Calibrate an optimal single-arm two-stage ROPE design — design_singlearm_twostage

Finds a single-arm two-stage Bayesian design based on the region of practical equivalence (ROPE) for a binary endpoint, with a single interim analysis allowing early stopping for futility. The design covers three decision types via the direction argument:

"equivalence": Posterior mass inside the two-sided ROPE \([p_0 - \delta,\, p_0 + \delta]\) must exceed \(\gamma_{\mathrm{eq}}\).
"noninferiority": Posterior probability \(\Pr(p \ge p_0 - \delta \mid Y)\) must exceed \(\gamma_{\mathrm{eq}}\).
"superiority": Posterior probability \(\Pr(p > p_0 + \delta \mid Y)\) must exceed \(\gamma_{\mathrm{eq}}\).

Usage

design_singlearm_twostage_rope(
  p0,
  delta,
  analysis_prior = c(1, 1),
  design_prior_h0,
  design_prior_h1,
  gamma_1 = 0.5,
  gamma_eq = 0.9,
  gamma_diff = gamma_eq,
  alpha = 0.1,
  power = 0.8,
  pce = NULL,
  alpha_freq = NULL,
  power_freq = NULL,
  p_t1e = NULL,
  p_power = NULL,
  nmax = 300L,
  direction = c("equivalence", "noninferiority", "superiority"),
  minimax = FALSE,
  progress = TRUE
)

Arguments

p0: Benchmark response probability.
delta: ROPE half-width ("equivalence"), non-inferiority margin ("noninferiority"), or superiority margin ("superiority"). Must be a single positive number.
analysis_prior: Numeric vector c(a, b) for the \(\mathrm{Beta}(a, b)\) analysis prior on \(p\). Defaults to c(1, 1) (uniform).
design_prior_h0: Numeric vector c(a, b) for the null design prior.
design_prior_h1: Numeric vector c(a, b) for the alternative design prior.
gamma_1: Interim futility threshold in \((0, 1)\) applied to the posterior support for \(H_0\). In the equivalence design, the trial stops early for futility if \(\Pr(p \notin \mathcal{R}_p \mid Y_1) \ge \gamma_1\), equivalently if the interim posterior ROPE probability is at most \(1-\gamma_1\). Continuation to stage 2 occurs otherwise.
gamma_eq: Final evidence threshold in \((0.5, 1)\): the appropriate posterior ROPE probability must exceed gamma_eq to declare equivalence, non-inferiority, or superiority.
gamma_diff: Threshold for compelling evidence for \(H_0\): the complementary posterior ROPE probability must exceed gamma_diff. In the two-stage design, compelling evidence for \(H_0\) may be obtained either at the interim analysis or at the final analysis. Defaults to gamma_eq.
alpha: Target predictive type-I error level (upper bound).
power: Target predictive power (lower bound).
pce: Optional lower bound on predictive PCE(H0).
alpha_freq: Optional upper bound on the frequentist type-I error.
power_freq: Optional lower bound on the frequentist power.
p_t1e: Point at which the frequentist type-I error is evaluated.
p_power: Point at which the frequentist power is evaluated.
nmax: Upper bound on the fixed-sample size \(n^*\) searched in step 1. An informative error is raised if no feasible size is found.
direction: Character string specifying the decision type. One of "equivalence" (default), "noninferiority", or "superiority".
minimax: Logical. If TRUE, minimise \(n\) (minimax criterion); if FALSE (default), minimise \(\mathrm{EN}_0\) (optimal criterion).
progress: Logical. If TRUE (default), print progress messages.

Value

An object of class "singlearm_rope_twostage_design".

Details

The search proceeds in two steps: (1) find the minimum fixed-sample size \(n^*\) at which the one-stage constraints are satisfied; (2) enumerate all two-stage splits \(n_1 + n_2 = n^*\) and retain those satisfying the two-stage constraints. The optimal design minimises \(\mathrm{EN}_0\) (or \(n^*\) under the minimax criterion) among all feasible splits.