R/gs_design_npe.R

In gsDesign2: Group Sequential Design with Non-Constant Effect

#  Copyright (c) 2024 Merck & Co., Inc., Rahway, NJ, USA and its affiliates.
#  All rights reserved.
#
#  This file is part of the gsDesign2 program.
#
#  gsDesign2 is free software: you can redistribute it and/or modify
#  it under the terms of the GNU General Public License as published by
#  the Free Software Foundation, either version 3 of the License, or
#  (at your option) any later version.
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#  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
#  GNU General Public License for more details.
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#  You should have received a copy of the GNU General Public License
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#' Group sequential design computation with non-constant effect and information
#'
#' Derives group sequential design size,
#' bounds and boundary crossing probabilities based on proportionate
#' information and effect size at analyses.
#' It allows a non-constant treatment effect over time,
#' but also can be applied for the usual homogeneous effect size designs.
#' It requires treatment effect and proportionate statistical information
#' at each analysis as well as a method of deriving bounds, such as spending.
#' The routine enables two things not available in the gsDesign package:
#' 1) non-constant effect, 2) more flexibility in boundary selection.
#' For many applications, the non-proportional-hazards design function
#' `gs_design_nph()` will be used; it calls this function.
#' Initial bound types supported are 1) spending bounds,
#' 2) fixed bounds, and 3) Haybittle-Peto-like bounds.
#' The requirement is to have a boundary update method that
#' can each bound without knowledge of future bounds.
#' As an example, bounds based on conditional power that
#' require knowledge of all future bounds are not supported by this routine;
#' a more limited conditional power method will be demonstrated.
#' Boundary family designs Wang-Tsiatis designs including
#' the original (non-spending-function-based) O'Brien-Fleming and Pocock designs
#' are not supported by [gs_power_npe()].
#'
#' @param theta Natural parameter for group sequential design
#'   representing expected incremental drift at all analyses;
#'   used for power calculation.
#' @param theta0 Natural parameter used for upper bound spending;
#'   if `NULL`, this will be set to 0.
#' @param theta1 Natural parameter used for lower bound spending;
#'   if `NULL`, this will be set to `theta`
#'   which yields the usual beta-spending.
#'   If set to 0, spending is 2-sided under null hypothesis.
#' @param info Proportionate statistical information at
#'   all analyses for input `theta`.
#' @param info0 Proportionate statistical information
#'   under null hypothesis, if different than alternative;
#'   impacts null hypothesis bound calculation.
#' @param info1 Proportionate statistical information
#'   under alternate hypothesis;
#'   impacts null hypothesis bound calculation.
#' @param info_scale Information scale for calculation. Options are:
#'   - `"h0_h1_info"` (default): variance under both null and alternative hypotheses is used.
#'   - `"h0_info"`: variance under null hypothesis is used.
#'   - `"h1_info"`: variance under alternative hypothesis is used.
#' @param alpha One-sided Type I error.
#' @param beta Type II error.
#' @param binding Indicator of whether futility bound is binding;
#'   default of `FALSE` is recommended.
#' @param upper Function to compute upper bound.
#' @param lower Function to compare lower bound.
#' @param upar Parameters passed to the function provided in `upper`.
#' @param lpar Parameters passed to the function provided in `lower`.
#' @param test_upper Indicator of which analyses should include
#'   an upper (efficacy) bound; single value of `TRUE` (default) indicates
#'   all analyses; otherwise, a logical vector of the same length as `info`
#'   should indicate which analyses will have an efficacy bound.
#' @param test_lower Indicator of which analyses should include an lower bound;
#'   single value of `TRUE` (default) indicates all analyses;
#'   single value `FALSE` indicates no lower bound; otherwise,
#'   a logical vector of the same length as `info` should indicate which
#'   analyses will have a lower bound.
#' @param r Integer value controlling grid for numerical integration
#'   as in Jennison and Turnbull (2000); default is 18, range is 1 to 80.
#'   Larger values provide larger number of grid points and greater accuracy.
#'   Normally `r` will not be changed by the user.
#' @param tol Tolerance parameter for boundary convergence (on Z-scale).
#'
#' @return A tibble with columns analysis, bound, z, probability, theta, info, info0.
#'
#' @details
#' The inputs `info` and `info0` should be
#' vectors of the same length with increasing positive numbers.
#' The design returned will change these by some constant scale
#' factor to ensure the design has power `1 - beta`.
#' The bound specifications in `upper`, `lower`, `upar`, `lpar`
#' will be used to ensure Type I error and other boundary properties are as specified.
#'
#' @section Specification:
#' \if{latex}{
#'  \itemize{
#'    \item Validate if input info is a numeric vector  or NULL, if non-NULL validate if it
#'    is strictly increasing and positive.
#'    \item Validate if input info0 is a numeric vector or NULL, if non-NULL validate if it
#'     is strictly increasing and positive.
#'    \item Validate if input info1 is a numeric vector or NULL, if non-NULL validate if it
#'    is strictly increasing and positive.
#'    \item Validate if input theta is a real vector and has the same length as info.
#'    \item Validate if input theta1 is a real vector and has the same length as info.
#'    \item Validate if input test_upper and test_lower are logical and have the same length as info.
#'    \item Validate if input test_upper value is TRUE.
#'    \item Validate if input alpha and beta are positive and of length one.
#'    \item Validate if input alpha and beta are from the unit interval and alpha is smaller than beta.
#'    \item Initialize bounds, numerical integration grids, boundary crossing probabilities.
#'    \item Compute fixed sample size for desired power and Type I error.
#'    \item Find an interval for information inflation to give correct power using \code{gs_power_npe()}.

