#' Design an optimal group-sequential clinical trial for a normally distributed
#' primary outcome
#'
#' \code{des_opt()} determines optimal group-sequential clinical trial designs
#' assuming the primary outcome variable is normally distributed, using the
#' approach proposed in Wason \emph{et al} (2012).
#'
#' @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>α</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>β</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>δ</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>σ</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>σ</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 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) for further details. 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 parallel See \code{\link[GA]{ga}}.
#' @param popSize See \code{\link[GA]{ga}}.
#' @param maxiter See \code{\link[GA]{ga}}.
#' @param run See \code{\link[GA]{ga}}.
#' @param seed A variable to be passed to \code{\link{set.seed}}, to make the
#' calculations reproducible.
#' @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{GA}: A \code{\link{list}} containing the output from the call to
#' \code{\link[GA]{ga}} and each of the \code{\link[GA]{ga}} specific input
#' parameters.
#' \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>τ</i> = 0}}{\eqn{\tau = 0}},
#' \ifelse{html}{\out{<i>τ</i> = <i>δ</i>}}{\eqn{\tau = \delta}}, and
#' \ifelse{html}{\out{<i>τ</i> =
#' argmax<sub>θ</sub><i>ESS</i>(<i>θ</i>)}}{
#' \eqn{\tau = argmax_{\theta}ESS(\theta)}}.
#' }
#' @examples
#' # The optimal group-sequential design for the default parameters
#' des <- des_opt()
#' # A three-stage optimal design
#' des_3 <- des_opt(J = 3)
#' # Optimal under the alternative hypothesis
#' des_alt <- des_opt(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_opt <- function(J = 2, alpha = 0.05, beta = 0.2, delta = 0.2, sigma0 = 1,
sigma1 = sigma0, ratio = 1, w = c(1, 0, 0, 0),
quantile_sub = FALSE, integer_n = TRUE, parallel = 1,
popSize = 50, maxiter = 100, run = maxiter,
seed = Sys.time(), summary = FALSE) {
set.seed(seed)
##### Check input variables ##################################################
J <- check_integer_range(J, "J", c(1, Inf), 1)
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)
w <- check_w(w)
check_logical(quantile_sub, "quantile_sub")
check_logical( integer_n, "integer_n")
check_logical( summary, "summary")
##### Print summary ##########################################################
if (summary) {
#summary_des_opt(J, alpha, beta, delta, sigma0, sigma1, ratio, w,
# quantile_sub, integer_n)
message("")
}
##### Perform main computations ##############################################
CovZ <- covariance(sqrt(1:J))
penalty <-
(1 + ratio)*((stats::qnorm(1 - alpha)*sigma0*sqrt(1 + 1/ratio) +
stats::qnorm(1 - beta)*
sqrt(sigma0^2 + sigma1^2/ratio))/delta)^2
fitness <- function(...) { -optimal(...) }
lower <- c(0, rep(-20, J), numeric(J - 1))
upper <- c(penalty, rep(20, 2*J - 1))
ga <- GA::ga(type = "real-valued",
fitness = fitness,
J = J,
alpha = alpha,
beta = beta,
delta = delta,
sigma0 = sigma0,
sigma1 = sigma1,
ratio = ratio,
w = w,
penalty = penalty,
CovZ = CovZ,
lower = lower,
upper = upper,
maxiter = maxiter,
parallel = parallel,
popSize = popSize,
run = run)
n0 <- ga@solution[1]
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
}
f <- ga@solution[2:(J + 1)]
e <- c(f[1:(J - 1)] + ga@solution[(J + 2):(2*J)], f[J])
sqrt_I <- sqrt(I <- information(n0, J, sigma0, sigma1, ratio))
n <- (n <- n0 + n1)*(1:J)
if (quantile_sub) {
e <- stats::qt(stats::pnorm(e), (1:J)*(n*(1 + ratio) - 2))
f <- stats::qt(stats::pnorm(f), (1:J)*(n*(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 = NA,
e = e,
f = f,
GA =
list(ga = ga,
ga_options = list(parallel = parallel,
popSize = popSize,
maxiter = maxiter,
run = run,
seed = seed)),
I = I,
integer_n = integer_n,
J = J,
method = NA,
n = n,
n_fixed = n_fixed,
n0 = n0,
n1 = n1,
name = paste0("Optimal: w = (",
paste(w, collapse = ", "), ")"),
opchar = opchar,
quantile_sub = quantile_sub,
ratio = ratio,
shape = NA,
sigma0 = sigma0,
sigma1 = sigma1,
summary = summary,
w = w)
class(output) <- c(class(output), "OptGS_des")
output
}
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