gsDesign2 (version 1.1.8)

gs_design_npe: Group sequential design computation with non-constant effect and information.

Description

The following two functions allow a non-constant treatment effect over time, but also can be applied for the usual homogeneous effect size designs. They require treatment effect and statistical information at each analysis as well as a method of deriving bounds, such as spending. Initial bound types supported are spending bounds and fixed bounds. These routines enables two things not available in the gsDesign package: 1) non-constant effect, 2) more flexibility in boundary selection.

gs_power_npe() derives group sequential bounds and boundary crossing probabilities for a design, given treatment effect and information at each analysis and the method of deriving bounds, such as spending.

gs_design_npe() derives group sequential design size, bounds and boundary crossing probabilities based on proportionate information and effect size at analyses, as well as the method of deriving bounds, such as spending.

The only differences in arguments between the two functions are the alpha and beta parameters used in the gs_design_npe().

Usage

gs_design_npe(
  theta = 0.1,
  theta0 = 0,
  theta1 = theta,
  info = 1,
  info0 = NULL,
  info1 = NULL,
  info_scale = c("h0_h1_info", "h0_info", "h1_info"),
  alpha = 0.025,
  beta = 0.1,
  upper = gs_b,
  upar = qnorm(0.975),
  lower = gs_b,
  lpar = -Inf,
  test_upper = TRUE,
  test_lower = TRUE,
  binding = FALSE,
  r = 18,
  tol = 1e-06
)
gs_power_npe(
  theta = 0.1,
  theta0 = 0,
  theta1 = theta,
  info = 1,
  info0 = NULL,
  info1 = NULL,
  info_scale = c("h0_h1_info", "h0_info", "h1_info"),
  upper = gs_b,
  upar = qnorm(0.975),
  lower = gs_b,
  lpar = -Inf,
  test_upper = TRUE,
  test_lower = TRUE,
  binding = FALSE,
  r = 18,
  tol = 1e-06
)

Value

A tibble with columns of

• analysis: analysis index.

• bound: either of value "upper" or "lower", indicating the upper and lower bound.

• z: the Z-score bounds.

• probability: cumulative probability of crossing the bound at or before the analysis.

• theta: same as the input.

• theta1: same as the input.

• info: statistical information at each analysis.

  • If it is returned by gs_power_npe, the info, info0, info1 are same as the input.

  • If it is returned by gs_design_npe, the info, info0, info1 are changed by a constant scale factor. factor to ensure the design has power 1 - beta.

• info0: statistical information under the null at each analysis.

• info1: statistical information under the alternative at each analysis.

• info_frac: information fraction at each analysis, i.e., info / max(info).

Arguments

theta

Natural parameter for group sequential design representing expected cumulative drift at all analyses; used for power calculation. It can be a scalar (constant treatment effect) or a vector (non-constant treatment effect). The user must provide a value for theta.

theta0

Natural parameter for null hypothesis. It can be a scalar (constant treatment effect) or a vector (non-constant treatment effect). The default is 0. If a value other than 0 is provided, it affects upper bound computation.

theta1

Natural parameter for alternate hypothesis, if needed for lower bound computation. It can be a scalar (constant treatment effect) or a vector (non-constant treatment effect). The default is the same as theta, which yields the usual beta-spending. If set to 0, spending is 2-sided under the null hypothesis.

info

Statistical information at all analyses for input theta. It is a vector of positive numbers in increasing order. The user must provide values corresponding to theta.

info0

Statistical information under null hypothesis. It is a vector of all positive numbers with increasing order. Default is set to be the same as info. If info0 is different than info, it impacts null hypothesis bound calculation.

info1

Statistical information under hypothesis used for futility bound calculation. It is a vector of all positive numbers with increasing order. Default is set to be the same as info. If info1 is different from info, it impacts futility bound calculation.

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. This is often used for testing methods that use local alternatives, such as the Schoenfeld method.

• "h1_info": variance under alternative hypothesis is used.

alpha

One-sided Type I error.

beta

Type II error.

upper

Function to compute upper bound.

• gs_spending_bound(): alpha-spending efficacy bounds.

• gs_b(): fixed efficacy bounds.

upar

Parameters passed to upper.

• If upper = gs_b, then upar is a numerical vector specifying the fixed efficacy bounds per analysis.

• If upper = gs_spending_bound, then upar is a list including

  • sf for the spending function family.

  • total_spend for total alpha spend.

