Distributed mediation analysis with  communication efficiency
Authors: Shaomin Li
Venue: NeurIPS 2025 poster ()
Keywords: Distributed inference
mediation analysis
indirect causal effects
communication efficiency
Primary area: probabilistic_methods
TL;DR: Distributed mediation analysis

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.

Distributed mediation analysis with communication-efiiienyy
Shaomin Li
School of Mathematics and Statistics Beijing Jiaotong University Beijing, 100044 smli1@bjtu edu. cn
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 's test and MaxP test, we in- troduce the distributed Sobel's test and distributed MaxP test, respectively. These tests are both communication--ffiiiee.. and easy to implement. Theoretical analy- sis and numerical experiments show that, compared to the global test obtained by pooling all data together, the proposed tests achieve nearly identical power, inde- pendent of the number of machines. Furthermore, based 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 educational study. testing whether the effect of high school mathematics on college-level Probability and Mathematical Statistics courses is mediated by Cal- culus. Our method successfully detects the mediation effect, which would not be identifiable using data from only the first or second class, highlighting the advan- tage of our approach.
Introduction
Mediation analysis is regarded as a prevalent tool to dissect mediation relationship between expo- sures and outcomes, and has been widely applied in different fields, such as epidemiology [[9]], economics [M]] psychology [[8], education [[]]], and many others. Baron and Kenny[3]] provided the basis for the advance of mediation analysis. They proposed conventional regression- based approach, commonly referred to the causal steps method, to examine the logical relationships among exposure, mediator, and outcome variables linking a causal chain. While the causal steps method established necessary conditions for causal inference, it did not provide joint test of the indirect effect of exposure on the outcome through a mediator. To test the mediation effect, the classic tests are Sobel's test [3I]] and the MaxP test [24]. However, these tests suffer from low sta- tistical power, as it overlooks the impact of the composite null structure mediation analysis [25]. To address the aforementioned issues, Nuijten et al. [2]] proposed approach based on Bayesian models. Zhang[S7]] developed two data-adaptive tests that outperform both Sobel's 's test and MaxP test. These studies were limited small number of mediators. For sigh-dimensional mediators, one usually first estimates the proportions of sub-null hypotheses and generates weighted p-values [7) R. III 77 [X]]
In many applications, data is collected independently by numerous agents and organizations. Due to limitations in resources and concerns about privacy, only summary statistics be shared between sites, while raw data remains inaccessible. To address this, distributed learning algorithms have gained attention in multi-source studies, enabling collaborative analyses without compromising the
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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privacy of individual-level data. There has been substantial research on distributed estimation, which can be categorized into two main approaches: the one-shot average method [G. 13. 14, 32. [8] and the multi-round iteration method [022 19, 20, 3.5, 36)] However, research on distributed statistical testing is still in its early stages. For example, Battey et al. explored distributed versions of Wald and Rao's score tests for sparse high-dimensional schemes. Zhao et al. al.[39]] proposed a distributed specification test for massive datasets and analyzed the impact of divide-and-conuue strategies on nonparametric testing. Du et al. [X] addressed one-sample mean testing within distributed frame- works. et al.[22] introduced a distributed conditional independence test aimed at discovering causal relationships. Cai et al.][S] studied distributed nonparametric goodness-of of-fit testing under differential privacy constraints in a white-noise-wire siee--withhiidd model. In the context of mediation anal- ysis research, meta-analysis is often employed to combine results from multiple studies. This is typically done by either computing a weighted average of p-values or test statistics [30], or by av- eraging the estimated mediation effects using weighted approach [2I]]. However, these methods often lack a detailed and rigorous statistical framework to support their use.
The distributed tests discussed above all rely simple weighted average approach. One limitation of this method is that, when the total sample size is fixed, the power of the test declines sharply as the number of data blocks K increases, as shown in the simulation. In this paper, we develop distributed framework for testing mediation effects, which overcomes this limitation. Specifically, main contributions can be summarized follows:
We propose communication-effiiini distributed versions of the classic Sobel's 's test statistic Tsobel and the MaxP test statistic TMazP, denoted T and T respectively. Af- ter presenting the theoretical properties of these statisties, we provide an easy-to-implement algorithm. Both theoretical analysis and numerical experiments demonstrate that the pro- posed tests achieve nearly identical power to the global test, which is obtained by pooling all data togethera process difficult implement practice due data-sharing limitations.
Our method lays the foundation for the further development of distributed mediation anal- ysis. Although the classic Sobel's and MaxP tests are conservative and underpowered, primarily because they do not account for the composite null nature of the mediation effect, many enhanced mediation tests are build upon Sobel's or MaxP tests. Consequently, the distributed versions of these enhanced tests can be derived from our distributed Sobel's or MaxP tests.
