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Sequential Randomization Tests Using e-values: Applications for trial monitoring

This paper introduces a family of nonparametric, anytime-valid sequential randomization tests (e-RT) for diverse trial endpoints that leverage a betting framework to construct test martingales, thereby guaranteeing strict Type I error control solely through the randomization mechanism without relying on parametric assumptions or asymptotic approximations.

Original authors: Fernando G Zampieri

Published 2026-03-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Fernando G Zampieri

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 casino. You have two tables: Table A (the new treatment) and Table B (the standard treatment). You want to know if Table A is actually "cheating" the odds in a good way, or if the results are just random luck.

Traditionally, to catch a cheater, you have to wait until the end of the night, count all the chips, and run a complex math formula. If you peeked at the chips halfway through and stopped the game because it looked like Table A was winning, the casino would say, "You can't do that! You changed the rules mid-game, so your math is invalid."

This paper introduces a new way to monitor clinical trials called "e-RT" (e-value Randomized Trial).

Instead of waiting until the end, it uses a "Betting Framework" that lets you check the results anytime you want, without breaking the rules.

Here is how it works, using simple analogies:

1. The Core Idea: The "Smart Gambler"

Imagine a detective (the statistician) who is betting on which table a player came from, based on how they played.

The Setup: Every time a patient finishes the trial, we see their result (e.g., "Did they get better?"). We don't know yet which table they were at.
The Bet: The detective looks at the result and says, "Hmm, that result looks like it came from Table A. I'll bet $1 that it did."
The Reveal: Then, we reveal the truth.
- If the detective guessed right: Their "wealth" (evidence) grows.
- If the detective guessed wrong: Their wealth shrinks.

The Magic Rule: If the new treatment is useless (the Null Hypothesis), the detective is just guessing like a coin flip. Over time, their wealth will bounce up and down but will never grow significantly. It's like a gambler playing a fair game; they might get lucky for a while, but they can't win forever.

However, if the new treatment actually works, the results will start to look different. The detective will start guessing correctly more often. Their wealth will start to grow exponentially.

2. Why This is a Game-Changer

In traditional trials, you have to decide in advance: "We will check the results at 50 patients, then 100, then 150." If you check too early, you might get a "false alarm" (Type I error).

With e-RT, you can check every single day.

If the detective's wealth suddenly skyrockets to a massive number (like 20x their starting money), you know the treatment is working. You can stop the trial immediately and save money/time.
If the wealth stays low, you keep going.
Crucially: Because the math is built on "betting," it doesn't matter when you stop. The guarantee that you aren't being fooled by luck holds true no matter when you decide to quit.

3. The Different "Games" (Variants)

The paper explains that different types of medical data need different betting strategies, just like different casino games need different rules.

Binary (e-RTb): The simplest game. Did the patient have an event (Yes/No)?
- Analogy: A coin flip. If the treatment works, "Heads" happens more often on Table A.
- Strategy: The detective bets a little bit more on the side that seems to be winning, but stays cautious.
Event-Only (e-RTe): Sometimes, tracking who didn't have an event is hard (e.g., tracking who is still alive is easy, but tracking who is "healthy" requires calling them every week).
- Analogy: You only get to see the "deaths" or "failures." You ignore the survivors.
- Strategy: This is like betting only on the rare, dramatic moments. It's less efficient (you need more data) but much easier to run because you don't have to chase down every single patient.
Continuous (e-RTc): The outcome is a number (e.g., "How many days on a ventilator?").
- Analogy: Instead of a coin flip, you are betting on a dartboard. If a patient stays on the ventilator for 2 days (very low), and the treatment usually helps, the detective bets heavily that this patient was on the treatment table.
- Strategy: This is tricky. If the detective bets too aggressively on a weird number, they could lose everything. So, the strategy is very careful and "adaptive."
Time-to-Event (e-RTs): When the timing of an event matters (e.g., survival time).
- Analogy: A race. You only get to bet when someone crosses the finish line (dies).
- Strategy: Since you only get to bet rarely (only when someone dies), you can afford to be a bit more aggressive with your bets because you don't have to make thousands of bets in a row.
Multi-State (e-RTms): Patients move around (ICU -> Ward -> Home).
- Analogy: A board game where players move forward (good) or backward (bad).
- Strategy: Every time a patient moves, you get a chance to bet. If they move forward, you bet that they are on the "good" table.

4. The "Wage Asymmetry" (Why some bets are safer than others)

The paper makes a fascinating point about how often you bet.

