Composed Pipeline Type I Error Simulator

2.1 Trial structure

Two-arm Phase II oncology design with binary endpoint (e.g., ORR). The default schedule mirrors the JSM 2026 reference design: monitoring looks at $n \in \{30, 45, 60, 90\}$, planned final at $n_{\text{base}} = 90$, cap at $N_{\max} = 150$, and SSR firing at $n = 45$ via blinded pooled response. Treatment and control arms accrue concurrently; allocation is either fixed 1:1 alternation (when RAR is off) or Thompson Sampling after a 10-patient burn-in (when RAR is on).

2.2 MAP prior

The control prior is a two-component Beta mixture

with default parameters $a_{\text{inf}} = 6.5,\ b_{\text{inf}} = 19.5,\ w = 0.70$ (informative component, ESS ≈ 26, prior mean ≈ 0.25) and $a_{\text{vag}} = b_{\text{vag}} = 0.5$ (Jeffreys robustification). Mixture weights are updated at each interim via the log marginal-likelihood ratio. The treatment-arm posterior uses an independent Jeffreys prior — historical borrowing applies only to control.

2.3 Bayesian monitoring

At each scheduled look the calculator computes

via Monte Carlo (default 2,000 samples per look). Stop for efficacy if this posterior probability exceeds $\gamma^*$. The default $\gamma^* = 0.970$ is the calibrated value that yields T1E ≈ α = 0.025 under H₀ with no conflict and no trend.

2.4 Blinded SSR

At $n_{\text{ssr,trig}}$ the pooled response count $x_{\text{pool}} = x_e + x_c$ is used in a Wald-test conditional-power formula to compute the smallest $N \le N_{\max}$ achieving the target conditional power (default 80%). Futile interims ($\hat p_{\text{pool}} \le p_0$) leave $N_{\text{final}}$ unchanged.

2.5 Response-Adaptive Randomization

Thompson Sampling: at each enrollment after a 10-patient burn-in, sample one Beta variate from each arm's posterior and allocate to whichever variate is larger. This is the canonical TS reference implementation; it converges toward complete allocation to the better arm (a fundamentally different limit from the Rosenberger DBCD target).

2.6 Time trend

Linear drift over fractional enrollment time: $p_{\text{arm}}(t) = p_{\text{arm,base}} + \delta \cdot t$, with $t \in [0, 1]$. Both arms drift in parallel under H₀, so the data-generating null is preserved. RAR coupled with this drift induces allocation imbalance that biases the final analysis.

Each preset on the calculator runs a multi-scenario batch and reproduces a specific table from the JSM paper.

4.1 The estimand

The reported rejection rate is the frequentist Type I error rate under the data-generating null $p_e = p_c$ — not a posterior predictive error rate conditional on prior beliefs. The prior–data conflict scenarios are realistic situations where the historical prior is misspecified relative to the current trial population, but H₀ still holds in the current data. The inflation we report is the false-positive rate under the current trial's data-generating null — the regulatory-relevant quantity, not a Bayesian robustness diagnostic.

4.2 Monte Carlo uncertainty

The simulator reports the rejection rate, its binomial Monte Carlo standard error $\text{MCSE} = \sqrt{\hat p (1 - \hat p) / N_{\text{sim}}}$, and a 95% Wald CI. At $N_{\text{sim}} = 5{,}000$ with $\hat p \approx 0.08$, MCSE ≈ 0.0038 and the CI half-width is about ±0.0075. To halve MCSE, run at $N_{\text{sim}} = 20{,}000$ in the Advanced panel.

4.3 Why a single-seed Python run can drift from the paper

The paper's reference numbers were generated using R's default Mersenne Twister PRNG with specific seeds (e.g., 2028 for the mild-conflict headline). This calculator uses NumPy's PCG64 PRNG. Same seed values produce different random streams across the two engines, so a Python single-seed run can drift by 2–3 MCSE from the paper's single-seed value even when the engines are mathematically identical. The reported σ-deviation column is the right diagnostic: across multiple seeds the engine reproduces the paper's population values within MCSE.

4.4 Super-additive interaction term

When the Mechanism Isolation 2×2 preset finishes, the calculator displays $\Delta_{\text{interaction}}$ below the results table. A positive value means the joint effect (conflict + trend) exceeds the sum of independent contributions — the mechanisms amplify each other beyond additivity. The paper's reference value is +0.0278 (~2.78pp). A null-or-negative interaction term indicates the perturbations roughly add; a strongly positive one (as observed) is the key qualitative finding.

• Two-arm, binary endpoint. Continuous and survival are on the roadmap but not implemented in v1; the JSM paper itself is binary-only.
• Blinded SSR with pooled response. The calculator uses aggregate counts at the SSR look (the "single-arm in the sense that SSR uses only aggregate counts, not treatment-arm allocation" phrasing from the paper). Unblinded treatment-arm-specific SSR is not modeled.
• Thompson Sampling RAR only. DBCD, Neyman, and other allocation rules are not exposed in this calculator. Use the dedicated RAR calculator for those.
• MAP prior is pre-pooled. Users supply $(a_{\text{inf}}, b_{\text{inf}}, w_{\text{inf}}, a_{\text{vag}}, b_{\text{vag}})$ directly via the Advanced panel. To fit a MAP prior from historical cohort data, use the Bayesian Borrowing calculator first and copy the fitted hyperparameters across.
• Linear time trend only. Sigmoidal, step, or non-linear trends would require code changes; the JSM paper's empirical observation is that linear drift is sufficient to demonstrate the super-additive interaction.