Bayesian Sequential Monitoring

Bayesian sequential monitoring extends fixed-sample Bayesian designs by allowing repeated interim analyses during patient enrollment. At each pre-specified look, the posterior probability that the treatment is superior is compared against stopping thresholds — enabling early termination for either efficacy or futility. Zhou & Ji [1] provide a comprehensive framework comparing three Bayesian sequential approaches, of which this calculator implements the posterior probability (PP) method.

Why Bayesian Sequential?

Supported Endpoint Types

Zhou & Ji [1] describe three distinct Bayesian approaches to sequential clinical trial design. Each defines stopping rules differently, leading to different boundary shapes and operating characteristics.

Binary Endpoint (Beta-Binomial)

Each arm has an independent Beta prior on the response rate:

At interim look $k$, having observed $s_{T,k}$ successes in $n_{T,k}$ treatment patients, the posterior is:

The posterior probability of treatment superiority is estimated via Monte Carlo:

Continuous Endpoint (Normal-Normal)

The treatment effect $\theta$ has a Normal prior:

With known data variance $\sigma^2$, after observing the z-statistic $z_k$ at look $k$ with cumulative sample size $n_k$:

At each interim analysis $k = 1, 2, \ldots, K$, the trial may stop based on the posterior probability that treatment is better:

Efficacy Stopping

where $\gamma_{\text{eff}}$ is the efficacy threshold (typically 0.95-0.995). Higher values are more conservative, reducing Type I error at the cost of lower power.

Futility Stopping (Optional)

where $\gamma_{\text{fut}}$ is the futility threshold (typically 0.05-0.20). Futility stopping reduces the expected sample size under the null hypothesis by terminating trials unlikely to succeed.

Continuous Endpoints: Analytical Boundaries

For Normal-Normal models, the stopping boundary at look $k$ with cumulative per-arm sample size $n_k$ is a z-score threshold derived analytically from the posterior probability criterion ([1], Equation 5):

Stop for efficacy if the observed z-statistic $z_k > c_k$. For futility, a symmetric lower boundary is derived using $\gamma_{\text{fut}}$.

Binary Endpoints: Simulation-Based

For Beta-Binomial models, there is no closed-form boundary. Instead, at each look:

1. Update Beta posteriors for each arm using observed data
2. Draw $M$ Monte Carlo samples from each posterior
3. Compute the posterior probability:
   $$P(\theta_T > \theta_C | \text{data}_k) \approx \frac{1}{M}\sum \mathbb{1}(\theta_T^{(m)} > \theta_C^{(m)})$$
4. Compare directly against $\gamma_{\text{eff}}$ and $\gamma_{\text{fut}}$

Operating characteristics are evaluated via Monte Carlo simulation under both the null and alternative hypotheses.

Simulation Algorithm

for sim in 1...N_simulations:
    # Generate full trial data under H0 and H1
    data_h0 = generate(θ_T = θ_C)
    data_h1 = generate(θ_T = θ_T_design)

    for look k = 1...K:
        # Check stopping at each look
        P_h0 = posterior_prob(data_h0[1:n_k])
        P_h1 = posterior_prob(data_h1[1:n_k])

        if P_h0 >= γ_eff → type1_count++, record stop at k
        if P_h1 >= γ_eff → power_count++, record stop at k
        if P ≤ γ_fut    → record futility stop at k

# Outputs:
type1_error = type1_count / N_simulations
power = power_count / N_simulations
expected_N_h0 = mean(stop_time_h0) × n_per_look
expected_N_h1 = mean(stop_time_h1) × n_per_look

Additional Metrics

Key Findings from Simulation Studies

Zhou & Ji [1] evaluated OC across 72 scenarios (combinations of true effect distribution $\nu_0$, prior width $\nu$, and number of looks $K$) with $N = 1000$, $\gamma = 0.95$, and 10,000 simulated trials:

• Type I error control: With a well-calibrated prior ($\nu \approx \nu_0$), the false positive rate (FPR) stays below 5% even with frequent monitoring ($K = 1000$ continuous looks).
• Prior-data conflict: When the prior is much wider than the true effect distribution ($\nu \gg \nu_0$), the FPR can inflate above the nominal level. Conversely, overly narrow priors reduce power.
• Number of looks: More interim analyses generally does not inflate error rates in the Bayesian framework (unlike repeated frequentist testing), provided the posterior probability threshold is held constant.
• Coverage: Posterior credible interval coverage remains near 95% across scenarios, with slight under-coverage when the prior is strongly misspecified.

Power Curve

The power curve shows the probability of declaring efficacy as a function of the true treatment effect. For binary endpoints, this is plotted against the treatment rate; for continuous, against the true effect size.

• At $\theta_T = \theta_C$: Should equal the Type I error (typically 2-5%)
• At design alternative: Should meet target power (typically 80-90%)
• Steep transition: Indicates good discriminatory ability

Average Sample Number (ASN) Curve

The ASN curve shows the expected sample size per arm as a function of the true effect. This quantifies the efficiency gains from sequential monitoring:

• Under H₀ (no effect): ASN well below max N indicates effective futility stopping
• Under H₁ (large effect): ASN well below max N indicates early efficacy detection
• Near the boundary: ASN approaches max N (ambiguous evidence requires full enrollment)

SAP Documentation Checklist

• Number & Timing of Looks: Pre-specify K and equally-spaced information fractions
• Stopping Rules: Efficacy threshold $\gamma_{\text{eff}}$, futility threshold $\gamma_{\text{fut}}$ (if applicable), and whether futility is binding or non-binding
• Prior Specification: Prior parameters with clinical justification, effective sample size, and sensitivity to prior choice
• Operating Characteristics: Type I error, power, expected N under both hypotheses, stopping probabilities at each look
• Simulation Details: Number of Monte Carlo simulations, random seed, software version
• Boundary Table: Full table of boundaries, information fractions, and stopping probabilities at each look

Calibration Guidance