The "High-Frequency" Trap: If you bet on every single patient (like in the Binary or Continuous games), you are playing a game where you make 1,000 bets in a row. If you bet too aggressively and get unlucky, your wealth gets crushed to zero very fast. You must bet conservatively.
The "Low-Frequency" Freedom: If you only bet when a rare event happens (like in the Survival game), you might only make 50 bets. Even if you bet aggressively and lose a few times, you have time to recover. You can afford to be more aggressive.

5. The "Universal Engine" (e-RTu)

Finally, the author proposes a "Universal Translator." Imagine a robot that doesn't care if you are betting on coin flips, dart throws, or board game moves. As long as you feed it a stream of "Good" or "Bad" signals, it runs the betting algorithm automatically. This makes it easy to apply this method to any new type of medical data in the future.

Summary

The e-RT method is like a "Real-Time Truth Detector" for clinical trials.

Old Way: Wait until the end, hope you didn't peek too early, and run a complex math test.
New Way (e-RT): Bet on the results as they come in. If the "wealth" of evidence grows big enough, you know the treatment works, and you can stop the trial immediately.
The Benefit: It's safer, more flexible, and doesn't require strict assumptions about how the data behaves. It works for almost any type of medical outcome, from simple "yes/no" events to complex patient journeys.

It turns the clinical trial from a static exam into a dynamic, living conversation between the data and the researchers.

1. Problem Statement

Traditional sequential monitoring of randomized controlled trials (RCTs) relies on parametric assumptions (e.g., normality, proportional hazards) and pre-specified stopping boundaries (e.g., $\alpha$ -spending functions). These methods face significant limitations:

Assumption Dependence: If parametric assumptions fail, Type I error control is compromised.
Rigidity: They require fixed interim analysis schedules. Evidence accumulating between looks is ignored, potentially delaying the detection of safety signals or efficacy.
Data Gaps: In many clinical settings (e.g., ICU mortality, oncology progression), tracking "non-events" (survivors or stable patients) is operationally difficult or impossible, rendering standard methods inapplicable.

The paper proposes a solution based on e-values and e-processes, which offer "anytime-valid" inference. This means Type I error control is guaranteed at any stopping time, regardless of the stopping rule, without requiring parametric assumptions.

2. Methodology: The e-RT Framework

The author introduces e-RT (e-value Randomized Trial), a family of nonparametric sequential tests. The core philosophy is a betting framework:

The Game: A gambler (the statistician) sequentially bets on the treatment assignment ( $T_i$ ) of a patient based on their observed outcome ( $Y_i$ ).
The Null Hypothesis ( $H_0$ ): Treatment assignment has no effect on the outcome ( $Y_i \perp T_i$ ). Under $H_0$ , outcomes provide no information about the arm assignment; the assignment is effectively a coin flip with probability $p$ .
The Wealth Process ( $W_n$ ): The statistician starts with wealth $W_0 = 1$ $W_{0} = 1$ . At each step $i$ $i$ , they choose a betting fraction $\lambda_i$ $λ_{i}$ (predicting the probability the patient is in the treatment arm).
- If the prediction is correct, wealth grows by a factor of $\lambda_i/p$ (if $T_i=1$ ) or $(1-\lambda_i)/(1-p)$ (if $T_i=0$ ).
- If incorrect, wealth shrinks.
Validity: Under $H_0$ , the expected wealth multiplier is exactly 1. Thus, $(W_n)$ is a non-negative martingale. By Ville's Inequality, the probability that wealth ever exceeds $1/\alpha$ is at most $\alpha$ . This allows rejection of $H_0$ the moment $W_n \geq 1/\alpha$ , controlling Type I error at level $\alpha$ regardless of when monitoring stops.

3. Key Variants (The e-RT Family)

The paper details six specific variants tailored to different data types, all sharing the same validity proof but differing in how they translate data into wagers:

A. Binary Outcomes (e-RTb)

Data: Event/No-event for every patient.
Strategy: Uses an adaptive half-Kelly strategy. The bet $\lambda_i$ is based on the running difference in event rates ( $\hat{\delta}$ ) between arms.
Mechanism: If the intervention arm has fewer events, the strategy bets that a new event likely came from the control arm.
Constraint: Uses a "burn-in" and "ramp" period to prevent over-betting when sample sizes are small.

B. Event-Only Monitoring (e-RTe)

Data: Only the stream of events (e.g., deaths) is observed; non-events are ignored.
Strategy: Treats the sequence of events as a Bernoulli coin flip: "Did this event come from the treatment arm?"
Advantage: Highly efficient when non-event tracking is impractical. It concentrates the signal (the "event coin" tilt is larger than the raw risk difference).
Trade-off: Requires approximately 2.5 $\times$ more events than full-sample methods to achieve similar power, but gains operational simplicity.

