Single-Arm Sample Size Re-estimation (SSR)

Single-arm Phase II trials are the dominant design in early oncology drug development. Enrolling all patients onto the experimental arm accelerates evidence generation when a randomized comparison is ethically or practically infeasible, and the observed objective response rate (ORR) is compared against a historical control rate $p_0$ drawn from prior trials of standard-of-care.

The FDA accelerated approval pathway permits drugs that demonstrate a meaningful improvement over existing therapies on a surrogate endpoint (frequently ORR) to reach patients while confirmatory Phase III trials are ongoing. Single-arm designs powered against a well-characterized $p_0$ are the workhorse of this pathway.

Why adaptive SSR? The initial sample size depends on a minimally-clinically-important alternative $p_1$ that sponsors often specify with significant uncertainty. If interim data suggest the true effect is smaller than $p_1$ but still clinically meaningful, a modest expansion can preserve power. Conversely, a very large observed effect supports an efficacy interim stop, and a very small effect supports futility termination—both sparing patients and resources.

Let $X_1, \ldots, X_n$ be i.i.d. Bernoulli responses with unknown true rate $p$. We test:

where $p_0$ is the historical control rate (null) and $p_1$ is the target alternative. Under a normal approximation to the one-sample binomial, the required fixed-design sample size is:

with $z_\alpha = \Phi^{-1}(1-\alpha)$ and $z_\beta = \Phi^{-1}(\text{power})$. The standard test at the final analysis rejects $H_0$ if the observed $\hat{p}$ exceeds a critical value derived from the binomial (or its normal approximation).

At the interim look with $n_1$ patients enrolled and $x_1$ responses, the design chooses between (i) early efficacy stop, (ii) early futility stop, or (iii) continuation—optionally with a re-estimated target sample size $n^*$ bounded by a pre-specified cap $n_{\max}$.

The Bayesian mode uses a conjugate Beta–Binomial framework. With prior $p \sim \text{Beta}(\alpha_0, \beta_0)$ and interim data $x_1$ responses in $n_1$ patients, the posterior is:

Posterior efficacy stopping. Stop early for efficacy at the interim if the posterior probability that $p$ exceeds the null rate clears the interim bar:

Two thresholds, not one. The design uses two distinct posterior-probability bars: gamma_efficacy at the interim (typically high, e.g., 0.97–0.99) and gamma_final at the final analysis (defaults to $1 - \alpha$, e.g., 0.975 for $\alpha = 0.025$). The interim bar is the stop-early gate; the final bar is the success criterion. Conflating the two depresses simulated power because predictive probability then projects to an inflated final bar.

Predictive futility stopping. Compute the Bayesian predictive probability (PPoS) that the trial will clear gamma_final at the final analysis given current data:

Stop for futility if $\text{PPoS} \leq \delta_\text{futility}$ (typically around 0.05). Otherwise, continue—optionally recalculating the final $n^*$ up to $n_{\max}$.

The conditional power (CP) mode adapts the Mehta–Pocock (2011) promising zone framework from two-arm to single-arm designs. Given interim statistic $z_1$ computed under the one-sample binomial:

the conditional power under the observed current trend (or under the target alternative, per SAP) is:

Zones are defined by CP thresholds:

• •Favorable (CP > promising upper): large effect; no re-estimation needed (or consider efficacy stop).
• •Promising (promising lower ≤ CP ≤ promising upper): re-estimate $n^*$ to restore planned CP, capped at $n_{\max}$.
• •Unfavorable (futility ≤ CP < promising lower): continue with planned sample size; do not inflate.
• •Futility (CP < futility threshold): consider stopping for futility.

The original Mehta–Pocock theorem (Chen, DeMets, Lan 2004; Gao, Ware, Mehta 2008) preserves Type I error in the two-arm normal/z-test setting when re-estimation is confined to the promising zone. For single-arm binomial designs this guarantee does not transfer analytically — the discrete sample space and exact-binomial final test mean Type I error must be confirmed via simulation (Tier 2 OC table) before fixing cp_promising_lower / cp_promising_upper for the protocol.