We apply our method an educational study, demonstrating that high school mathematics' effect on college-level Probability and Mathematical Statistics is mediated by Calculus. The data are sourced from three classes, with only summary information of local data available from each class due to student privacy concerns. Our distributed test successfully detects the mediation effect, which would be undetectable using local tests from just the first or second class.
2 Preliminaries: Mediation Analysis
Let A represent the exposure variable, Y the continuous outcome, M the continuous mediator, and X E IR? the vector of additional covariates used to adjust for potential confounding. In their seminal work, Baron and Kenny[G] proposed the following linear structural equation models for the outcome and mediator:
Y = 30 + BAA + BM + 3 + +
(I)
M 70 + + +/ , + EM.
(2)
where ey and Eas represent error terms with zero means and constant variances, assumed to be uncorrelated with one another.
The mediation effect is the Natural Indirect Effect (NIE), a concept introduced by [28] and [[7]] which quantifies the effect of the exposure on the outcome that is mediated through the mediator. For continuous mediator and outcome variables, the mediation effect is defined as the product By, where B and y are the coefficients from the respective equations. Graphically, the mediation effect corresponds to the NIE conveyed through the chain A + M + Y. as illustrated in the directed acyclic graph (DAG) in Figure The modern causal inference framework relies on the following standard identification assumptions to estimate the mediation effect [62]] [

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Center
X
7
7000
D
M
D
B
Worker
A
Y
Worker 2
Worker K
Figure The causal graph with treatment A, outcome Y, mediator M, and confounder X. is the ef- feet of A M, and is the effect of M on Y.
Figure distributed framework with K worker machines connected central processor. The k- th machine computes a local statistie T() using its local data ( and then transmits it to the central processor. The central processor subsequently com- putes the distributed statistic Dis based on the K local statistics.
Assumptions
A1. There are unmeasured exposure-outcome confounders given X;
A2. There are unmeasured mediator-outcome confounders given (X; A);
A3. There are unmeasured exposure-mediator confounders given X;
A4. There is no effect of exposure that confounds the mediator-outcome relationship;
A5. There is no exposure and mediator interaction the outcome.
To test the existence of a mediation effect, we can formalize the problem as a hypothesis testing problem:
Ho 87 = 0 versus Hi By + 0. The null hypothesis is composite and can be decomposed into three disjoint cases:
Case 1. 8 * 0, = 0; Case 2. 8 = 0, * 0: Case 3. B = 0, = 0.
The classic tests, Sobel's 's test and MaxP test, are constructed from the t-test statistics, T3 B/ag. and T, = 5/0,, where 3 and y are least squared estimators, and ā and o, are the estimated standard errors for 39 and 9t. respectively. Under mild regularity conditions, we have T3 - 1), 3 = 0, and T. S (0, 1) y = 0. Building on these two t-statistics, a Wald-type Sobel's test statistic [GT] is constructed
39
TrT,
T Sohel
v /9203 + 3203
\ I3 + T?
Alternatively, the MaxP test statistic 22 can be defined as:
TMaMP max 2 (TT(TTl): 24h (IT,
where (b-) is the standard normal cumulative distribution function.
According to [23]], in cases and Tsobel asymptotically follows N0, 1), while in case 3, Tsabet asymptotically follows (0, 1/4). In cases 1 and 2, TMaP asymptotically follows uniform distribu- tion on [0, 1], and case 3, T MaxP asymptotically follows a Bela(2, 1) distribution. In practice, these two tests do not account for case 3. making them overly conservative and underpowered. Numer- ous enhanced tests have been proposed based on these methods, such as those in [17, 7 2% 8. [0]] though this lies beyond the scope of our discussion.

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3 Distributed Mediation Analysis
In the distributed framework, the total N data are splitted and stored in K worker ma- chines (locations) connected to a central processor shown in Figure where D (Y A(k) M(k), ( {(Yss, Aki, Mass X denotes the data in the k-th machine, &c 1, K, and N = DE=1 ni. Assume that there exist positive constants с₁ and e2 such that S infx, M2 Rky /nk, S iufay,Ay / S 02, and K can be either finite or diverging to infinity as long as K/ mine T&g + as 00.
oo. Each worker machine extracts statistics based on the local data and transmits it to the central machine. The final statistic is obtained by suitably aggregating the local statistics.
Note that both the Sobel's s and MaxP tests are based on T's and Ty To derive the distributed versions of these tests, we first compute the distributed versions of T3 and Ty, Specifically, we calculate the local values _ and TT( on each machine, transmit them to the central machine, and then obtain the distributed statistics by weighted averaging:
K T,=
T3
Σ
(k)
Thk N
(4)
k=1
The properties of the distributed T3 and T, are outlined below, based on Lemma 25 of (16)..
Proposition 3.1. Under assumptions Al-A5 and with K o(vN), as N + oo,
K
K
T3
T N
(R) 1(,,),
T -
Σ
Tuk Y 4 1(),
N 07
(K)
3
Based on T3 and Ty we can construct Sobel's test and MaxP test the distributed framework, as shown below:
T&T,
T Sobel
(5)
+ +
and
D MaaP max (2 2 (T (T)))-
(6)
The asymptotic properties of these distributed statistics are analogous to those of the global statisties, as demonstrated in the following theorem.