C. Continuous Outcomes (e-RTc)

Data: Continuous measurements (e.g., biomarkers, ventilator-free days).
Strategy: Uses a doubly adaptive approach.
1. Standardizes the new observation against the running median and MAD (Median Absolute Deviation) to determine "extremeness."
2. Combines this with the running Cohen's $d$ (effect size estimate) to determine direction.
Constraint: Requires very conservative betting (half-Kelly) because updates occur at every patient; over-betting leads to rapid wealth destruction.

D. Time-to-Event (e-RTs / e-Survival)

Data: Survival times with censoring.
Strategy: Updates wealth only at event times using a sequential Log-Rank score.
Mechanism: Uses a fixed wager ( $\lambda = 0.25$ ) with an adaptive direction (based on the cumulative Log-Rank score).
Rationale: Because updates are sparse (only at deaths/failures), the wealth process is robust to moderate over-betting, allowing for a fixed, aggressive wager size.

E. Multi-State Trajectories (e-RTms)

Data: Complex patient trajectories (e.g., ICU $\to$ Ward $\to$ Home/Dead).
Strategy: Classifies each transition as "Good" (recovery) or "Bad" (deterioration) and bets on the binary outcome of the transition quality.
Significance: Captures clinical improvement even when mortality is unchanged, avoiding complex competing risks modeling.

F. Universal Abstraction (e-RTu)

Concept: A domain-agnostic engine that accepts any stream of binary-classified signals (Good/Bad, Treatment/Control). It unifies the logic of the other variants, separating signal generation from the betting engine.

4. Key Contributions

Anytime-Valid Monitoring: Provides a rigorous, assumption-free method for continuous trial monitoring that guarantees Type I error control at any stopping time.
Operational Flexibility: Introduces e-RTe, enabling valid monitoring in settings where only events are tracked (e.g., EMR-based mortality studies), a scenario where traditional methods fail.
Principled Betting Asymmetry: The paper rigorously analyzes why different variants require different betting strategies (Section 9).
- Dense updates (Binary/Continuous): Require conservative, adaptive strategies (Half-Kelly) because over-betting compounds rapidly across thousands of steps, destroying wealth.
- Sparse updates (Survival/Event-Only): Can tolerate fixed or full-Kelly strategies because the limited number of updates prevents catastrophic compounding of errors.
Unified Framework: Demonstrates that diverse clinical endpoints (binary, continuous, survival, multi-state) can be reduced to a single betting logic based on randomization exchangeability.

5. Simulation Results

Type I Error: All variants maintained strict control, with observed error rates consistently between 2% and 4% (below the nominal 5%), confirming the martingale property.
Power & Early Stopping:
- In trials designed for 80% power using traditional methods, e-RT variants achieved a ~50% probability of early stopping (typically around 45–56% of the planned sample size) if a true effect existed.
- e-RTe (Event-Only): Outperformed e-RTb when baseline event rates were low (15–25%) due to signal concentration, despite seeing fewer data points.
- e-RTms: Showed high power (89%) in detecting trajectory improvements even when mortality differences were negligible.
Robustness: The methods performed well across various effect sizes and censoring scenarios without requiring distributional assumptions.

6. Significance and Implications

Complement to Traditional Methods: e-RT is not necessarily a replacement for fixed-sample analysis but a powerful complement. It allows for "free" early stopping options without alpha-spending penalties. If the threshold is not crossed, the trial proceeds to the planned analysis.
Real-World Applicability: Particularly valuable for critical care and oncology trials where data collection is asynchronous, non-events are hard to verify, or patient trajectories are complex.
Philosophical Shift: Moves the focus from "estimating a parameter" to "testing exchangeability." It asks: Can we predict treatment assignment based on outcomes? If yes, the treatment works.
Limitations & Future Work: The author notes these are experimental methods. They currently lack point estimates and confidence intervals. The adaptive learning requires a burn-in period, and the methods may struggle with non-stationary effects (e.g., if a treatment effect reverses mid-trial). Future work includes adaptive weighting schemes to handle time-varying effects.

Conclusion:
The paper establishes e-RT as a robust, assumption-free toolkit for sequential trial monitoring. By leveraging the betting framework and e-values, it offers a flexible alternative to parametric group-sequential designs, capable of handling diverse data types and operational constraints while guaranteeing statistical validity under the randomization mechanism alone.