#'    \item
#'    \item If there is no interim analysis, return a tibble including Analysis time, upper bound, Z-value,
#'    Probability of crossing bound, theta, info0 and info1.
#'    \item If the design is a group sequential design, return a tibble of Analysis,
#'     Bound, Z, Probability,  theta, info, info0.
#'   }
#' }
#' \if{html}{The contents of this section are shown in PDF user manual only.}
#'
#' @author Keaven Anderson \email{keaven_anderson@@merck.com}
#'
#' @export
#'
#' @examples
#' library(dplyr)
#' library(gsDesign)
#'
#' # Example 1 ----
#' # Single analysis
#' # Lachin book p 71 difference of proportions example
#' pc <- .28 # Control response rate
#' pe <- .40 # Experimental response rate
#' p0 <- (pc + pe) / 2 # Ave response rate under H0
#'
#' # Information per increment of 1 in sample size
#' info0 <- 1 / (p0 * (1 - p0) * 4)
#' info <- 1 / (pc * (1 - pc) * 2 + pe * (1 - pe) * 2)
#'
#' # Result should round up to next even number = 652
#' # Divide information needed under H1 by information per patient added
#' gs_design_npe(theta = pe - pc, info = info, info0 = info0)
#'
#'
#' # Example 2 ----
#' # Fixed bound
#' x <- gs_design_npe(
#'   alpha = 0.0125,
#'   theta = c(.1, .2, .3),
#'   info = (1:3) * 80,
#'   info0 = (1:3) * 80,
#'   upper = gs_b,
#'   upar = gsDesign::gsDesign(k = 3, sfu = gsDesign::sfLDOF, alpha = 0.0125)$upper$bound,
#'   lower = gs_b,
#'   lpar = c(-1, 0, 0)
#' )
#' x
#'
#' # Same upper bound; this represents non-binding Type I error and will total 0.025
#' gs_power_npe(
#'   theta = rep(0, 3),
#'   info = (x %>% filter(bound == "upper"))$info,
#'   upper = gs_b,
#'   upar = (x %>% filter(bound == "upper"))$z,
#'   lower = gs_b,
#'   lpar = rep(-Inf, 3)
#' )
#'
#' # Example 3 ----
#' # Spending bound examples
#' # Design with futility only at analysis 1; efficacy only at analyses 2, 3
#' # Spending bound for efficacy; fixed bound for futility
#' # NOTE: test_upper and test_lower DO NOT WORK with gs_b; must explicitly make bounds infinite
#' # test_upper and test_lower DO WORK with gs_spending_bound
#' gs_design_npe(
#'   theta = c(.1, .2, .3),
#'   info = (1:3) * 40,
#'   info0 = (1:3) * 40,
#'   upper = gs_spending_bound,
#'   upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
#'   lower = gs_b,
#'   lpar = c(-1, -Inf, -Inf),
#'   test_upper = c(FALSE, TRUE, TRUE)
#' )
#'
#' # one can try `info_scale = "h1_info"` or `info_scale = "h0_info"` here
#' gs_design_npe(
#'   theta = c(.1, .2, .3),
#'   info = (1:3) * 40,
#'   info0 = (1:3) * 30,
#'   info_scale = "h1_info",
#'   upper = gs_spending_bound,
#'   upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
#'   lower = gs_b,
#'   lpar = c(-1, -Inf, -Inf),
#'   test_upper = c(FALSE, TRUE, TRUE)
#' )
#'
#' # Example 4 ----
#' # Spending function bounds
#' # 2-sided asymmetric bounds
#' # Lower spending based on non-zero effect
#' gs_design_npe(
#'   theta = c(.1, .2, .3),
#'   info = (1:3) * 40,
#'   info0 = (1:3) * 30,
#'   upper = gs_spending_bound,
#'   upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
#'   lower = gs_spending_bound,
#'   lpar = list(sf = gsDesign::sfHSD, total_spend = 0.1, param = -1, timing = NULL)
#' )
#'
#' # Example 5 ----
#' # Two-sided symmetric spend, O'Brien-Fleming spending
#' # Typically, 2-sided bounds are binding
#' xx <- gs_design_npe(
#'   theta = c(.1, .2, .3),
#'   info = (1:3) * 40,
#'   binding = TRUE,
#'   upper = gs_spending_bound,
#'   upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
#'   lower = gs_spending_bound,
#'   lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL)
#' )
#' xx
#'
#' # Re-use these bounds under alternate hypothesis
#' # Always use binding = TRUE for power calculations
#' gs_power_npe(
#'   theta = c(.1, .2, .3),
#'   info = (1:3) * 40,
#'   binding = TRUE,
#'   upper = gs_b,
#'   lower = gs_b,
#'   upar = (xx %>% filter(bound == "upper"))$z,
#'   lpar = -(xx %>% filter(bound == "upper"))$z
#' )
gs_design_npe <- function(
    theta = .1, theta0 = NULL, theta1 = NULL, # 3 theta
    info = 1, info0 = NULL, info1 = NULL, # 3 info
    info_scale = c("h0_h1_info", "h0_info", "h1_info"),
    alpha = 0.025, beta = .1,
    upper = gs_b, upar = qnorm(.975),
    lower = gs_b, lpar = -Inf,
    test_upper = TRUE, test_lower = TRUE, binding = FALSE,
    r = 18, tol = 1e-6) {
  # Check & set up parameters ----
  n_analysis <- length(info)