Theorem 3.1. Under Assumptions A1-A5, the null hypothesis, and K = olvN), as N + oc, we have the following results.
In Cases and 2,
Dis Sobel
4 NO, 1), and Maze 4 Uniform((.1)
In Case 3,
Dis Sobel
(O0, and Dis 4 Beta(2,
The proof of Theorem 3 follows similar approach the proofs of Results 1(b) and 2(b) in [X]]. and is therefore omitted here. According the theorem, given the significance level a, we reject Ho if DT Sobell Z1-a/2 Die Meas a, where Z1-a/2 denotes the 1 a/2 percentile of the standard normal distribution. The distributed test procedure is outlined in Algorithm 1. From the algorithm. we can observe that each machine only needs to transmit two scalars the central machine. This not only minimizes transmission costs but also significantly protects data privacy. The following theorem demonstrates that the asymptotic power of the distributed test, based on the local summary statistics, is equivalent to the asymptotic power of the global test that aggregates all the data.
To investigate the power of the two distributed statistics, we first introduce the necessary nota-
tions. Let μ3 [T = Σ
THE
ETT and μ., = =ET,] = Σk=1
Th N
E Ac-
N
cording to [23]] ( x /
OM
R2 R
M E z vnk7
a
1
R2 x
where
σY

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Algorithm 1: The Distributed Test of Mediation Effects.
Step 1. For k = 11 ....., K, compute the local statistics T and (M) then transmit them to the central machine.
Step 2. In the central machine, compute the distributed statistics T3 and T,, using (9), then compute the distributed Sobel statistic TDiia and MaxP statistic TiD respectively.
MoxP using (5) and (6).
Sobel
Step 3. Given the significance level a, if [T > Z1-0/2 or T Taat
MoxP < reject Ho.
0A = VVar(A), RAx and R K are the coefficients of determination by regressing A X, and regressing M on (A,X?)? respectively. Thus, As
VNB TM or V R RM
My 2 VN
OA JM
VI-
Theorem 3.2. Under the same assumptions as Theorem can be calculated analytically
the powers of Dio Sobel and Dis MazP
J
( (
(-
2
drdy
1 21-0/2
2
and
( Zi 21/222 + ( Zi 21-a/2){(0(44 - Zi + Z Z1-a/2)h
respectively.
The proof of Theorem 2 is analogous to the proofs of Results (cc) and 2(c) in [ and is therefore not repeated here. Furthermore, the analytical powers of the proposed distributed tests are identical to the powers of the global tests in [23)] provided that = o( Nv This property is also corrob- orated by subsequent simulations. Notably, for given total sample size N, the power of tests remains unaffected by the number of machines K, as long as K o( (V).. This behavior can be
attributed to the fact that (R)) and TT(k) (for k = 1, K) exhibit an approximately linear form. In contrast, many test statistics for other hypothesis testing problems (e.g., goodness-of-fe tests)
degenerate U-statistics, which leads to reduction in power for their distributed versions as K increases, as discussed in [39] 2. G.
I5L 5J].
Remark 3.1. The condition == o(VN) is commonly used in distributed statistical inference and federated learning. This is equivalent to K/ng- + as - oc, meaning the number of sites K is much smaller than the sample size within each site. This setup is frequently encountered in practice, particularly in meta-analysis, where researchers aggregate conclusions from dozens of studies, each with hundreds or even thousands of data points. When K = of /N) is not satisfied, test power may decrease, and Type I error could become wncontrollable. quite challenging. Our simulations reveal that when N 21l and K 32 (less than VN 45), our method yields results similar to the global method. However, when K 64, the difference between our method and the global method increases. This suggests that our method somewhat tolerant of larger values of K. Therefore, we recommend ensuring that K VN in practice, which should be sufficiently effective.
practice, determining this boundary is
Simulation Studies
In this section, conduct extensive simulation studies evaluate the performance of the pro- posed distributed Sobel test and MaxP test. We generated the p-dimensional exposure variable A (ernouli(0.5), the covariate X NO0 Σx) with p 3 and Ex (10.5)--1) pxp- The mediator and the outcome Y were simulated as follows
Y = A 3 +
N(022).
(7)
M = M + xx + CM. CM N(01)),
(8)
5

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where 3x (1.-0.5.1)², 7x (0.,,,, Under the null hypothesis, we consider three scenarios: (1) (0.2.))) (B.7) (0,0.)) (3.r) = (0. 0). Under the alternatives, we set (B, 7) (0.2, (.000, and (0.05, 0.2). ). We compared the proposed distributed tests with the global tests (denoted T and TG Mazr). which use all the data pooled together, the local tests Sobel and T(l1]
from the first machine (denoted as (1)1)
MMap)) and the simple average tests that average the K local statistics (denoted as T Σ / T(() significance level a 0.05. We conducted the tests in Studio a MacBook Pro with M2 CPU, and each experiment was repeated 2,000 times to calculate the empirical sizes and powers of the tests.
