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Exit Probabilities for Phase 2/3 Seamless Design

Source: R/RcppExports.R
exitprob_seamless.Rd

Computes the upper and lower exit probabilities for a phase 2/3 seamless design. In Phase 2, multiple active arms are compared against a common control arm. If the test statistic for the arm ranked rankp0 at the end of Phase 2 crosses the efficacy boundary, the trial stops early for efficacy; if it falls below the futility boundary, the trial stops early for futility. Otherwise, the arm is selected to proceed to Phase 3, where it is tested against the control over multiple looks with upper and optional lower stopping boundaries.

Usage

exitprob_seamless(
  M = NA_integer_,
  r = 1,
  theta = NA_real_,
  corr_known = TRUE,
  K = NA_integer_,
  b = NULL,
  a = NULL,
  I = NULL,
  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.

theta

A vector of length \(M\) representing the true treatment effects for each active arm versus the common control.

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

Number of sequential looks in Phase 3.

b

A vector of efficacy boundaries (length \(K + 1\)). The first element is the efficacy boundary for the Phase-2 test statistic; the remaining \(K\) elements are efficacy boundaries for the selected arm in Phase 3.

a

An optional vector of futility boundaries (length \(K + 1\)). The first element is the futility boundary for the Phase-2 test statistic; the remaining \(K\) elements are futility boundaries for the selected arm in Phase 3. If omitted, no futility stopping is applied.

I

A vector of information levels (length \(K + 1\)) for any active arm versus the common control. The first element is for Phase 2; the remaining \(K\) elements are for the looks in Phase 3.

rankp0

An integer between 1 and M specifying which ranked Phase-2 arm is carried forward when the trial continues to Phase 3. rankp0 = 1 selects the largest Phase-2 Z-statistic, rankp0 = 2 selects the second largest, and so on.

Value

A list containing the following components:

  • exitProbUpper: A vector of length \(K + 1\). The first element is the probability of stopping for efficacy in Phase 2; the remaining elements are the probabilities of stopping for efficacy at each look in Phase 3.

  • exitProbLower: A vector of length \(K + 1\). The first element is the probability of stopping for futility in Phase 2; the remaining elements are the probabilities of stopping for futility at each look in Phase 3.

  • exitProbByArmUpper: A \((K + 1) \times M\) matrix. The \((k, m)\)-th entry gives the probability of stopping for efficacy at look \(k\) given that arm \(m\) is selected at rank rankp0.

  • exitProbByArmLower: A \((K + 1) \times M\) matrix. The \((k, m)\)-th entry gives the probability of stopping for futility at look \(k\) given that arm \(m\) is selected at rank rankp0.

  • selectionProb: A vector of length \(M\) containing the probability that each active arm is selected at rank rankp0.

Details

The function assumes a multivariate normal distribution for the Wald statistics. Among designs that continue beyond the Phase-2 analysis, the carried-forward arm is the one with rank rankp0 based on the p-value of the Z-statistic at the end of Phase 2.

Decision Rules:

  • Phase 2 efficacy stop: reject if the Phase-2 test statistic for the arm selected at rank rankp0 satisfies \(Z_{[rankp0]}(I_0) \ge b_0\).

  • Phase 2 futility stop: stop for futility if the Phase-2 test statistic for the arm selected at rank rankp0 satisfies \(Z_{[rankp0]}(I_0) \le a_0\).

  • Continue to Phase 3: if \(a_0 < Z_{[rankp0]}(I_0) < b_0\), continue with the arm selected at rank rankp0 only.

  • Phase 3 efficacy stop: at look \(k\), reject if the selected arm's Z-statistic exceeds the efficacy boundary and no earlier stop has occurred.

  • Phase 3 futility stop: at look \(k\), stop for futility if the selected arm's Z-statistic is below the futility boundary and no earlier stop has occurred.

Design Assumptions:

  • All active arms share the same information level in Phase 2.

  • Exactly one active arm is selected at the end of Phase 2 based on the rankp0-th largest observed Z-statistic when the trial continues to Phase 3.

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


# Setup: 2 active arms vs control in Phase 2; 1 selected arm vs control
# in Phase 3. Phase 3 has 2 sequential looks.

# Information levels: equal spacing over 3 looks based on a maximum of
# 110 patients per arm, SD = 1.0
I <- c(110 / (2 * 1.0^2) * seq(1, 3)/3)

# O'Brien-Fleming efficacy boundaries
b <- c(3.776605, 2.670463, 2.180424)

# No futility stopping
p0 <- exitprob_seamless(M = 2, theta = c(0, 0), K = 2, b = b, I = I)
cumsum(p0$exitProbUpper)
#> [1] 0.0001572756 0.0066431322 0.0250000060

# Add futility stopping
a <- c(0, 0.5, b[3])
p1 <- exitprob_seamless(
  M = 2, theta = c(0.3, 0.5), K = 2, b = b, a = a, I = I)
cbind(
  cumulativeEfficacy = cumsum(p1$exitProbUpper),
  cumulativeFutility = cumsum(p1$exitProbLower)
)
#>      cumulativeEfficacy cumulativeFutility
#> [1,]         0.05477566        0.007803144
#> [2,]         0.62292767        0.014040545
#> [3,]         0.89800885        0.101991188