  # check alpha & beta
  check_alpha_beta(alpha, beta)

  # check theta, theta0, theta1
  if (length(theta) == 1) {
    theta <- rep(theta, n_analysis)
  }

  if (is.null(theta1)) {
    theta1 <- theta
  } else if (length(theta1) == 1) {
    theta1 <- rep(theta1, n_analysis)
  }

  if (is.null(theta0)) {
    theta0 <- rep(0, n_analysis)
  } else if (length(theta0) == 1) {
    theta0 <- rep(theta0, n_analysis)
  }

  check_theta(theta, n_analysis)
  check_theta(theta0, n_analysis)
  check_theta(theta1, n_analysis)

  # check test_upper & test_lower
  if (length(test_upper) == 1 && n_analysis > 1) test_upper <- rep(test_upper, n_analysis)
  if (length(test_lower) == 1 && n_analysis > 1) test_lower <- rep(test_lower, n_analysis)
  check_test_upper(test_upper, n_analysis)
  check_test_lower(test_lower, n_analysis)

  # Set up info ----
  if (is.null(info0)) {
    info0 <- info
  }

  if (is.null(info1)) {
    info1 <- info
  }

  # set up info_scale
  info_scale <- match.arg(info_scale)

  if (info_scale == "h0_info") {
    info <- info0
    info1 <- info0
  }
  if (info_scale == "h1_info") {
    info <- info1
    info0 <- info1
  }

  # check info, info0, info1
  check_info(info)
  check_info(info0)
  check_info(info1)
  if (length(info0) != length(info)) stop("gs_design_npe(): length of info, info0 must be the same!")
  if (length(info1) != length(info)) stop("gs_design_npe(): length of info, info1 must be the same!")