T and Mazz EE ne (TT(())
N p. The
4.1 Balanced Data
In this setting, we set the sample size in each machine are the same, that is, n N/K. We fix the total sample size N 2²,, and set the number of machines K 2, 4, 8, 16, 32, 64. The empirical sizes are presented in Figure B. We can see that all the tests can control the type I error, while the empirical sizes of the tests are close to when 3 = 7 = This because in this case, the six tests are too conservative. As the number of machine K' increases, the empirical sizes of (1( Sobel
and ( Mox tends decrease, while our proposed distributed test are not influenced by K. This because for fixed N, as K increases, the sample size in the first machine will decreases, thus, the local tests may distance away from their asymptotic null distribution. As contrast, the asymptotic null distributions of our distributed tests is not influenced by K as long as K o(n).
Figure M shows the empirical powers. As expected, the powers of the local tests (T Sobet and TT()
M p) and average tests (Sohet Sobeet and 7 Mag p)) decrease as K increases, while the proposed distributed test is less affected by K and performs similarly the global tests. also evident that the performance of Tave MaxP is even worse than that of ( Maa In the leftmost four plots of Figure EL we can observe that when K is large (e.g., K 64), the power of our method slightly lower than that of the global tests. This may be due to the fact that when N is fixed and K is large, the condition K of o(vN) might not be satisfied, causing the distribution of our test statistic to deviate somewhat from the asymptotic distribution.
Tave
The Empirical Sizes of Sobel Tests
P y=0.2
=0..2
Method
0.08
0.06
0.04
0.02
0.00
2
4
16
02
64
64
2
16
32
The Empirical Sizes of MaxP Tests
B m0 ym0
8=0. ym02
-m0... y-0
Method
0.100
0.075
0.050
0.025
0.000
2
16
92
64
2
16
32
16
32
K
Figure 3: The empirical sizes of the eight tests under three null hypothesis and seven choices of K, with the total sample size N fixed at 2
4.2 Unbalanced Data
In this setting, we first generate the local sample sizes n& randomly from 50 to 150 for k 1, 1,2....,3 ,2. Then, we generate total of Σ nu data points. Next, we calculate the local tests Sobet and (1) Mazr) based on nı samples, the global tests (Ti(G) Sobel and (((G) Mezp) based on Σk= ni Sobel and Trev p) based on the first K local test statistics, and the dis-
(T(1)
samples, the average tests (Tare

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The Empirical Powers of Sobel Tests
0.006 y-0.2
B B-0.1 V-01
m-2 y=0.05
Method
0.2
100 Sibe
16
64
16
32
16
Sibe
K
The Empirical Powers MaxP Tests
-0.05 y-0.2
B-0.1 Y-011
B 3-02, y-0.05
Method
0.6
0.4
K
Figure 4:
The empirical powers of the eight tests under three alternative hypothesis and seven choices of K, with total sample size N fixed 2
tributed tests Sobee and Dis Mazp) based on the first summary local statistics, for = 1,2,..... 32. The empirical sizes and powers presented in Figures E and f. respectively.
We observe that the sizes of the local tests are close zero, as they only use nn samples. As K increases, the total sample size N also increases, and the empirical sizes of the global tests and distributed tests approach the nominal significance levels scenarios and 2. However, in scenario 3 (8 0, Y 0), increasing the sample size does not affect the empirical sizes of the Sobel and MaxP tests. We can also see that the size and power of Tave are close to zero, indicating that averaging the local p-values does not improve the power of the test.
As expected, the powers of the local tests in the first machine are close zero, and the powers of the global and distributed tests increase as K increases. Notably, the power of the distributed tests nearly identical to that the global tests, despite using only the local summary statistics. This highlights the advantage of the proposed distributed tests: they do not require pooling all the data together, yet they achieve the same performance the global tests, which pool all the data.
The Empirical Sizes of Sobel Tests
B =000 y=0
=-0 y=0.
=02. y=
Method
0.08
0.04
Taa
0.02
0.00
10
20
30
10
30
The Empirical Sizes of MaxP Tests
8 800 ym
B m0 .
8m0.. ym0
Method
0.04
0.02
0.00
20
10
30
Figure 5: The empirical sizes of the six tests under three null hypothesis. The number of machines K increases from to 32, and the total sample size N increases with K.

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The Empirical Powers of Sobel Tests
=0.06 y.2
B-0.1 v-0.1
B-2 y=0.05
Method
Sibe
10
30
20
30
20
Sibe
K
The Empirical Powers MaxP Tests
=0.05, y-0.2
B=0.1,V=0.1 .V-0.1
B-02, 8-02.y-0.05
Method
0.8
0.6
0.4
K
Figure 6: The empirical powers of the six tests under three alternative hypothesis. The number of machines K increases from to 32, and the total sample size increases with K.
