Title: Distributed mediation analysis with  communication efficiency
Venue: NeurIPS 2025
Accepted: yes
Keywords: Distributed inference

mediation analysis

indirect causal effects

communication efficiency
Abstract: We study the mediation analysis under the distributed framework, where data are stored and processed across different worker machines due to storage limitations or privacy concerns. Building upon the classic Sobel's test and MaxP test, we introduce the distributed Sobel's test and distributed MaxP test, respectively. These tests are both communication-efficient and easy to implement. Theoretical analysis and numerical experiments show that, compared to the global test obtained by pooling all data together, the proposed tests achieve nearly identical power, independent of the number of machines. Furthermore, based on these two distributed test statistics, many enhanced mediation tests derived from the Sobel's or MaxP tests can be easily adapted to the distributed system. We apply our method to an educational study, testing whether the effect of high school mathematics on college-level Probability and Mathematical Statistics courses is mediated by Calculus. Our method successfully detects the mediation effect, which would not be identifiable using data from only the first or second class, highlighting the advantage of our approach.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates mediation analysis within a distributed framework. The authors propose two novel methods: a distributed Sobel's test and a distributed MaxP test, both developed under a split-and-conquer framework. Theoretical analyses and numerical experiments demonstrate the performance characteristics of the proposed methods.\n\nSTRENGTHS AND WEAKNESSES:\n### **Strengths**\n\n1. **Novel Contribution to Distributed Mediation Analysis**  \n   The paper makes a valuable contribution by addressing mediation analysis within a distributed framework, where data are stored and processed across multiple worker machines due to storage or privacy constraints. The authors propose distributed versions of Sobel’s test and the MaxP test, which are communication-efficient and easy to implement. \n\n2. **Clear and Well-Structured Writing**  \n   The paper is well-organized, with a logical flow from problem formulation to theoretical analysis, simulations, and real-world application. The abstract and introduction clearly outline the research objectives, methodology, and contributions. The theoretical results are presented rigorously, and the numerical experiments are well-documented, making the paper easy to follow.  \n\n### **Weaknesses**  \n\n1. **Limited Novelty in Methodology**  \n   While the paper applies distributed inference to mediation analysis, the underlying split-and-conquer framework is not new. Prior work (e.g., Battey et al. [4], Zhao et al. [35]) has extensively studied distributed statistical testing, and **this paper appears to be a direct extension of existing methods to mediation models**.  \n\n2. **Lack of Discussion on Extensions to Other Mediation Tests**  \n   The authors claim that their method can be adapted to other enhanced mediation tests (e.g., those based on Sobel’s or MaxP tests), but they **do not provide concrete examples or theoretical justifications**. \n\n3. **Restriction to Univariate Exposure and Mediator**  \n   The paper focuses exclusively on **one-dimensional exposure (A) and mediator (M) variables**, limiting its applicability to modern high-dimensional settings (e.g., genomics, where multiple mediators may exist). The authors acknowledge this limitation in Section 6 but do not propose solutions or extensions.\n\nQUESTIONS:\n1. How does the proposed distributed mediation testing methods differ fundamentally from existing split-and-conquer approaches for general statistical inference? Please give some discussions.\n2. What specific enhanced mediation tests (beyond Sobel/MaxP) could directly benefit from this distributed framework? Please give some detailed examples or discussions.\n3. What modifications would be needed to handle multivariate mediators or exposures? The authors can propose a conceptual extension to multidimensional mediators, even if theoretical analysis are deferred to future work.\n4. Please add comparisons with naive distributed implementations (e.g., averaging p-values) in simulation studies to emphasize the advantage of the proposed methods.\n\nLIMITATIONS:\nPlease see section questions.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-19"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of mediation hypothesis testing in a federated setting where raw observations cannot be centrally pooled. The framework is built upon classic Sobel's and MaxP tests, and hence proposes their distributed versions. The extension is comprehensive: many enhanced tests built upon those two classic tests can be easily extended to the distributed setting. The method is simple: the server computes the distributed statistics following Equations 5 and 6, but this simple method yields a closed-form solution of both tests (Theorem 3.2).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. This paper has strong novelty. To the best of my knowledge, this is the first distributed framework for testing mediation.\n2. The framework has low communication cost: each server only needs to communicate two scalars in one round without further iterations.\n3. The theoretical guarantees are sound and the limits are clearly stated (value of K).