📊 statistics

Adaptive clinical trials based on design-optimal e-values with automatic curtailment: An application to single-arm trials with binary data

This paper proposes and validates finite-horizon optimal e-value designs for single-arm binary clinical trials that utilize dynamic programming to maximize power or minimize sample size while ensuring anytime-validity and enabling automatic futility curtailment.

Original authors: Stef Baas, Judith ter Schure, Joost van Rosmalen

Published 2026-05-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Stef Baas, Judith ter Schure, Joost van Rosmalen

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 running a clinical trial to see if a new cancer drug works. In the old way of doing things, you set a rigid schedule: "We will check the results after 10 patients, then 20, then 30." If you miss a check-in or recruit patients faster than expected, the math gets messy, and you might accidentally claim the drug works when it doesn't.

This paper introduces a smarter, more flexible way to run these trials using a concept called e-values. Think of an e-value not as a dry statistic, but as a gambling score or a bank account balance that tracks how much evidence you have gathered against the idea that the drug is useless.

Here is the breakdown of the paper's ideas using simple analogies:

1. The Betting Game (The Core Idea)

Imagine you are a gambler who thinks the new drug does work. You start with $1 of "evidence capital."

The House (Null Hypothesis): The "House" believes the drug is useless. They set the odds based on the standard of care (e.g., a 10% success rate).
The Bet: Every time a patient takes the drug, you get to bet a portion of your capital.
- If the patient responds well (a "win"), your capital grows.
- If the patient doesn't respond (a "loss"), your capital shrinks.
The Goal: Your goal is to grow your capital from $1 to $20 (which represents strong proof that the drug works). If you hit $20, you win the trial.
The Safety Rule: If the drug is actually useless, the "House" ensures that, on average, your capital will never grow. It's a fair game for the House. But if the drug is good, your capital will grow rapidly.

2. The "Smart" vs. "Dumb" Betting Strategy

The paper compares two ways to play this game:

The "Dumb" Strategy (GROW/Kelly Betting): This is the standard method used in many modern trials. It's like playing a game where you assume you have infinite time to play. You bet a fixed percentage every time to maximize long-term growth.
- The Problem: Clinical trials have a hard stop (e.g., "We can only recruit 50 people"). If you are behind schedule near the end, the "infinite time" strategy is too cautious. It won't bet enough to catch up, and you might miss the win even if the drug works.
The "Smart" Strategy (Design-Optimal): This is what the authors invented. They use a GPS (called Dynamic Programming) that knows exactly how many patients are left.
- If you are ahead: The GPS tells you to bet conservatively to avoid losing your lead.
- If you are behind: The GPS tells you to bet aggressively (maybe even "all-in") to try to reach the $20 goal before the game ends.
- The Result: This strategy is much better at finding the drug's true effect in a limited number of patients.

3. Automatic "Stop Signs" (Curtailment)

One of the coolest features of this system is Automatic Curtailment.

The Bankruptcy Scenario: Imagine you are betting, but you lose so much that your capital drops to $0.
- In the old way, you might keep recruiting patients just to be "sure," even though you know you can't possibly win now.
- In this new system, if your capital hits $0, the game automatically stops. You know immediately that it is impossible to reach the $20 goal with the remaining patients. You save time and money by stopping a futile trial instantly.
The "Hopeless Zone": Even before you hit $0, the system can see if you are in a "Hopeless Zone" (where your capital is so low that even winning every single remaining bet wouldn't get you to $20). It signals you to stop early.

4. Why This is Better for Real Life

The paper argues that this method is more robust than current standards for three main reasons:

It Handles Chaos: Real trials are messy. Patients drop out, or recruitment is faster than planned. If you change your schedule, the old math breaks. The e-value system is "anytime-valid," meaning it works perfectly whether you check the results after 5 patients or 50. It doesn't care if you deviate from the plan.
It Combines Evidence: You can start a trial with a "head start" of evidence from a previous study (like starting with $1.50 instead of $1). The math handles this safely without inflating the risk of false alarms.
It's Flexible: You can design the trial to either maximize the chance of finding a working drug (Power) or minimize the number of patients needed (Efficiency), and the system adjusts the betting strategy to hit that specific goal.

Summary

The authors have built a new "smart betting" system for clinical trials. Instead of following a rigid script, the trial adapts its strategy based on how much evidence it has collected and how many patients are left. It stops immediately when a trial is hopeless (saving resources) and bets aggressively when it needs to catch up (finding effective drugs faster). It is a more flexible, robust, and efficient way to decide if a new treatment works.

Technical Summary: Adaptive Clinical Trials Based on Design-Optimal E-Values

Problem Statement
Standard adaptive clinical trials, while flexible, suffer from rigidity regarding type I error rate control under protocol deviations (e.g., changes in monitoring schedules or sample sizes) and often require stopping once a pre-specified maximum sample size is reached. Furthermore, existing e-value literature primarily focuses on asymptotic optimality (Growth Rate Optimal in the Worst-case, or GROW), which assumes an infinite horizon. This approach is often suboptimal for clinical trials where sample sizes are finite and fixed. There is a need for e-value-based designs that are optimized for finite horizons, offer robustness to protocol deviations, and facilitate automatic curtailment (stopping for futility) without sacrificing statistical power.