4.3 Heterogeneous
While the proposed test primarily focuses on the homogeneous case, we also investigate its behavior in the heterogeneous setting. In this scenario, consider situation where the distribution of X varies across different centers, and the parameters and , differ for each machine, while maintain- ing the consistency their signs across all centers. Specifically, we define X ~ N N(4 Σx), where f1,... PK ~i. (-33, 3), and 1p = (1, ..., 1)" a p-dimensional vector. The true parame- ter values for machine k are given by:
(R)
(k)
(k)
P 0.05
Eg,1 0.1 +
6 0.2 + 43,3
(k)
(k)
(k)
7
0.05 + 6,1 0.1 +
7,2:
0.2 6,,3
where (2) 1, 6,,1 wi.d U0-0.02, 0.02), 6,2 2,2 Riid U(-0.04, 0.04), and 6,33 64,3 midd U(-0.08, 0.08).
(k)
(K)
The results from this simulation shown in Figure and are consistent with those obtained in the homogeneous case (Figure 9) Specifically, our method performs similarly to the global test, with noticeable differences only when K is particularly large (K 64). This indicates that method is well-suited to handle heterogeneity across centers.
5 Real Data Analysis
We examined the impact of Gaokao mathematics (A) performance on college-level "Probability and Mathematical Statistics" course grades (Y). with college "Calculus" course grades serving as mediator (M). The Gaokao is the National Higher Education Entrance DExamination in China, and the results of the Gaokao are often seen key determinant of student's future academic and career opportunities in China. In this study, we sought to explore whether there is a mediating effect in which Gaokao mathematics performance influences calculus grades, which turn affect the grades in the probability and statistics course. This research holds important implications for both the learning and teaching of Probability and Mathematical Statisties. The confounders (X) include learning interest and classroom efficiency.
We distributed surveys to three classes university. To protect student privacy, the surveys were organized and the statistical calculations were conducted by the instructors of each class. The results follows.
Class 1: 34, I 4.654908, 1 1.723737:

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The Empirical Powers of Sobel Tests in Heterogeneous Settings
8 3-0.05 +0,
y-0.2 +0g
P-0.1 +eg. V=0.1 402
B 5-02 48,, y=0.05 40;
Method
0.5-1
0.4
0.2
100 Sibe
0.11
16
64
4
16
32
64
16
K
The Empirical Powers of MaxP Tests in Heterogeneous Settings
0 -0.05 +0,
y-0.++
P-0.1 +0p V-0.1 +0,
B B-02 +2,, y-0.05 to
Method
0.6
16
16
K
Figure 7: The empirical powers of the eight tests under three alternative hypothesis in heterogeneous settings, with the total sample size N fixed at 2
Class 2: n2 14, 133 0.530207, 2 0.981202:
Class 3: n1 17, T 4.829768, 1, 3.274552.
The sample sizes of Class 2 and Class 3 are relatively small. Fortunately, the total sample size N n1 + n2 + n3 = 65 is sufficient. Although we do not have the original data for each class, the test results obtained using our proposed method
very close to the results from pooling the original data from all three classes. Let Dis.; represent the distributed test using the local test statistics from classes i and j, for (i.j) (1, 2), (1, 3), (2, 3), and Dis represent the distributed test statistic using the local test statistics from all three classes. The corresponding p-values for these tests are
shown in Table M.
Table 1: The p-values of the tests.
Method
Class
Class2
Class3
Dis
Dis
Disga
Dis
Sobel 0.106
0.6409
0.0067
0.0732
0.0032
0.0152
0.0032
MaxP
0.0848
0.596
0.0011
0.0476
0.001
0.002
0.0007
From this table, we can see that for a given significance level of a = 0.05, when each class is tested individually, the results mediation effect with Calculus as the mediator, while the test for Class detects the mediation effeet. However, by combining the test statistics from Class I and Class 2 using resulting distributed MaxP test yields p-value < 0.05, indicating the presence of a mediation effect. This means that although the individual tests for Class 1 and Class 2 failed to detect the mediation effect, our method, which combines their local test statistics, successfully identified the mediation effect through the distributed test. Furthermore, the Sobel and MaxP tests for Dis Dis23, and Dis all detected the presence of the mediation effect, and the p-values of the Dis are the smallest.
follows: the tests for Class 1 and Class 2 both fail to detect the
proposed method, the
In conclusion, this study successfully identified the mediation effect between Gaokao mathematics performance, Calculus grades, and Probability and Mathematical Statistics course grades by pooling test statistics from multiple classes. Despite mixed results from individual class tests, our method overcame the limitations of small sample sizes and revealed that Gaokao mathematics performance influences statistics course grades through Calculus. This finding has important implications for teaching and learning in higher education mathematics courses.