The choice of prior $\text{Beta}(\alpha_0, \beta_0)$ materially affects interim decisions, particularly when $n_1$ is small. Zetyra offers three presets:

• •Jeffreys Beta(0.5, 0.5) — default. The Jeffreys prior is the invariant reference prior for a Bernoulli parameter, derived from the square root of the Fisher information. It is objective in the sense that it is invariant under reparameterization and has prior effective sample size (ESS) of 1.
• •Flat Beta(1, 1). The uniform prior on $[0, 1]$. Often preferred by sponsors for its intuitive interpretation; ESS of 2. Slightly more informative than Jeffreys in the tails.
• •Custom informative priors. Derived from prior trials via the MAP prior / bayesian-borrowing workflow or elicited from experts via prior elicitation. Use with caution: regulators scrutinize informative priors that favor efficacy claims.

For both modes, simulated operating characteristics are mandatory before fixing thresholds for the protocol. Bayesian stopping rules are not analytically tied to frequentist Type I error, and the two-arm Mehta–Pocock promising-zone theorem does not transfer analytically to single-arm binomial CP designs (FDA Adaptive Designs Guidance 2019, Section V).

Zetyra's OC table reports, for a grid of true rates $p \in \{p_0, \ldots, p_1, \ldots\}$:

• •Type I error at $p = p_0$: must be $\leq \alpha$. If inflated in Bayesian mode, raise gamma_efficacy (typically toward 0.97–0.99) and re-simulate. If inflated in CP mode, do not assume tighter promising-zone bounds will help— Type I error is non-monotonic in cp_promising_lower for this single-arm binomial design (see the warning note below). Grid-search a few neighbouring values and re-simulate, or switch to Bayesian mode.
• •Simulated power at $p = p_1$: should match the planned power target.
• •Expected sample size $\mathbb{E}[N \mid p]$: shows the adaptive design's efficiency gain over fixed-N under each true rate, together with quantiles $N_{10}, N_{50}, N_{90}$.
• •Stopping probabilities: Pr(efficacy stop), Pr(futility stop), Pr(N hits cap) at each true rate.

Interpret the table jointly: a design with 5% Type I error, 82% power at $p_1$, and $\mathbb{E}[N \mid p_0]$ substantially below the fixed-N is well-tuned. An 8% Type I error means the thresholds are too liberal.

• •FDA Adaptive Designs Guidance (2019), Section IV.B. Sample size re-estimation is a well-characterized adaptation provided the rule, timing, and caps are pre-specified and Type I error is verified by simulation.
• •FDA Accelerated Approval. Single-arm ORR trials supporting accelerated approval must enroll a pre-specified population, use a locked analysis plan, and demonstrate a meaningful effect over historical control.
• •Project Optimus (2023). FDA oncology dose-optimization initiative emphasizes adequate sample sizes for dose selection and characterization of tolerability in Phase II, which SSR directly supports by expanding cohorts under promising interim trends.
• •Pre-specification requirements. The SAP must fix $p_0, p_1, \alpha, \text{power}$, the interim timing $n_1$, the prior (if Bayesian), the thresholds $(\gamma, \delta)$ or CP zones, the cap $n_{\max}$, and include simulation-based OC evidence.
• •SAP text generation. The Zetyra report exports an SAP-ready decision rule description plus the OC table and sensitivity scenarios directly suitable for inclusion in a protocol and SAP submission.