\n\nWeaknesses:\n1. The asymptotic restriction of $K$ seems strong ($K = o(\\sqrt{N})$). This restricts that on average, each site cannot have a high number of samples. Such a restriction may make the results less applicable.\n2. The scope is a bit narrow: It only considers single continuous mediator. Although this paper is the first work of this topic so this is not a severe problem.\n3. Privacy is still a major concern in distributed settings, but this paper did not provide privacy guarantees, and information leakage was not discussed.\n\nQUESTIONS:\nPlease see points 1 and 3 in Weaknesses. In particular,\n1. Does the author have any idea or predictions on how the results would be if K is larger than $o(\\sqrt{N})$?\n3. Was the privacy discussion omitted due to technical difficulties or was it taken care of inside the framework already?\n\nLIMITATIONS:\nYes, in section 6.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe manuscript proposes a framework for mediation analysis when data is stored across multiple locations and cannot be pooled together due to various reasons. The proposed solution is distributed and communication-efficient. The core idea is to use distributed versions of the classic Sobel's and MaxP tests by having each local machine compute t-test statistics and sending only these two scalars to a central server for an average aggregation. The manuscript shows that the proposed tests achieve asymptotic power nearly identical to that of a global test performed on pooled data. The paper also presents promising results from simulations and a real-data educational study, where the distributed approach successfully identifies a mediation effect not initially found in local analyses.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n* Theoretical foundation for an unaddressed application area. The manuscript is one of the first to formalize a solution for conducting mediation analysis in a distributed data environment. Specifically, Theorems 3.1 and 3.2 are relevant as they provide a principled approach for aggregating siloed statistics into a global test under conditions over N (the amount of data) and K (the number of distributed machines). The asymptotic nature of the result in Theorem 3.2 allows the proposed method to be communication-efficient (as only two statistics are needed).\n* Simple and practical. The proposed method is theoretically simple to follow, and its implementation is straightforward. This accessibility has added practical value for adoption.\n\n**Weaknesses**\n\n* The primary concern with the proposed method is the assumption of homogeneity among distributed sites. In Section 3, Equation 4, there is an implicit assumption that beta and gamma causal effects are homogeneous across all K data clusters. The manuscript should either address or comment on this assumption, since this \"average\" assumption is critical and should be considered case-by-case.\n* Another weakness of the method is the underlying Sobel and MaxP tests themselves, as highlighted in Section 2. The paper's impact could be enhanced by considering additional baseline tests.\n\nQUESTIONS:\n1. Regarding the \"average\" effect (see weaknesses section), how would the proposed method perform in cases where parameters beta and gamma are considerably different among K clusters? Are there any thoughts on how to adapt the framework for these cases, or a method for testing heterogeneity that would enable one to decide whether to apply the framework or not?\n2. As noted in Section 4.1, when the condition K = o(sqrt(N)) is not satisfied, the power of the method is slightly lowered. Could the authors elaborate on this practical boundary? Is it possible to estimate the rate of power degradation?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
This paper develops federated / distributed variants of classical statistical algorithms for hypothesis testing / mediation analysis such as Sobel’s test and MaxP test in a single communication round. Each client computes some sufficient statistics locally, which are then aggregated at the server to compute the final test result. The power of the final test is shown to be asymptotically identical to that of the centralized variants.

STRENGTHS AND WEAKNESSES:
Strengths:

- Federated statistics is a highly understudied yet critical problem since most sensitive datasets such as in healthcare or science require statistical analysis - not just training ML models. Hence, I find the problem studied (though simple) well motivated and potentially impactful.
- The algorithm requires only a single communication round  
- The authors show that their distributed variant has asymptotically identical power to that of the centralized variants.  

Weakness:  

- The authors don't discuss how this relates to classical meta analysis methods that similarly aim to combine results from multiple sites.  
-  The authors also don't discuss existing work on federated causal inference ([[Guo et al. 24](https://arxiv.org/abs/2404.15746)], [[Khellaf et al. 24](https://arxiv.org/abs/2410.16870)], and related works) and how their method methodologically relates.
- No discussion is made about the distribution of the data across the centers - what happens if there is heterogeneity?

QUESTIONS:
1. How do your methods relate to meta analysis techniques?  
2. How do they technically and methodologically relate to existing federated causal inference works? What are the new techniques introduced here?  
3. How would data heterogeneity affect the results and the framework?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-10