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gs_design_npe.R
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gs_design_npe.R
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# 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.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
#' 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}
#'
#' @importFrom tibble tibble
#' @importFrom stats qnorm uniroot
#'
#' @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) %>%
dplyr::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)
}