  • param for the parameter of the spending function.

  • timing specifies spending time if different from information-based spending; see details.

lower

Function to compute lower bound, which can be set up similarly as upper. See this vignette.

lpar

Parameters passed to lower, which can be set up similarly as upar.

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.

test_lower

Indicator of which analyses should include a lower bound; single value of TRUE (default) indicates all analyses; single value of FALSE indicated no lower bound; otherwise, a logical vector of the same length as info should indicate which analyses will have a lower bound.

binding

Indicator of whether futility bound is binding; default of FALSE is recommended.

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.

tol

Tolerance parameter for boundary convergence (on Z-scale); normally not changed by the user.

Specification

The contents of this section are shown in PDF user manual only.

The contents of this section are shown in PDF user manual only.

Author

Keaven Anderson

Details

The bound specifications (upper, lower, upar, lpar) of gs_design_npe() will be used to ensure Type I error and other boundary properties are as specified. See the help file of gs_spending_bound() for details on spending function.

Examples

library(gsDesign)

# Example 1 ----
# gs_design_npe with 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 ----
# gs_design_npe with with 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 |> dplyr::filter(bound == "upper"))$info,
  upper = gs_b,
  upar = (x |> dplyr::filter(bound == "upper"))$z,
  lower = gs_b,
  lpar = rep(-Inf, 3)
)

# Example 3 ----
# gs_design_npe with spending bound
# 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 ----
# gs_design_npe with 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 ----
# gs_design_npe with 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 |> dplyr::filter(bound == "upper"))$z,
  lpar = -(xx |> dplyr::filter(bound == "upper"))$z
)

# Example 6 ----
# Default of gs_power_npe (single analysis; Type I error controlled)
gs_power_npe(theta = 0) |> dplyr::filter(bound == "upper")

# Example 7 ----
# gs_power_npe with fixed bound
gs_power_npe(
  theta = c(.1, .2, .3),
  info = (1:3) * 40,
  upper = gs_b,
  upar = gsDesign::gsDesign(k = 3, sfu = gsDesign::sfLDOF)$upper$bound,
  lower = gs_b,
  lpar = c(-1, 0, 0)
)

# Same fixed efficacy bounds, no futility bound (i.e., non-binding bound), null hypothesis
gs_power_npe(
  theta = rep(0, 3),
  info = (1:3) * 40,
  upar = gsDesign::gsDesign(k = 3, sfu = gsDesign::sfLDOF)$upper$bound,
  lpar = rep(-Inf, 3)
) |>
  dplyr::filter(bound == "upper")

# Example 8 ----
# gs_power_npe with fixed bound testing futility only at analysis 1; efficacy only at analyses 2, 3
gs_power_npe(
  theta = c(.1, .2, .3),
  info = (1:3) * 40,
  upper = gs_b,
  upar = c(Inf, 3, 2),
  lower = gs_b,
  lpar = c(qnorm(.1), -Inf, -Inf)
)

# Example 9 ----
# gs_power_npe with spending function bounds
# Lower spending based on non-zero effect
gs_power_npe(
  theta = c(.1, .2, .3),
  info = (1:3) * 40,
  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)
)

# Same bounds, but power under different theta
gs_power_npe(
  theta = c(.15, .25, .35),
  info = (1:3) * 40,
  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 10 ----
# gs_power_npe with two-sided symmetric spend, O'Brien-Fleming spending
# Typically, 2-sided bounds are binding
x <- gs_power_npe(
  theta = rep(0, 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)
)

# 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,
  upar = (x |> dplyr::filter(bound == "upper"))$z,
  lpar = -(x |> dplyr::filter(bound == "upper"))$z
)

# Example 11 ----
# Different values of `r` and `tol` lead to different numerical accuracy
# Larger `r` and smaller `tol` give better accuracy, but leads to slow computation
n_analysis <- 5
gs_power_npe(
  theta = 0.1,
  info = 1:n_analysis,
  info0 = 1:n_analysis,
  info1 = NULL,
  info_scale = "h0_info",
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  lower = gs_b,
  lpar = -rep(Inf, n_analysis),
  test_upper = TRUE,
  test_lower = FALSE,
  binding = FALSE,
  # Try different combinations of (r, tol) with
  # r in 6, 18, 24, 30, 35, 40, 50, 60, 70, 80, 90, 100
  # tol in 1e-6, 1e-12
  r = 6,
  tol = 1e-6
)