Efficacy Boundaries for Phase 2/3 Seamless Design

Calculates the efficacy stopping boundaries for a phase 2/3 seamless design, accounting for rank-based treatment selection at the end of Phase 2 and sequential testing in Phase 3.

Usage

getBound_seamless(
  M = NA_integer_,
  r = 1,
  corr_known = TRUE,
  k = NA_integer_,
  informationRates = NA_real_,
  alpha = 0.025,
  typeAlphaSpending = "sfOF",
  parameterAlphaSpending = NA_real_,
  userAlphaSpending = NA_real_,
  spendingTime = NA_real_,
  efficacyStopping = NA_integer_,
  rankp0 = 1L
)

Arguments

M

Number of active treatment arms in Phase 2.

r

Randomization ratio of each active arm to the common control in Phase 2.

corr_known

Logical. If TRUE, the correlation between Wald statistics in Phase 2 is derived from the randomization ratio \(r\) as \(r / (r + 1)\). If FALSE, a conservative correlation of 0 is used, which is only valid when rankp0 = 1 (i.e., the arm with the largest Phase-2 Z-statistic is selected for Phase 3).

k

The index of the current look in Phase 3.

informationRates

A numeric vector of information rates up to the current look. Values must be strictly increasing and \(\le 1\).

alpha

The significance level. Defaults to 0.025.

typeAlphaSpending

The type of alpha spending. One of the following: "OF" for O'Brien-Fleming boundaries, "P" for Pocock boundaries, "WT" for Wang & Tsiatis boundaries, "sfOF" for O'Brien-Fleming type spending function, "sfP" for Pocock type spending function, "sfKD" for Kim & DeMets spending function, "sfHSD" for Hwang, Shi & DeCani spending function, "user" for user defined spending, and "none" for no early efficacy stopping. Defaults to "sfOF".

parameterAlphaSpending

The parameter value for the alpha spending. Corresponds to \(\Delta\) for "WT", \(\rho\) for "sfKD", and \(\gamma\) for "sfHSD".

userAlphaSpending

The user defined alpha spending. Cumulative alpha spent up to each stage.

spendingTime

A numeric vector of length \(k+1\) specifying the error spending time at each analysis. Values must be strictly increasing and \(\le 1\). If omitted, defaults to informationRates.

efficacyStopping

Indicators of whether efficacy stopping is allowed at each stage. Defaults to TRUE if left unspecified.

rankp0

An integer between 1 and M specifying the rank of the arm to be selected at the end of Phase 2 for the purpose of determining the boundaries. For example, if rankp0 is 1, the boundaries are determined based on the arm with the largest Z-statistic at the end of Phase 2; if rankp0 is 2, the boundaries are determined based on the arm with the second largest Z-statistic at the end of Phase 2, and so on. The default is 1, which corresponds to the common practice of determining boundaries based on the top-ranked arm at the end of Phase 2.

Value

A numeric vector of length \(k + 1\) containing the critical values (on the standard normal Z-scale) for each analysis up to the current look.

Details

The function determines critical values by solving for the boundary that satisfies the alpha-spending requirement, given the selection of the arm at rank rankp0 at the end of Phase 2.

If typeAlphaSpending is "OF", "P", "WT", or "none", then informationRates, efficacyStopping, and spendingTime must be of full length kMax, and informationRates and spendingTime must end with 1.

References

Ping Gao, Yingqiu Li. Adaptive two-stage seamless sequential design for clinical trials. Journal of Biopharmaceutical Statistics, 2025, 35(4), 565-587.

Author

Kaifeng Lu, kaifenglu@gmail.com

Examples

# Determine O'Brien-Fleming boundaries for a seamless design with
# 2 active arms in Phase 2 and 2 looks in Phase 3 (3 looks total).
getBound_seamless(M = 2, k = 2, informationRates = seq(1, 3)/3,
                  alpha = 0.025, typeAlphaSpending = "OF")
#> [1] 3.776605 2.670463 2.180424