The above results highlight a key application distributed testing, especially in healthcare settings. Many hospitals face limitations in data availability due to factors like privacy concerns and regula- tory restrictions, preventing them from sharing raw data. Our distributed method addresses this by only requiring hospitals to share test statistics rather than raw data. This allows the detection of me-

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instead of the linear model equation (M), and compute the local maximum likelihood estimate (MLE) of 3, from which could derive the distributed 7a based on the MLE. However, as shown diation effects that might be missed in individual analyses, making it a valuable tool for healthcare institutions with limited data and privacy constraints.
Conclusion
In this article, we construct distributed framework for mediation analysis. Building on Sobel's and MaxP tests, we introduce their distributed version, which are communication-effiien and easy to implement. Theoretical analysis and experiments show that the distributed tests achieve nearly identical power to global tests long as K o( N(VN). We apply our method to an educational study, demonstrating its ability to detect mediation effect that would not be identifiable using data from only the first or second class, highlighting the advantage of our approach.
There are still several limitations. First, the Sobel's s and MaxP tests are conservative and under- powered, and focus on one mie-dimensiona mediator, so the proposed distributed test inherits these disadvantages. Fortunately, there are several enhanced tests based on Sobel's or MaxP tests, and therefore, their distributed versions can be easily derived from our distributed Sobel's 's or MaxP tests. For instance, suppose there are candidate mediator variables M,,..., M,, leading to J hypothesis tests Huj: 8yy Let T MarP, represents the global test statistics for the j-th test. Dai al. [ propose the HDMT method, which constructs the test statistic:
THDMT5 = (fo1 + #10) MazP.j + o0 (T (MMEP3))
where #01. #10. and Aoo are the estimated proportions of (3, 0, + 0), (3j * 0, 0), and (83 = 0, % = 0) across all J tests, respectively. In the distributed framework, T MasP.j can
be replaced by the proposed distributed test statistic TDi Maxpy. and कp be approximated by the simple average x Ek=1 #01
(*(k)
Second, our study assumes that the outcome Y is continuous. is binary, we could a logistic model
logit(Pr(Y= = 1|A, [| = fu + BAA + BB + 3
in [19], the properties of the distributed s may be influenced by K. Therefore, improvements to our method are needed. These directions will be explored in future work.
7 Acknowledgment
This research is supported by the National Natural Science Foundation of China (12101056) National Statistical Science Research Project (2022LY040), and the Talent Fund of Beijing Jiaotong University (2023XKRC008).
References
[1] Susan Athey and Guido W Imbens. The state of applied econometrics: Causality and policy evaluation. Journal of Economic Perspectives, (1(2:33-3, 2017.
[2] Ernest Atta-Asiamah and Mingao Yuan. Distributed inference for degenerate u-statistics. Stat, 8(1)xe234, 2019.
[3] Reuben M Baron and David A Kenny, The moderator-mediator variable distinction in so- cial psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, (1))): 1et::1133111 1182, 1986.
[4] Heather Battey, Jianqing Fan, Han Liu, Junwei Lu, and Ziwei Zhu. Distributed testing and estimation under sparse high dimensional models. Annals of Statistics, 46(3):1352-1383, 2018
[5] T Tony Cai, Abhinav Chakraborty, and Lasse Vuursteen. Federated nonparametric hypothesis testing with differential privacy constraints: Optimal rates and adaptive tests. arXiv preprint 2rXiv:2406.067499 2024.
10

---

[6] Song Xi Chen and Liuhua Peng. Distributed statistical inference for massive data. The Annals of Statistics, 49(5)2251122869 2021.
[7] James Y Dai, Janet L Stanford, and Michael LeBlanc. A multiple-testing procedure for high-dimensional mediation hypotheses. Journal of the American Statistical Association, 1176537):188-133, 2022.
[8] Jiarong Ding and Xuehu Zhu. Amdp: adaptive detection procedure for false discovery rate control in dighediimesioona mediation analysis. Advances in Neural Information Processing Systems, 36:65906-6559999 2023.
[9] Bin Du, Junlong Zhao, and Xin Zhang- Hypothesis testing of one sample mean vector in distributed frameworks. Communications in Statistics-Simulation and Compulation, pages I- 18, 2024.
[10] Jiacong Du, Xiang Zhou, Dylan and Bhramar Mukherjee. Methods for large-scale single mediator hypothesis testing: Possible choices and comparisons. Genetic Epidemiology, 1772):167-184 2023.
Wei Hao, Yongmei Liu, Jennifer A Smith,
[1]] Altay Eren. Uncovering the links between prospective teachers personal responsibility, aca- demic optimism, hope, and emotions about teaching: A mediation analysis. Social Psychology of Education, 17:73-104, 2014.
