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A Bayesian hierarchical push-forward method for cluster Randomized Controlled Trial power calculations in complex humanitarian settings

This paper introduces a Bayesian hierarchical push-forward method that integrates prior information on cluster and individual dropout to improve power calculations and trial design for cluster-randomized controlled trials in complex humanitarian settings, demonstrated through a forthcoming study in Bangladesh and a publicly available R Shiny application.

Original authors: Margaret Elliott, Sonia Jain, Adam Richards, Brendan Fries, Ravi Goyal

Published 2026-10-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Margaret Elliott, Sonia Jain, Adam Richards, Brendan Fries, Ravi Goyal

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

In the chaotic aftermath of a humanitarian crisis, where displaced families move unpredictably and resources are scarce, designing a medical study is an exercise in navigating the unknown. Researchers often turn to cluster randomized trials, a method where entire groups of people, such as villages or refugee camps, are assigned to receive a new health intervention or a standard one. This approach is vital for testing treatments in places where individual tracking is impossible, but it carries a unique risk: entire groups can vanish from the study due to conflict, displacement, or safety concerns. When these groups disappear, the data becomes incomplete, and the study may lose the ability to detect whether a treatment actually works. To solve this, scientists must move beyond simple guesses about how many people will stay in the study and instead embrace a framework that treats uncertainty not as a flaw, but as a central part of the calculation.

A team of statisticians and public health experts has developed a new way to plan these difficult studies, one that acknowledges that the future is rarely a single, fixed point. Instead of calculating a single number for how likely a study is to succeed, they created a method that generates a full range of possibilities, showing how the odds of success shift as conditions change. Their work was driven by an upcoming trial in Cox's Bazar, Bangladesh, which aims to help refugees from Myanmar manage heart disease and hepatitis C. In this setting, the researchers knew that people might leave the study area or that entire camps might become inaccessible, making traditional planning methods unreliable. By using a Bayesian approach, which allows them to weave in what is already known from past studies while explicitly modeling the chaos of the present, they built a tool that predicts the strength of a trial under many different scenarios of loss and disruption.

The researchers began by looking at a similar trial recently completed in Myanmar, where they analyzed how many people stayed in the study and how the loss of participants was distributed across different villages. They found that sometimes people left individually, but other times, entire villages were lost to follow-up due to conflict. They used this real-world data to build a mathematical model that simulates thousands of potential versions of the upcoming Bangladesh trial. In each simulation, they varied the number of clusters, the size of the groups, and the rate at which people might drop out, including scenarios where whole clusters disappear. This process, which they call a "push-forward" method, takes the uncertainty about how many people will remain and pushes it through the study design to see what the final results might look like.

The results of their simulations revealed a critical flaw in the way studies are often planned. Traditional methods usually provide a single number, such as an 80 percent chance of success, which can give a false sense of security. The new method showed that even when the average chance of success looks good, there is often a significant risk that the actual power of the study could fall much lower. For instance, in some scenarios, while the average power looked sufficient, there was still a thirty-five percent chance that the study would be too weak to detect a real effect. This happens because the loss of entire groups creates a different kind of uncertainty than losing individuals one by one. The researchers found that simply increasing the number of people in each group does not always fix the problem; sometimes, adding more clusters is far more effective at ensuring the study can withstand the loss of a few villages.

When they applied this new framework to the Bangladesh trial, using data from the Myanmar study to inform their predictions, the results were reassuring but nuanced. They tested six different scenarios for how people might leave the study, ranging from random individual departures to the loss of multiple entire clusters. In every scenario, the median chance of the study succeeding remained high, often above 87 percent. However, the method also highlighted that in the worst-case scenario, where several clusters were lost, there was still a 35 percent chance the study would fall short of the standard 80 percent power threshold. This level of detail allows researchers to make more informed decisions, understanding not just if a study will likely work, but how vulnerable it is to the specific types of disruption common in humanitarian crises.

To make this complex tool accessible to other scientists and public health workers, the team built a free, interactive software application that anyone can use to design their own trials. This tool allows investigators to input their own assumptions about how many people might leave and see a visual representation of the risks and rewards of different study sizes. By shifting the conversation from a single point estimate to a range of possibilities, this work helps researchers design studies that are robust enough to survive the unpredictability of crisis zones. It offers a way to plan with eyes wide open, acknowledging that in humanitarian settings, the only certainty is that things will change, and that a good study design must be flexible enough to account for that reality.

Technical Summary: A Bayesian Hierarchical Push-Forward Method for cRCT Power Calculations in Complex Humanitarian Settings

Problem Statement
Designing Cluster Randomized Controlled Trials (cRCTs) in humanitarian crises (e.g., refugee camps, conflict zones) presents unique challenges that standard frequentist power calculations often fail to address adequately. These settings are characterized by extreme uncertainty regarding key design parameters, including intervention efficacy, intracluster correlation coefficients (ICC), and, most critically, attrition (loss to follow-up). In such environments, displacement, resource constraints, and safety concerns can lead to heterogeneous follow-up rates and the complete loss of entire clusters. Traditional methods typically rely on point estimates for these parameters, yielding a single power value that obscures the substantial risk that the actual trial power may fall below acceptable thresholds (e.g., 80%) due to unanticipated data loss. There is a recognized gap in the literature for Bayesian design methods that explicitly incorporate prior information and uncertainty about missingness at the design stage of cRCTs.

