Two-dose vs. Three-Dose Optimization Under Sample Size Constraint

Dose optimization is a hallmark of Project Optimus for oncology drug development. The number of doses to include in a dose optimization study depends on the totality of evidence, which is often unclear in early-phase development. With equal sample sizes per dose, carrying three doses is clearly more advantageous than two for optimization. In this paper, we show that, even when the total sample size is fixed, it is still preferable to carry three unless there is very strong evidence that one can be dropped. A mathematical approximation is applied to guide the investigation, followed by a simulation study to complement the theoretical findings. Semi-quantitative guidance is provided for practitioners, addressing both randomized and non-randomized dose optimization while considering population homogeneity.

💡 Research Summary

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The manuscript tackles a practical question that has become central to oncology drug development under the FDA’s Project Optimus: when the total number of patients available for a dose‑optimization study is fixed, should investigators allocate those patients to two dose arms or three dose arms? The authors approach the problem from a statistical perspective, first deriving a simple analytical approximation for the relationship between Type I error (α), Type II error (β), effect size (δ) and sample size (n) in a two‑arm trial with equal allocation. By expressing the required sample size as (n \approx 4,\Phi^{-1}(α/2)^2 / δ^2) and introducing the “relative signal strength” (r = δ\sqrt{n}), they show that, for the range of powers typical of early‑phase dose‑optimization studies (≈25 %–75 %), the normal CDF can be approximated linearly around zero with less than 2 % error. This yields a convenient proportionality: the power of a three‑arm design that compares the highest and lowest doses directly is roughly 1.5 times the power of a two‑arm design that uses the same total sample size, while a two‑arm design that compares adjacent doses (low vs. middle or middle vs. high) attains only about 0.375 of the three‑arm power.

The authors then incorporate prior belief about the middle dose’s optimality, denoted λ. Two decision rules are examined: (1) a two‑arm trial that includes only the low and high doses, and (2) a two‑arm trial that includes two adjacent doses. By solving the inequalities derived from the power approximations, they find that the low‑high design is justified only when λ exceeds roughly 0.63 (i.e., there is at least a 60 % prior probability that the middle dose is not optimal). For the adjacent‑dose design, a stronger prior confidence (λ > 0.78) is required. These thresholds quantify the intuitive notion that excluding a dose is only sensible when the investigator is fairly certain it will not be the best choice.

To validate the analytical results, a Bayesian model‑selection simulation is performed. Four plausible dose‑response shapes (linear increase, early plateau, delayed plateau, and steep rise) are defined, and priors are calibrated to reflect realistic oncology scenarios. For each shape, the authors simulate a three‑arm trial with 30 patients per dose (total N = 90) and compare it to a two‑arm trial that keeps the total N constant. The primary metric is the probability of correctly selecting the optimal dose (PCS). Under the linear scenario (the most favorable for detecting differences), the three‑arm design achieves a PCS of about 74 %, whereas the two‑arm design reaches only 51 %. The gap widens when the two‑arm design happens to omit the true optimal dose (e.g., when the high dose is excluded). In plateau scenarios, PCS values for the two designs become more comparable, but the three‑arm design still offers the advantage of characterizing the full dose‑response curve in a single study, something the two‑arm design cannot provide.

Practical guidance is distilled into a five‑star rating system that evaluates different implementation strategies: randomization versus back‑filling, and the number of doses included. Randomized three‑arm designs receive the highest rating, while back‑filling in a homogeneous patient population (e.g., same disease subtype or biomarker status) can achieve comparable information with fewer resources if the low dose is added as a “backfill” after the escalation phase. The authors also discuss hybrid approaches (e.g., enrolling extra patients at the low dose while randomizing only middle and high doses) and suggest using different strategies across indications or geographic regions to balance speed, cost, and statistical rigor.

In the discussion, the authors reiterate that the mathematical approximation assumes a linear dose‑response relationship, which is a reasonable default for early‑phase oncology trials where the true shape is unknown. The simulation results, based on a Bayesian framework, corroborate the analytical insight that three doses generally provide higher power and more reliable dose selection unless there is strong prior evidence (≈60 %–80 % confidence) that the middle dose is sub‑optimal. They advocate incorporating all available data—from dose‑escalation, back‑fill, and early efficacy signals—into an adaptive Phase 2/3 design, potentially allowing the dose‑optimization data to feed directly into the pivotal trial analysis.

Overall, the paper delivers a clear, quantitative message: when the total sample size is constrained, including three dose levels is statistically preferable to limiting the study to two levels, unless investigators possess compelling prior evidence that one of the doses (typically the middle one) is unlikely to be optimal. The combination of a tractable analytical formula, intuitive prior‑belief thresholds, and supportive simulation results makes the work directly applicable for clinical development teams planning dose‑optimization studies under Project Optimus.