[12] Jianqing Fan, Yongyi Guo, and Kaizheng Wang. Communication-effiicnn accurate statistical estimation. Journal of the American Statistical Association, 118(542):1000-0111 2023.
[13] Jianqing Fan, Dong Wang, Kaizheng Wang, and Ziwei Zhu, Distributed estimation of principal eigenspaces, Annals of Statistics, 4766)33009-3031, 2019.
[14] Tianyu Guo, Sai Praneeth Karimireddy, and Michael Jordan. Collaborative heterogencous causal inference beyond meta-analysis. arXiv preprint arXiv:2404.15746, 2024.
[15] Bingyao Huang, Yanyan Liu, and Liuhua Peng. Distributed inference for two-sample statistics massive data analysis. Scandinavian Journal of Statistics, 5063):1000-1115, 2023.
[16] Cheng Huang and Xiaoming Huo. A distributed one-step estimator. Mathematical Program- ming, 174:41-76. 2019.
[17] Yen-Tsung Huang. Genome-wide analyses of sparse mediation effects under composite null hypotheses. The Annals of Applied Statistics, 13(1):60-84, 2019.
[18] Kosuke Imai, Luke Keele, and Dustin Tingley. A general approach to causal mediation analysis. Psychological Methods, 15(4):309-334 2010.
[19] Michael Jordan, Jason D Lee, and Yun Yang. Communication-effiienn distributed statistical inference. Journal of the American Statistical Association, 1 114(526):668-66) 661, 2019.
[20] Rémi Khellaf, Aurélien Bellet, and Julie Josse. Federated causal inference: Multi Multi-study estimation beyond anal-sna arXiv preprint arXiv:2410.16870 2024.
[21] Hopin Lee, Markus Hübscher, G Lorimer Moseley, Steven Kamper, Adrian C Traeger, Gemma Mansell, and James H McAuley. How does pain lead to disability? a systematic review and meta-analysis of mediation studies in people with back and neck pain. Pain, 156(6):988- 997, 2015.
[22] Loka Li, Ignavier Ng, Gongxu Luo, Biwei Huang, Guangyi Chen, Tongliang Liu, Bin Gu, and Kun Zhang. Federated causal discovery from heterogeneous data. The Twelfth International Conference Learning Representations, 2024.
[23] Zhonghua Liu, Jincheng Shen, Richard Barfield, Joel Schwartz, Andrea A Baccarelli, and Xihong Lin. Large-scale hypothesis testing for causal mediation effects with applications in genome-wide epigenetic studies. Journal of the American Statistical Association, 11tiiiiinnn1111531 117(537):67- 81 2022.

---

[24] David P MacKinnon, Chondra M Lockwood, Jeanne M Hoffman, Stephen G West, and Virgil Sheets. A comparison of methods to test mediation and other intervening variable effects. Psychological Methods, 7(1):83-104, 2002.
[25] David P MacKinnon, Chondra M Lockwood, and Jason Williams. Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate Behavioral Research, (9(1):99-122) 2004.
[26] Michèle B Nuijten, Ruud Wetzels, Dora Matzke, Conor V Dolan, and Eric-Jan Wagenmakers. A default bayesian hypothesis test for mediation. Behavior Research Methods, 47:85-97, 2015.
[27] Judea Pearl. Direct and indirect effects. In Probabilistic and causal inference: the works of Judea Pearl, pages 373-392. 2022.
[28] James M Robins and Sander Greenland. Identifiability and exchangeability for direct and indirect effects. Epidemiology, 3(2): 155-155, 1992.
[29] Kenneth J Rothman and Sander Greenland. Causation and causal inference in epidemiology. American Journal of Public Health, 95(S1): 2005.
[30] Katherine RK Saunders, Sabine Landau, Louise Howard, Helen L Fisher, Louise Arseneault, Geraldine FH McLeod, and Sian Oram. Past-year intimate partner violence perpetration among people with and without depression: an individual participant data (ipd) meta-mediation anal- ysis. Social psychiatry and psychiatric epidemiology. 58(12):1735-1747 2023.
[31] Michael E Sobel. Asymptotic confidence intervals for indirect effects in structural equation models. Sociological Methodology, 13:290-311 -129 1982.
[32] Zhenheng Tang, Yonggang Zhang. Peijie Dong, Yiu-ming Cheung, Amelie Zhou, Bo Han, and Xiaowen Chu. Fusefl: One-shot federated learning through the lens causality with progressive model fusion. Advances in Neural Information Processing Systems, 37:28393- 28429, 2024.
[33] Linda Valeri and Tyler J VanderWeele. Mediation analysis allowing for exposure interactions and causal interpretation: theoretical assumptions and implementation with and spss macros. Psychological Methods, 18(2):137-150, 2013.
[34] Tyler VanderWeele and Stijn Vansteelandt, Conceptual issues concerning mediation, interven- tions and composition. Statistics and its Interface, 2:457-468, 2009.
