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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 phase-2 max-Z statistic 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 best-performing arm is selected to proceed to Phase 3, where it is tested against the common 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
)

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.

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 max-Z 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 max-Z 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.

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 as best.

  • 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 as best.

  • selectAsBest: A vector of length \(M\) containing the probability that each active arm is selected to move on to Phase 3.

For backward compatibility, the list also contains:

  • exitProb: identical to exitProbUpper.

  • exitProbByArm: identical to exitProbByArmUpper.

Details

The function assumes a multivariate normal distribution for the Wald statistics. The "best" arm is defined as the active arm with the largest Z-statistic at the end of Phase 2 among designs that continue beyond the phase-2 analysis.

Decision Rules:

  • Phase 2 efficacy stop: reject if the phase-2 max-Z statistic satisfies \(\max_m Z_m(I_0) \ge b_0\).

  • Phase 2 futility stop: stop for futility if the phase-2 max-Z statistic satisfies \(\max_m Z_m(I_0) \le a_0\).

  • Continue to Phase 3: if \(a_0 < \max_m Z_m(I_0) < b_0\), select the arm with the largest phase-2 Z-statistic and continue with that arm 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 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,]         0.05477567        0.007803144
#> [2,]         0.62292767        0.014040545
#> [3,]         0.89800885        0.101991188