigma1     = sigma1,
                                ratio      = ratio,
                                w          = w,
                                CovZ       = CovZ,
                                b_fac      = b_fac,
                                sqrt_I_fac = sqrt_I_fac,
                                n_fac      =
                                  (sigma0^2 + sigma1^2/ratio)*(1 + ratio),
                                seq_j      = seq_j,
                                seq_jm1    = seq_jm1)$par
  }
  e_fac         <- b_fac^(Delta[1] - 0.5)
  f_fac         <- b_fac^(Delta[2] - 0.5)
  C             <- stats::optim(par        = c(0.5, 0.5),
                                fn         = eval_C_pf,
                                J          = J,
                                alpha      = alpha,
                                beta       = beta,
                                delta      = delta,
                                CovZ       = CovZ,
                                e_fac      = e_fac,
                                f_fac      = f_fac,
                                sqrt_I_fac = sqrt_I_fac,
                                seq_j      = seq_j,
                                seq_jm1    = seq_jm1)$par
  I             <- (seq_J/J)*(sum(C)/delta)^2
  e             <- C[2]*(seq_J/J)^(Delta[1] - 0.5)
  f             <- delta*sqrt(I) - C[1]*(seq_J/J)^(Delta[2] - 0.5)
  n0            <- I[1]*(sigma0^2 + sigma1^2/ratio)
  if (integer_n) {
      n0        <- ceiling(n0)
      n1        <- n0*ratio
      while (n1%%1 != 0) {
        n0      <- n0 + 1L
        n1      <- n0*ratio
      }
      n0        <- as.integer(n0)
      n1        <- as.integer(n1)
  } else {
    n1          <- n0*ratio
  }
  sqrt_I        <- sqrt(I <- information(n0, J, sigma0, sigma1, ratio))
  n             <- (n0 + n1)*seq_J
  if (quantile_sub) {
    e           <- stats::qt(stats::pnorm(e), seq_J*(n[1]*(1 + ratio) - 2))
    f           <- stats::qt(stats::pnorm(f), seq_J*(n[1]*(1 + ratio) - 2))
  }
  argmax_ess    <- stats::optim(par    = 0.5*delta,
                                fn     = minus_ess,
                                method = "Brent",
                                lower  = 0,
                                upper  = delta,
                                e      = e,
                                f      = f,
                                sqrt_I = sqrt_I,
                                CovZ   = CovZ,
                                n      = n)$par
  opchar        <- opchar_int(sort(c(0, argmax_ess, delta)), e, f, sqrt_I, CovZ,
                              n)
  n_fixed       <- des_fixed(alpha = alpha, beta = beta, delta = delta,
                             sigma0 = sigma0, sigma1 = sigma1, ratio = ratio,
                             integer_n = integer_n)$n

  ##### Output results #########################################################

  output        <- list(alpha        = alpha,
                        beta         = beta,
                        CovZ         = CovZ,
                        delta        = delta,
                        Delta        = Delta,
                        e            = e,
                        f            = f,
                        GA           = NA,
                        I            = I,
                        integer_n    = integer_n,
                        J            = J,
                        method       = method,
                        n            = n,
                        n_fixed      = n_fixed,
                        n0           = n0,
                        n1           = n1,
                        name         = paste0("Power-family: Delta = (",
                                              paste(Delta, collapse = ", "),
                                              ")"),
                        opchar       = opchar,
                        quantile_sub = quantile_sub,
                        ratio        = ratio,
                        shape        = "power_family",
                        sigma0       = sigma0,
                        sigma1       = sigma1,
                        summary      = summary,
                        w            = w)
  class(output) <- c(class(output), "OptGS_des")
  output

}