  # Check design type ----
  if (identical(lower, gs_b) && (!is.list(lpar))) {
    if (all(test_lower == FALSE)) {
      two_sided <- FALSE
      lpar <- rep(-Inf, n_analysis)
    } else {
      two_sided <- ifelse(identical(lpar, rep(-Inf, n_analysis)), FALSE, TRUE)
    }
  } else {
    two_sided <- TRUE
  }

  # Initialization ----
  a <- rep(-Inf, n_analysis) # bounds
  b <- rep(Inf, n_analysis)
  hgm1_0 <- NULL # numerical integration grids
  hgm1_1 <- NULL
  upper_prob <- rep(NA, n_analysis) # boundary crossing probabilities
  lower_prob <- rep(NA, n_analysis)

  # Fixed design ----
  # compute fixed sample size for desired power and Type I error.
  min_x <- ((qnorm(alpha) / sqrt(info0[n_analysis]) + qnorm(beta) / sqrt(info[n_analysis])) / theta[n_analysis])^2
  # for a fixed design, this is all you need.
  if (n_analysis == 1) {
    ans <- tibble(
      analysis = 1, bound = "upper", z = qnorm(1 - alpha),
      probability = 1 - beta, probability0 = alpha, theta = theta,
      info = info * min_x, info0 = info0 * min_x, info1 = info1 * min_x,
      info_frac = info / max(info)
    )
    return(ans)
  }

  # Search for the inflation factor to info ----
  # ensure `min_x` gives power < 1 - beta
  # and `max_x` gives power > 1 - beta
  min_temp <- gs_power_npe(
    theta = theta, theta0 = theta0, theta1 = theta1,
    info = info * min_x, info0 = info0 * min_x, info1 = info * min_x, info_scale = info_scale,
    upper = upper, upar = upar, test_upper = test_upper,
    lower = lower, lpar = lpar, test_lower = test_lower,
    binding = binding, r = r, tol = tol
  )
  min_power <- (min_temp[min_temp$bound == "upper" & min_temp$analysis == n_analysis, ])$probability

  # a flag indicates if max_x can be found
  flag <- FALSE
  if (min_power < 1 - beta) {
    # if min_power < 1 - beta
    # then find a max_power > 1 - beta
    # by increasing `min_x` to `max_x` until `max_power` > 1 - beta
    max_x <- 1.05 * min_x

    for (i in 1:10) {
      max_temp <- gs_power_npe(
        theta = theta, theta0 = theta0, theta1 = theta1,
        info = info * max_x, info0 = info0 * max_x, info1 = info * max_x, info_scale = info_scale,
        upper = upper, upar = upar, test_upper = test_upper,
        lower = lower, lpar = lpar, test_lower = test_lower,
        binding = binding, r = r, tol = tol
      )
      max_power <- (max_temp[max_temp$bound == "upper" & max_temp$analysis == n_analysis, ])$probability

      if (max_power < 1 - beta) {
        min_x <- max_x
        max_x <- 1.05 * max_x
      } else {
        flag <- TRUE
        break
      }
    }
    if (!flag) stop("gs_design_npe: could not inflate information to bracket power before root finding!")
  } else {
    # if min_power  > 1 - beta
    # then find a micro_power < 1 - beta
    # by decreasing `min_x` to `micro_x` until `micro_power` < 1 - beta
    micro_x <- min_x / 1.05

