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

This paper demonstrates that under a fixed total sample size, including three doses in an oncology dose optimization study is generally preferable to two, unless there is very strong evidence to exclude one, a conclusion supported by mathematical approximation and simulation to guide practitioners in both randomized and non-randomized settings.

Original authors: Linda Sun, Yixin Ren, Cong Chen

Published 2026-01-29

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Linda Sun, Yixin Ren, Cong Chen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a chef trying to find the perfect amount of salt for a new soup recipe. You have three candidates: a little salt (Low), a medium amount (Middle), and a lot of salt (High). You have a limited budget for ingredients and time (a fixed "sample size"), and you need to decide how to run your taste tests to find the winner.

This paper, written by statisticians at Merck, tackles a common dilemma in drug development: Should you test two doses or three doses when you have a fixed number of patients?

Here is the breakdown of their findings using simple analogies:

The Big Question: Two Arms or Three?

In the old days, drug developers often just picked the highest dose they could tolerate (the "Maximum Tolerable Dose"). But for new, smarter drugs, the relationship between dose and effect isn't always a straight line up. Sometimes, too much is bad, or the "sweet spot" is in the middle.

The FDA's "Project Optimus" wants companies to find this sweet spot early. But with limited patients, you face a choice:

The Two-Arm Race: Put all your patients into two groups (e.g., Low vs. High). You get more people per group, making the comparison very loud and clear.
The Three-Arm Race: Split your patients into three groups (Low, Middle, High). You have fewer people per group, so the signal is quieter, but you have a better chance of catching the winner if it happens to be the Middle dose.

The Mathematical "Rule of Thumb"

The authors did some math to see when it's okay to drop the Middle dose.

The Analogy: Imagine you are betting on a horse race. If you bet on the favorite (High) and the underdog (Low), you have a good chance of seeing a clear winner. But if the winner is actually the horse in the middle, and you didn't include it in your race, you lose.
The Finding: The math shows that unless you are 60% to 80% sure that the Middle dose is a "loser" (either too weak or too toxic), you should keep all three doses.
- If you drop the Middle dose, you might miss the optimal treatment entirely.
- Even though the three-dose study has fewer people per group, the ability to compare the Low and High doses directly (which is very powerful) usually outweighs the benefit of having more people in a two-dose study.

The Simulation: A Virtual Taste Test

To prove their math, the authors ran a computer simulation. They created four different "flavors" of how a drug might work (e.g., a straight line where more is better, or a plateau where it stops working after a certain point).

The Result: When the drug worked in a straight line (Low < Middle < High), the Three-Dose study picked the best dose (High) about 74% of the time. The Two-Dose study only got it right 51% of the time.
Why? In the Two-Dose study, if the researchers happened to pick Low and Middle to test, they would never see the High dose, and they would miss the true winner. The Three-Dose study covers all bases.

Practical Advice: How to Run the Race

The paper offers a "5-Star Rating System" to help drug developers decide how to organize their study. They suggest three main ways to get your three doses in:

Randomization (The Gold Standard): Randomly assign patients to Low, Middle, and High. This is the most scientific but takes time.
Backfilling (The Flexible Approach): You start with a small group to find a safe range. Then, you "backfill" by adding more patients to specific doses later.
- The Catch: This only works well if the patients are very similar to each other (homogeneous). If the patients are very different, backfilling can lead to unfair comparisons (like comparing apples to oranges).
The Hybrid: Randomize the Middle and High doses, but just add more patients to the Low dose if you aren't sure about it yet. This gives you flexibility to drop the Low dose if it looks bad without wasting the whole study.

The Bottom Line

The paper concludes with a simple message for drug developers: Don't be afraid to test three doses.

Unless you have very strong evidence that the middle option is useless, testing three doses is statistically smarter than testing two. It gives you a better map of the "dose-response landscape" and ensures you don't accidentally leave the best dose behind. If you must choose two, make sure they are far apart (Low vs. High) rather than close together (Low vs. Middle), because comparing extremes gives you more information.

Technical Summary: Two- vs. Three-Dose Optimization Under Sample Size Constraint

Problem Statement
In the context of FDA's Project Optimus, which seeks to reform dose selection paradigms in oncology, a critical challenge arises during early-phase development: determining the optimal number of doses to include in a dose optimization study when the total sample size is fixed. While traditional cytotoxic chemotherapy often relies on the Maximum Tolerable Dose (MTD), novel therapies require a more nuanced approach to identify the optimal dose within a range. When three candidate doses (low, middle, high) are available, investigators face a trade-off: a two-arm study allows for 50% more patients per arm compared to a three-arm study, potentially increasing power for specific pairwise comparisons. However, excluding a dose risks missing the optimal candidate. The central question addressed is whether a two-dose study is statistically preferable to a three-dose study under a fixed total sample size, and under what conditions (e.g., strength of prior evidence) one might justify excluding a dose.

