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Bayesian sample size determination using robust commensurate priors with interpretable discrepancy weights

This paper addresses the nonmonotonicity and interpretability issues in Bayesian sample size determination using commensurate priors by proposing a linearization technique that yields an analytical formula and interpretable weights representing the probability of historical data relevance, thereby facilitating expert elicitation for trial design.

Original authors: Lou E. Whitehead, James M. S. Wason, Oliver Sailer, Haiyan Zheng

Published 2026-06-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lou E. Whitehead, James M. S. Wason, Oliver Sailer, Haiyan Zheng

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

The Big Picture: Cooking with Old Recipes

Imagine you are a chef planning a new, high-stakes cooking competition. You need to decide how many judges (participants) to invite. If you invite too few, you might not get a clear winner; if you invite too many, you waste money and time.

Usually, chefs rely on new data to make this decision. But what if you have old cookbooks (historical studies) from similar competitions? It would be smart to use those old recipes to help you decide how many judges you need. This is the core idea of the paper: using past data to design a new medical trial more efficiently.

However, there is a catch. You can't just copy-paste the old recipes. You have to ask: "How similar is this old recipe to what I'm cooking today?"

The Problem: The "Volume Knob" is Broken

In the past, statisticians used a "volume knob" (called a discrepancy weight) to control how much of the old data to listen to.

Turn the knob to 0: Listen to the old data 100%.
Turn the knob to 1: Ignore the old data completely.

The authors discovered that in previous methods, this volume knob was broken.

It wasn't linear: If you turned the knob just a tiny bit (from 0 to 0.1), the volume dropped drastically. But if you turned it from 0.5 to 0.6, the volume barely changed. It was like a light switch that was too sensitive at the start and useless at the end.
It wasn't predictable: Sometimes, turning the knob up (to ignore data) actually made the system listen more to the data. This is called non-monotonicity. It's like trying to turn down the heat on a stove, but the flame suddenly gets bigger.

Because of this broken knob, it was impossible for experts to say, "I think this old study is 50% relevant," because the math wouldn't actually give them 50% relevance. It made communication between statisticians and doctors very difficult.

The Solution: A New, Smooth Knob

The authors propose a two-step fix to make the knob work properly.

Step 1: A Better Way to Mix the Ingredients

They changed the mathematical recipe for how the old data is mixed together. Instead of the messy, unpredictable mixing method used before, they used a cleaner method (based on the work of Winkler, 1981).

The Result: Now, the relationship between the knob and the volume is smooth and predictable. Turning the knob up always lowers the volume. No more surprises.

Step 2: The "Magic Translator"

Even with the new mixing method, the relationship between the knob and the final result (the number of judges needed) is still a bit curved.

The Fix: They created a "Magic Translator" (a mathematical transformation).
How it works: You tell the translator, "I want to use 50% of the old data." The translator takes that 50%, does a quick calculation, and tells the computer, "Okay, set the internal knob to this specific tiny number."
The Benefit: Now, when an expert says, "I'm 50% skeptical of this old data," the system actually uses exactly 50% of the information. The relationship becomes linear (a straight line).

The Real-World Test: Alzheimer's Exercise Study

To prove this works, the authors applied their method to a hypothetical study about Alzheimer's disease.

The Goal: Test if exercise helps improve memory.
The Data: They looked at 7 past studies. Some were very similar to the new plan; others were quite different.
The Expert: A doctor looked at the 7 studies and said, "Study #5 is very relevant (low weight), but Study #6 is not relevant at all (high weight)."
The Outcome:
- Without their new method, the math would have misunderstood the doctor's weights, leading to a trial with 332 participants (too big).
- With their new "Magic Translator," the math correctly interpreted the weights and calculated a trial size of 176 participants.
- This saved resources while still ensuring the trial was big enough to be scientifically valid.

Why This Matters

The paper doesn't claim to cure diseases or invent new drugs. Instead, it fixes the toolkit used to design the experiments that test those drugs.

By making the "volume knob" for historical data intuitive and predictable, they hope that:

Doctors and Statisticians can talk better: Doctors can give honest opinions on how relevant old data is without worrying the math will break.
Trials are more efficient: We can stop wasting money on trials that are too big, or risking failure on trials that are too small.

In short, they built a better ruler for measuring how much we should trust the past when planning the future.

Technical Summary: Bayesian Sample Size Determination Using Robust Commensurate Priors with Interpretable Discrepancy Weights

Problem Formulation
In the design of randomized controlled trials (RCTs), incorporating historical data can improve efficiency, particularly in settings with limited patient populations (e.g., rare diseases or pediatrics). The Bayesian paradigm offers a framework for this via "commensurate priors," where historical information is downweighted based on a discrepancy parameter representing the similarity between historical and current trials.