Methodology
The authors propose a framework for single-arm multi-stage clinical trials with binary data (e.g., tumor response) using design-optimal e-values. Unlike standard GROW e-values, which maximize the asymptotic growth rate of evidence, these designs optimize performance over a finite horizon $n$ (the maximum sample size).

Betting Analogy and E-Processes: The methodology utilizes the betting analogy where an e-value represents the capital of a bettor wagering against the null hypothesis $H_0: \theta \le \theta_0$ in favor of $H_1: \theta \ge \theta_1$ . The capital process $M_t$ is a test supermartingale under $H_0$ .
Markov Decision Process (MDP): The optimal betting strategy is derived by solving a finite-horizon MDP. The state space is defined by the current time $t$ and the current e-value (capital) $M_t$ . The action space consists of the betting fraction $B_t \in [0, 1]$ .
Optimization Objectives: The authors define two primary reward functions to solve via dynamic programming (Bellman equations):
- Power-Maximizing (P-max): Maximizes the probability of rejecting $H_0$ (reaching $M_n \ge 1/\alpha$ ) at the final analysis.
- Expected Sample Size-Minimizing (ESS-min): Minimizes the expected sample size under $H_1$ , subject to a constraint that the overall power remains above a specified threshold (e.g., 80%).
Automatic Curtailment: The framework naturally incorporates non-stochastic curtailment. If the e-value drops to zero (or enters a "hopeless zone" where the threshold $1/\alpha$ can no longer be reached), the trial is automatically stopped for futility.
Blocked Designs: The approach accommodates blocked designs (analyzing data in groups rather than continuously) by treating outcomes as fully sequential within blocks. This ensures that the minimum power constraint is met even if interim analysis points deviate from the plan, provided the maximum sample size is reached.

Key Contributions

Finite-Horizon Optimality: The paper introduces e-values optimized for specific trial designs with finite sample sizes, moving beyond the infinite-horizon GROW criterion.
Dynamic Programming Construction: It provides a computational method to construct optimal betting strategies that either maximize power or minimize expected sample size under power constraints.
Robustness to Protocol Deviations: By relying on the anytime-valid property of e-values, the designs maintain type I error control even if the monitoring schedule changes or the trial is continued beyond the planned size, provided the maximum sample size is respected.
Automatic Futility Stopping: The method provides a rigorous, automatic mechanism for futility stopping (when $M_t = 0$ ) that is integrated into the design optimization, rather than being an ad-hoc addition.
Comparison with Standard Designs: The authors demonstrate that design-optimal e-values can outperform standard GROW e-values in finite samples and offer competitive operating characteristics compared to standard stochastic curtailment (SC) designs and Simon's two-stage designs.

Results
Numerical comparisons were conducted for single-arm trials with binary data ( $n=50$ , $\theta_0=0.1$ , $\theta_1=0.242$ , $\alpha=0.05$ ) across fully sequential, blocked, and two-stage designs:

Type I Error Control: All e-value-based designs maintained type I error rates below the nominal level.
Power: The P-max e-value-based design achieved higher final power than standard stochastic curtailment (SC) and the GROW e-value, though it exhibited lower early power. The ESS-min design achieved power comparable to SC.
Expected Sample Size (ESS): The ESS-min e-value-based design resulted in a lower median expected sample size under $H_1$ compared to SC.
Futility Stopping: The e-value-based designs demonstrated a higher probability of early termination for futility under the null hypothesis compared to SC, particularly in two-stage designs. The GROW e-value (Kelly betting) did not exhibit non-stochastic curtailment.
Blocked Designs: The "fully sequential in blocks" approach preserved the power properties of the fully sequential design, ensuring the minimum power constraint was met regardless of block size variations.

Significance and Claims
The paper claims that design-optimal e-values provide a viable and competitive alternative to the current state-of-the-art in single-arm binary trials. By shifting focus from asymptotic optimality to finite-horizon optimality, these designs overcome the rigidity of standard adaptive trials regarding protocol deviations and fixed stopping rules. The authors assert that e-value-based designs offer:

Robustness: Valid inference under unspecified data monitoring and optional continuation.
Efficiency: Competitive or superior operating characteristics (power and ESS) compared to standard adaptive designs.
Flexibility: The ability to incorporate historical data and adapt to changing trial conditions without inflating type I error rates.
Interpretability: A clear betting interpretation that facilitates the introduction of anytime-valid inference and provides intuitive indicators for trial continuation or futility.

The authors conclude that while these designs lose the direct connection to likelihood ratios found in GROW e-values, they offer superior practical performance in finite-sample settings and warrant extension to multi-arm and response-adaptive trial settings.