exitprob_seamless: Exit Probabilities for Phase 2/3 Seamless Design

In lrstat: Power and Sample Size Calculation for Non-Proportional Hazards and Beyond

Exit Probabilities for Phase 2/3 Seamless Design

Description

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.

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.

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.

Author(s)

Kaifeng Lu, [e-mail]

References

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

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)

# 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)
)

lrstat documentation built on May 13, 2026, 9:06 a.m.