Methodology
The authors employ a dual-methodological approach combining mathematical approximation and simulation to evaluate the statistical power and probability of correct selection (PCS) across different study designs.

Mathematical Approximation:
- The authors derive an approximation for the relationship between Type I/II errors and sample size in two-arm studies.
- They define a relative signal strength parameter, $r$ , adjusted for sample size, and demonstrate that for underpowered studies (typical of Phase 2 optimization) with power between 25% and 75%, the power function is approximately linear with respect to the signal strength. This linearity arises from the approximation of the standard normal CDF ( $\Phi$ ) near zero.
- Using this linear relationship, the authors model the power of a three-arm study (driven by the contrast between low and high doses via a Cochran-Armitage trend test) versus various two-arm configurations (Low vs. High, Low vs. Middle, Middle vs. High).
- They calculate the threshold of prior belief ( $\lambda$ ) required to justify excluding the middle dose or restricting a study to two adjacent doses.
Simulation Study:
- A Bayesian model-selection framework is implemented to supplement the theoretical findings.
- The simulation spans four plausible dose-response shapes (ranging from flat to linear to plateau) using priors calibrated to realistic clinical expectations.
- Designs compared include a 3-dose design ( $N=30$ per dose) versus 2-dose comparisons, holding the total sample size constant.
- The primary metric is the Probability of Correctly Selecting the right dose (PCS). The selection rule prioritizes higher efficacy, with lower doses chosen if efficacy is comparable (safety consideration).

Key Contributions and Results

Power Dynamics and Prior Belief:
- The mathematical analysis reveals that a three-arm study generally offers superior power for dose selection compared to a two-arm study with the same total sample size, provided the optimal dose is not known with high certainty.
- Specifically, the power of a two-arm study comparing the low and high doses is approximately $\sqrt{3/2}$ times the power of a three-arm study, while comparing adjacent doses yields only $\sqrt{3/8}$ of the three-arm power.
- Thresholds for Exclusion: The authors derive that the middle dose may only be excluded in a two-arm study (Low vs. High) if there is a prior belief ( $\lambda$ ) of approximately 63% that the middle dose is not optimal. Conversely, justifying a study of two adjacent doses requires a much stronger prior belief of approximately 78% that the optimal dose lies strictly between them.
Simulation Findings:
- Under a linear dose-response assumption, the 3-dose design achieved a PCS of ~74% for selecting the high dose, compared to ~51% for a 2-dose design (where the high dose might be excluded entirely).
- Even in scenarios where a 2-dose design consistently selects the correct pair (e.g., Middle vs. High), it fails to characterize the full dose-response relationship, a limitation the 3-dose design overcomes.
- In plateau scenarios, both designs yield comparable PCS for selecting the optimal dose, but the 3-dose design remains superior for establishing the shape of the dose-response curve.
Practical Guidance and Rating System:
- The paper proposes a "5-star rating system" to guide practitioners in selecting dose optimization strategies based on population homogeneity and randomization methods.
- 3-dose optimization is rated one star higher than 2-dose optimization generally.
- Dose randomization is rated two stars higher than non-randomized "backfilling" (expanding dose escalation cohorts) due to the risk of baseline imbalances in the latter.
- However, the authors note that extended backfilling in highly homogeneous populations (e.g., specific biomarker-defined groups) with sufficient sample sizes can provide comparable optimization information to randomized designs, offering a cost-effective alternative.

Significance and Claims
The paper claims to provide semi-quantitative guidance for practitioners navigating the complexities of Project Optimus. Its primary significance lies in challenging the assumption that maximizing sample size per arm (via a 2-dose study) is always the optimal strategy. Instead, the authors argue that unless there is very strong evidence to exclude a candidate dose, a three-dose study is preferable even with a fixed total sample size.

The findings support the intuition that spacing doses apart (Low vs. High) is more informative than comparing adjacent doses, but they quantify the high level of confidence required to justify such a reduction in dose levels. The paper emphasizes that a 3-dose design not only facilitates better dose selection but also characterizes the dose-response relationship, which is critical for regulatory decision-making. The authors position their work as a "rule-of-thumb" derived from mathematical approximation and simulation, intended to balance statistical rigor with the practical constraints of early-phase drug development.