    for (i in 1:10) {
      micro_temp <- gs_power_npe(
        theta = theta, theta0 = theta0, theta1 = theta1,
        info = info * micro_x, info0 = info0 * micro_x, info1 = info * micro_x, info_scale = info_scale,
        upper = upper, upar = upar, test_upper = test_upper,
        lower = lower, lpar = lpar, test_lower = test_lower,
        binding = binding, r = r, tol = tol
      )
      micro_power <- (micro_temp[micro_temp$bound == "upper" & micro_temp$analysis == n_analysis, ])$probability

      if (micro_power > 1 - beta) {
        min_x <- micro_x
        micro_x <- micro_x / 1.05
      } else {
        flag <- TRUE
        break
      }
    }

    if (!flag) stop("gs_design_npe: could not deflate information to bracket targeted power before root finding!")
    max_x <- min_x
    min_x <- micro_x
  }

  # use root finding with the above function to find needed sample size inflation
  # now we can solve for the inflation factor for the enrollment rate to achieve the desired power
  res <- try(uniroot(errbeta,
    lower = min_x, upper = max_x,
    theta = theta, theta0 = theta0, theta1 = theta1,
    info = info, info0 = info0, info1 = info1, info_scale = info_scale,
    z_upper = upper, upar = upar, test_upper = test_upper,
    z_lower = lower, lpar = lpar, test_lower = test_lower,
    beta = beta, n_analysis = n_analysis, binding = binding, r = r, tol = tol
  ))
  if (inherits(res, "try-error")) {
    stop("gs_design_npe(): Sample size solution not found!")
  } else {
    inflation_factor <- res$root
  }

  # Return the output ----
  # calculate the probability under H1
  ans_h1 <- gs_power_npe(
    theta = theta, theta0 = theta0, theta1 = theta1,
    info = info * inflation_factor, info0 = info0 * inflation_factor, info1 = info1 * inflation_factor,
    info_scale = info_scale,
    upper = upper, upar = upar,
    lower = lower, lpar = lpar,
    test_upper = test_upper, test_lower = test_lower,
    binding = binding, r = r, tol = tol
  )

  # calculate the probability under H0
  ans_h0 <- gs_power_npe(
    theta = 0, theta0 = theta0, theta1 = theta1,
    info = info0 * inflation_factor, info0 = info0 * inflation_factor, info1 = info1 * inflation_factor,
    info_scale = info_scale,
    upper = upper, upar = upar,
    lower = if (!two_sided) {
      gs_b
    } else {
      lower
    },
    lpar = if (!two_sided) {
      rep(-Inf, n_analysis)
    } else {
      lpar
    },
    test_upper = test_upper, test_lower = test_lower,
    binding = binding, r = r, tol = tol
  )

  # combine probability under H0 and H1
  suppressMessages(
    ans <- ans_h1 %>%
      full_join(
        ans_h0 %>%
          select(analysis, bound, probability) %>%
          rename(probability0 = probability)
      )
  )

  ans <- ans %>% select(analysis, bound, z, probability, probability0, theta, info_frac, info, info0, info1)

  ans <- ans %>% arrange(analysis)

  return(ans)
}


## Create a function that uses gs_power_npe to compute difference from targeted power
## for a given sample size inflation factor
errbeta <- function(x = 1, n_analysis = 1,
                    beta = .1,
                    theta = .1, theta0 = 0, theta1 = .1,
                    info = 1, info0 = 1, info1 = 1, info_scale = "h0_h1_info",
                    z_upper = gs_b, upar = qnorm(.975),
                    z_lower = gs_b, lpar = -Inf,
                    test_upper = TRUE, test_lower = TRUE,
                    binding = FALSE, r = 18, tol = 1e-6) {
  x_temp <- gs_power_npe(
    theta = theta, theta0 = theta0, theta1 = theta1,
    info = info * x, info0 = info0 * x, info1 = info1 * x, info_scale = info_scale,
    upper = z_upper, upar = upar, test_upper = test_upper,
    lower = z_lower, lpar = lpar, test_lower = test_lower,
    binding = binding, r = r, tol = tol
  )

  x_power <- (x_temp[x_temp$bound == "upper" & x_temp$analysis == n_analysis, ])$probability

  ans <- 1 - beta - x_power
  return(ans)
}