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Exit Probabilities for Two-Stage Seamless Sequential Design (TSSSD)

Source: R/RcppExports.R
exitprob_tsssd.Rd

Computes the exit (rejection) probabilities for a two-stage selection and testing design. In Phase 2, multiple active arms are compared against a common control arm. The best-performing arm is selected to proceed to Phase 3, where it is tested against the common control over multiple looks.

Usage

exitprob_tsssd(
  M = NA_integer_,
  r = 1,
  theta = NA_real_,
  corr_known = TRUE,
  K = NA_integer_,
  b = NA_real_,
  I = NA_real_
)

Arguments

M

Number of active treatment arms in Phase 2 (\(M \ge 2\)).

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 critical values (length \(K+1\)). The first element is for Phase 2; the remaining \(K\) elements are for the looks in Phase 3.

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:

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

  • exitProbByArm: A \((K+1) \times M\) matrix. The \((k, m)\)-th entry represents the probability that the global null is rejected at look \(k\) given that arm \(m\) was selected as the best arm.

  • selectAsBest: A vector of length \(M\) containing the probability that each active arm is selected to move on to 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 score statistic at the end of Phase 2.

Decision Rules:

  • Phase 2: The global null hypothesis is rejected if the Wald statistic for the best arm, \(Z(I_0)\), satisfies \(Z(I_0) \ge b_0\).

  • Phase 3: If the trial continues, the hypothesis is rejected at look \(k\) if \(Z(I_k) \ge b_k\) and all previous looks (including Phase 2) failed to reject.

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 statistic.

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 max 110 patients
# per arm, SD = 1.0
I <- c(110 / (2 * 1.0^2) * seq(1, 3)/3)

# O'Brien-Fleming critical values
b <- c(3.776606, 2.670463, 2.180424)

# Type I error under the global null hypothesis
p0 <- exitprob_tsssd(M = 2, theta = c(0, 0), K = 2, b = b, I = I)
cumsum(p0$exitProb)
#> [1] 0.000157278 0.006643135 0.025000008

# Power under alternative: Treatment effects of 0.3 and 0.5
p1 <- exitprob_tsssd(M = 2, theta = c(0.3, 0.5), K = 2, b = b, I = I)
cumsum(p1$exitProb)
#> [1] 0.05477555 0.62309680 0.90160747