The paper identifies a critical flaw in the existing methodology proposed by Zheng et al. (2023a,b) regarding the aggregation of multiple historical sources. In that framework, discrepancy weights ( $w_q \in [0,1]$ ) are intended to represent the probability of irrelevance of a specific historical source. However, the authors demonstrate that the relationship between these weights and the resulting prior precision (and consequently the required sample size) is non-monotonic and highly nonlinear.

Non-monotonicity: Increasing a weight $w_q$ (intended to discount a source) does not always lead to a decrease in prior precision. Local minima and maxima exist, particularly when aggregating multiple sources ( $Q > 1$ ).
Nonlinearity: Small changes in weights near zero result in disproportionately large changes in information incorporation, while changes near one have negligible effects.
These properties render the weights uninterpretable as probabilities, making it difficult for subject-matter experts to elicit sensible values during the trial design stage.

Methodology
The authors propose a two-part solution to resolve these interpretability issues:

Alternative Prior Aggregation:
Instead of the convolution-based aggregation used in previous work (which led to complex weighting functions), the authors propose aggregating predictive priors using the method suggested by Winkler (1981). This involves forming a collective prior where the precision is the sum of the precisions of the individual predictive distributions.
- The collective prior mean is a weighted sum of historical means.
- The collective prior variance ( $\sigma^2_{CP*}$ ) is the reciprocal of the sum of the precisions ( $\xi^{-2}_q$ ) of the individual sources.
- Crucially, under this new aggregation, the relationship between the discrepancy weight $w_q$ and the collective precision becomes monotonically decreasing. As $w_q$ increases (indicating greater irrelevance), the precision decreases uniformly.
Linearization via Functional Transformation:
Even with monotonicity, the relationship between $w_q$ and the sample size remains nonlinear. To achieve uniform shrinkage, the authors introduce a linearization technique:
- They isolate the nonlinear component of the sample size formula corresponding to each source.
- A linear interpolation is performed between the precision at $w_q=0$ (full borrowing) and $w_q=1$ (no borrowing).
- An inverse transformation function $f(w_q) = w'_q$ is derived.
- This transformation maps the expert-elicited probability of irrelevance ( $w_q$ ) to a transformed weight ( $w'_q$ ) such that the resulting sample size changes linearly with respect to the transformed weight. This ensures that specifying a weight of 0.5 results in incorporating exactly 50% of the information from that source relative to the range of possible information.

Key Contributions and Results

Analytical Sample Size Formula: The authors derive a closed-form Bayesian sample size formula for a two-arm RCT with normally distributed outcomes, incorporating the new aggregation method and linearized weights. The formula is:
$n \ge \frac{\sigma^2_0}{R(1-R)} \left[ \left(\frac{z_\eta + z_\zeta}{\delta}\right)^2 - \sum_{q=1}^Q \xi^{-2}_q(w'_q) \right]$
where $\xi^{-2}_q$ depends on the transformed weights.
Interpretability: The proposed method allows experts to elicit $w_q$ as a direct probability of irrelevance (e.g., "I am 60% skeptical that this historical trial is relevant"). The transformation handles the nonlinearity, ensuring the statistical behavior matches the expert's intent.
Simulation Study: A simulation study across four configurations of historical data (varying in effect size and variance) confirms that the proposed method preserves pre-specified Bayesian decision properties. Specifically, when using the calculated sample sizes, 100% of simulated trials reached a decisive conclusion regarding efficacy or futility, meeting the pre-specified posterior probability thresholds ( $\eta$ and $\zeta$ ).
Application Example: The method is applied to a hypothetical Alzheimer's disease trial using data from seven historical RCTs. The authors demonstrate that using "raw" weights leads to over-discounting (resulting in a sample size of 332), whereas using the linearized transformation yields a more appropriate sample size (176) that reflects the intended degree of information borrowing.

Significance and Claims
The paper claims that its primary contribution is resolving the non-monotonicity and nonlinearity issues inherent in previous commensurate prior aggregation methods. By ensuring that discrepancy weights behave monotonically and can be linearized, the authors argue that:

Communication is facilitated: Statisticians and subject-matter experts can communicate more effectively regarding the relevance of historical data, as the weights are now interpretable as probabilities.
Design is robust: The method provides a coherent framework for determining sample sizes that incorporate historical data without the risk of unintended information discounting or inflation caused by non-monotonic behavior.
Generalizability: While focused on two-arm trials with continuous outcomes, the authors note the methodology can be extended to binary and time-to-event endpoints (detailed in supplementary materials) and potentially to other trial designs like basket trials.

The authors maintain a modest stance, acknowledging that while the method improves interpretability and design efficiency, it does not eliminate the need for careful selection of historical trials or extensive simulation to evaluate operating characteristics (such as Type I error rates) in specific practical applications. They emphasize that sample size determination is only one factor in deciding whether to borrow historical data.