Adjust the final futility bound with same value as the efficacy bound for asymmetric non-binding design

[LittleBeannie]

Example 1: a two-sided, asymmetric, beta-spending with non-binding lower bound design

The following code generate a two-sided, asymmetric, beta-spending with non-binding lower bound design.

# ------------------------------ #
#         parameters             #
# ------------------------------ #
alpha <- 0.025
beta <- 0.1
ratio <- 1

# Enrollment
enroll_rate <- define_enroll_rate(
  duration = c(2, 2, 10),
  rate = (1:3) / 3)

# Failure and dropout
fail_rate <- define_fail_rate(
  duration = Inf, fail_rate = log(2) / 9,
  hr = 0.6, dropout_rate = .0001)

# IA and FA analysis time
analysis_time <- c(12, 24, 36)

# Randomization ratio
ratio <- 1

# Spending
upper <- gs_spending_bound
lower <- gs_spending_bound
upar <- list(sf = sfLDOF, total_spend = alpha)
lpar <- list(sf = sfHSD, total_spend = 0.1, param = 3)

# ------------------------------ #
# design by gsDesign2   #
# ------------------------------ #
x_gsd2 <- gs_design_ahr(
  enroll_rate = enroll_rate, fail_rate = fail_rate,
  alpha = alpha, beta = beta, ratio = ratio,
  info_scale = "h0_h1_info",
  info_frac = 1:3/3,
  analysis_time = 36,
  upper = upper, upar = upar, test_upper = TRUE,
  lower = lower, lpar = lpar, test_lower = TRUE,
  binding = FALSE, h1_spending = TRUE
)

Issue

The final efficacy and futility bounds at the final analysis are not the same. However, we expect them to the be same value at the final analysis.

> x_gsd2$bound |> dplyr::filter(analysis == 3, bound == "upper") |> dplyr::pull(z)
[1] 1.993051
> x_gsd2$bound |> dplyr::filter(analysis == 3, bound == "lower") |> dplyr::pull(z)
[1] 1.789483

In gsDesign with the sample parameters, the efficacy and futility bounds are of same value at the final analysis.

# ------------------------------ #
# design by gsDesign    #
# ------------------------------ #
x_gsd <- gsSurv(k = 3, test.type = 4, alpha = alpha, beta = beta,
                astar = 0, timing = 1:3/3,
                sfu = sfLDOF, sfupar = 0,
                sfl = sfHSD, sflpar = 3,
                lambdaC = fail_rate$fail_rate, hr = fail_rate$hr, hr0 = 1,
                eta = fail_rate$dropout_rate,
                gamma = enroll_rate$rate,
                R = enroll_rate$duration,
                S = NULL, T = analysis_time[3],
                minfup = analysis_time[3] - sum(enroll_rate$duration),
                ratio = ratio)

> x_gsd$upper$bound[3]
[1] 1.99297
> x_gsd$lower$bound[3]
[1] 1.99297

Example 2: a two-sided, asymmetric, beta-spending with binding lower bound design

The following code generate a two-sided, asymmetric, beta-spending with binding lower bound design.

# ------------------------------ #
#         parameters             #
# ------------------------------ #
alpha <- 0.025
beta <- 0.1
ratio <- 1

# Enrollment
enroll_rate <- define_enroll_rate(
  duration = c(2, 2, 10),
  rate = (1:3) / 3)

# Failure and dropout
fail_rate <- define_fail_rate(
  duration = Inf, fail_rate = log(2) / 9,
  hr = 0.6, dropout_rate = .0001)

# IA and FA analysis time
analysis_time <- c(12, 24, 36)

# Randomization ratio
ratio <- 1

# Spending
upper <- gs_spending_bound
lower <- gs_spending_bound
upar <- list(sf = sfLDOF, total_spend = alpha)
lpar <- list(sf = sfHSD, total_spend = 0.1, param = 3)

# ------------------------------ #
# design by gsDesign2   #
# ------------------------------ #
x_gsd2 <- gs_design_ahr(
  enroll_rate = enroll_rate, fail_rate = fail_rate,
  alpha = alpha, beta = beta, ratio = ratio,
  info_scale = "h0_h1_info",
  info_frac = 1:3/3,
  analysis_time = 36,
  upper = upper, upar = upar, test_upper = TRUE,
  lower = lower, lpar = lpar, test_lower = TRUE,
  binding = TRUE, h1_spending = TRUE
)

Issue

The final efficacy and futility bounds at the final analysis are not the same. However, we expect them to the be same value at the final analysis.

> x_gsd2$bound |> dplyr::filter(analysis == 3, bound == "upper") |> dplyr::pull(z)
[1] 1.850081
> x_gsd2$bound |> dplyr::filter(analysis == 3, bound == "lower") |> dplyr::pull(z)
[1] 1.665859

In gsDesign with the sample parameters, the efficacy and futility bounds are of same value at the final analysis.

# ------------------------------ #
# design by gsDesign    #
# ------------------------------ #
x_gsd <- gsSurv(k = 3, test.type = 3, alpha = alpha, beta = beta,
                astar = 0, timing = 1:3/3,
                sfu = sfLDOF, sfupar = 0,
                sfl = sfHSD, sflpar = 3,
                lambdaC = fail_rate$fail_rate, hr = fail_rate$hr, hr0 = 1,
                eta = fail_rate$dropout_rate,
                gamma = enroll_rate$rate,
                R = enroll_rate$duration,
                S = NULL, T = analysis_time[3],
                minfup = analysis_time[3] - sum(enroll_rate$duration),
                ratio = ratio)

> x_gsd$upper$bound[3]
[1] 1.789885
> x_gsd$lower$bound[3]
[1] 1.789885

Objective

We will follow what gsDesign does to enable gsDesign2 to have the same efficacy and futility bound when it is a two-sided, asymmetric, beta-spending with non-binding/binding lower bound design.