• •Historical control stability. The entire design rests on $p_0$ being a stable, well-characterized historical rate. Drift in $p_0$ (e.g., supportive-care improvements, population shifts, selection bias in the historical source) inflates Type I error without detection.
• •Binary endpoint only. The v1 engine supports binary (response/no response) endpoints. Continuous and time-to-event single-arm designs are not implemented.
• •Historical control misspecification. Even modest (2–5 pp) drift in $p_0$ can materially shift achieved Type I error. Sensitivity scenarios in the report show how the recalculated N and CP change under plausible alternative $p_0$.
• •Not for confirmatory Phase III. Single-arm designs are exploratory; efficacy claims for full approval require randomized confirmatory evidence except in narrow accelerated-approval settings.
• •One interim look. The v1 engine supports a single interim analysis. Multi-look GSD-style boundaries for single-arm trials should use the group-sequential calculator instead.

Endpoint: POST /api/v1/calculators/ssr-single-arm

Request parameters

Example Request

{
  "ssr_method": "bayesian",
  "p0": 0.20,
  "p1": 0.40,
  "alpha": 0.025,
  "power": 0.80,
  "interim_fraction": 0.5,
  "n_max_factor": 1.5,
  "prior_alpha": 0.5,
  "prior_beta": 0.5,
  "gamma_efficacy": 0.95,
  "gamma_final": null,
  "delta_futility": 0.05,
  "pp_promising_upper": 0.50,
  "simulate": true,
  "simulation_seed": 42,
  "n_simulations": 10000
}

gamma_final: null defaults to 1 - alpha (e.g., 0.975 for alpha 0.025). Raise pp_promising_upper toward 0.70 to keep more trials in the SSR promising zone.

Response Schema (abridged)

{
  "calculation_id": "...",
  "tier": "analytical+simulation",
  "analytical_results": {
    "initial_n": 36,
    "interim_n": 18,
    "interim_fraction": 0.5,
    "ssr_method": "bayesian",
    "posterior_probability": 0.97,
    "predictive_probability": 0.81,
    "conditional_power": 0.82,
    "conditional_power_planned": 0.82,
    "zone": "",
    "z1": 1.96,
    "efficacy_stop": true,
    "futility_stop": false,
    "recalculated_n": 18,
    "inflation_factor": 0.5,
    "n_capped": false,
    "n_max_used": 54,
    "gamma_final_used": 0.975,
    "prior_description": "Jeffreys Beta(0.5, 0.5)",
    "decision_rule_description": "...",
    "recalculation_scenarios": [
      {
        "label": "Planned effect",
        "assumed_nuisance": 0.40,
        "recalculated_n_per_arm": 36,
        "recalculated_n_total": 36,
        "inflation_factor": 1.0,
        "conditional_power": 0.82,
        "decision": "continue_favorable"
      }
    ],
    "regulatory_notes": [...]
  },
  "metadata": {...},
  "simulation": {...},
  "warnings": [],
  "regulatory_citations": [...]
}

decision enum values: stop_efficacy, stop_futility, continue_ssr, continue_favorable, continue_unfavorable. Five sensitivity rows are returned by default (50%, 75%, 100%, 125%, 150% of planned effect).

• •Simon R. (1989). Optimal two-stage designs for phase II clinical trials. Controlled Clinical Trials, 10(1), 1–10.
• •Lee JJ, Liu DD. (2008). A predictive probability design for phase II cancer clinical trials. Clinical Trials, 5(2), 93–106.
• •Mehta CR, Pocock SJ. (2011). Adaptive increase in sample size when interim results are promising: A practical guide with examples. Statistics in Medicine, 30(28), 3267–3284.
• •FDA. (2019). Adaptive Designs for Clinical Trials of Drugs and Biologics: Guidance for Industry. U.S. Food and Drug Administration.
• •FDA. (2023). Project Optimus: Optimizing the Dosage of Human Prescription Drugs and Biological Products for the Treatment of Oncologic Diseases. U.S. Food and Drug Administration.
• •Jeffreys H. (1961). Theory of Probability (3rd ed.). Oxford University Press.
• •Thall PF, Simon R. (1994). Practical Bayesian guidelines for phase IIB clinical trials. Biometrics, 50(2), 337–349.