Methodology
The authors propose a Bayesian design framework that generates a Push-Forward Power Distribution (PFPD). This approach moves beyond point estimates to characterize the full distribution of trial power, accounting for uncertainty in design parameters. The methodology consists of three core components:

Hierarchical Modeling of Historical Data: The framework utilizes data from previously completed trials (or prior information) to estimate posterior distributions for key parameters: treatment effect ( $\delta$ ), outcome ICC ( $\rho$ ), follow-up rate ( $\pi$ ), and the ICC for attrition ( $\tau$ ). A hierarchical model allows for borrowing strength across multiple studies if available, modeling trial-specific parameters as drawn from a population distribution defined by hyperparameters.
Attrition and Outcome Models:
- Attrition: The model treats cluster-specific follow-up propensities as latent variables following a Beta distribution. This allows for the derivation of the mean follow-up rate ( $\pi$ ) and the attrition ICC ( $\tau$ ), which quantifies the correlation of missingness within clusters (ranging from random missingness to entire cluster loss).
- Outcome: A generalized linear mixed-effects model (specifically a Bernoulli outcome with a logit link) is used to model the binary outcome, incorporating a normally distributed random effect for clusters to estimate the outcome ICC.
Push-Forward Mapping: The core innovation is the "push-forward" mechanism. Instead of fixing parameters at point estimates, the method draws samples from the posterior distributions of the design parameters ( $\Theta$ ). These samples are propagated through a deterministic mapping function ( $f$ ), which is the standard sample size/power formula for cRCTs with attrition (incorporating a variance inflation factor $G$ ). This process generates an empirical distribution of power values ( $P(\xi | D, \Gamma)$ ), where $\xi$ represents power, $D$ is historical data, and $\Gamma$ represents fixed design constraints (e.g., number of clusters $n$ , individuals per cluster $m$ ).

Key Contributions

Novel Framework: The paper introduces a formal Bayesian strategy specifically tailored for cRCT design in complex settings, explicitly integrating uncertainty in attrition (both individual and cluster-level) into power calculations.
Push-Forward Distribution: It extends existing Bayesian ICC estimation methods (e.g., Turner et al.) to include attrition parameters, providing a distributional representation of power rather than a single point estimate.
Practical Implementation: The authors developed and disseminated a publicly available R Shiny application to make these complex calculations accessible to investigators, allowing for interactive exploration of how sample size and parameter uncertainty affect power.
Real-World Application: The method is applied to the design of a forthcoming cRCT in Cox's Bazar, Bangladesh, assessing a community health worker intervention for cardiovascular disease and hepatitis C among Rohingya refugees.

Results

Simulation Study: The authors conducted simulations varying sample sizes, treatment effects, and attrition rates.
- The simulations demonstrated that while frequentist point estimates and PFPD medians might suggest a trial is adequately powered (e.g., >80%), the PFPD often reveals a non-negligible probability (tail risk) that the actual power falls below this threshold.
- For instance, in scenarios where the frequentist estimate was 83% and the PFPD median was 92%, there remained a 35% probability that the true power was below 80%.
- The study highlighted that increasing cluster size yields diminishing returns in power stability once a certain total sample size is reached, a nuance captured by the distributional shape of the PFPD.
Bangladesh Application: Applying the method to the Bangladesh trial design using data from a similar completed trial in Myanmar:
- The authors evaluated six scenarios ranging from random individual missingness to complete cluster dropout.
- Despite fluctuations in attrition parameters (particularly in scenarios with cluster dropout), the PFPDs indicated a high probability of achieving sufficient power (medians $\ge$ 0.87).
- However, the analysis also quantified the risk: even in the most optimistic scenarios, there was a probability of up to 0.35 that the trial would be underpowered, a risk that would be invisible in a standard point-estimate calculation.

Significance and Claims
The paper claims that this work provides a foundation for more robust and efficient cRCTs in humanitarian contexts by formalizing strategies that anticipate data loss. The significance lies in shifting the paradigm from binary "powered/not powered" decisions to a probabilistic understanding of trial operating characteristics. By quantifying the uncertainty in power, the method allows investigators and stakeholders to make more informed decisions regarding sample size and resource allocation, acknowledging the inherent instability of humanitarian settings. The authors argue that this approach strengthens the evidence base for health interventions in resource-limited and crisis-affected populations by ensuring that trial designs are resilient to the high levels of attrition typical in these environments. The work is modest in its claims, noting limitations such as the current focus on binary outcomes and the reliance on single-source prior data in the application, while suggesting future extensions to continuous outcomes and multiple data sources.

Technical Summary: A Bayesian Hierarchical Push-Forward Method for cRCT Power Calculations in Complex Humanitarian Settings

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