[35] Shuyuan Wu, Danyang Huang, and Hansheng Wang. Quasi-newton updating for large-scale distributed learning. Journal of the Royal Statistical Society Series B: Statistical Methodology, 85(44:1326-1354 2023.
[36] Qiaoling Ye, Arash A Amini, and Qing Zhou. Federated learning of generalized linear causal networks. IEEE Transactions on Pattern Analysis and Machine Intelligence, 46(10):6623- 6636, 2024.
[37] Haixiang Zhang. Efficient adjusted joint significance test and sobel-type confidence interval for mediation effect. Structural Equation Modeling: A Multidisciplinary Journal, 32(1):93-104 2025.
[38] Yuchen Zhang, John C Duchi, and Martin J Wainwright. Communication-effiien algorithms for statistical optimization. The Journal of Machine Learning Research, 14(1):3321-334 2013.
[39] Yanyan Zhao, Changliang Zou, and Zhaojun Wang. A scalable nonparametric specification testing for massive data. Journal of Statistical Planning and Inference, 200:161-175, 2019.
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Reviews:
[{"review_id": "IldSW7DWnV", "reviewer": "Reviewer_pTFS", "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.", "questions": "1. How do your methods relate to meta analysis techniques?  \n2. How do they technically and methodologically relate to existing federated causal inference works? What are the new techniques introduced here?  \n3. How would data heterogeneity affect the results and the framework?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n- 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.\n- The algorithm requires only a single communication round  \n- The authors show that their distributed variant has asymptotically identical power to that of the centralized variants.  \n\nWeakness:  \n\n- The authors don't discuss how this relates to classical meta analysis methods that similarly aim to combine results from multiple sites.  \n-  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.\n- No discussion is made about the distribution of the data across the centers - what happens if there is heterogeneity?", "was_revised": true, "final_justification": "The response addressed my major concerns.", "rebuttal": "We thank the reviewer for the careful reading and thoughtful comments. We address the reviewer's questions in the following. We hope the responses below address the reviewer's concerns.\n\n**Weakness 1:** The authors don't discuss how this relates to classical meta analysis methods that similarly aim to combine results from multiple sites.\n\n**Answer:** Thank you very much for your insightful comment. The proposed method can be viewed as a form of meta-analysis. In the context of mediation analysis research, meta-analysis is often employed to combine results from multiple studies. This is typically done by either computing a weighted average of p-values or test statistics (Saunders et. al., 2023), or by averaging the estimated mediation effects ($\\widehat{ME_k}$, for $k = 1, \\dots, K$) using a weighted approach (Lee et. al., 2015), such as:\n\n$$\n\\widehat{ME_{Dis}} = \\frac{\\sum_{k=1}^{K} \\frac{\\widehat{ME_k}}{\\hat{\\sigma_k}}} {\\sum_{k=1}^{K} \\frac{1}{\\hat{\\sigma_k}}}.\n$$\n\nSubsequently, a confidence interval for $\\widehat{ME}_{Dis}$ is computed for statistical inference. However, it is important to note that these methods often lack a detailed and rigorous statistical framework to support their use. For example, these approaches require that $K = o(\\sqrt{N})$, yet this assumption is rarely discussed in the literature.\n\nIn contrast, the method proposed in this paper is grounded in a more rigorous statistical theory, offering a clear advantage in terms of precision and reliability. In the simulation of the revision, we included comparisons with two weighted averaging methods:\n$$T_{Sobel}^{ave} = \\sum_{k=1}^{K} \\sqrt{\\frac{n_k}{N}} T_{Sobel}^{(k)},$$\n$$T_{MaxP}^{ave} = \\sum_{k=1}^{K} \\sqrt{\\frac{n_k}{N}} T_{MaxP}^{(k)}.$$\nWe found that both of these naive methods showed much poorer performance compared to our proposed method. For a fixed $N$, as the value of $K$ increased, the power of both averaging methods noticeably declined. In fact, their performance only slightly surpassed the results obtained from using the $T_{Sobel}^{(1)}$ and $T_{MaxP}^{(1)}$ statistics from just the first machine. When $K = 8$, the power of the averaging methods dropped to below 0.1, while the power of our proposed method remained identical to the global test.\n\nWe hope this provides a clearer understanding of the value and novelty of our approach.\n\n**Weakness 2:** The authors also don't discuss existing work on federated causal inference ([Guo et al. 24], [Khellaf et al. 24], and related works) and how their method methodologically relates.\n\n**Answer:** Thank you for your valuable suggestion. We appreciate your reference to existing work on federated causal inference.While the mediation effect is indeed an indirect causal effect, there is a methodological distinction between causal inference and mediation analysis. In causal inference, the focus is primarily on estimating treatment effects, particularly total effects, within the potential outcomes framework. On the other hand, mediation analysis focuses more on hypothesis testing within the structural equation model framework, where the primary concern is testing for mediation effects, such as $\\beta\\gamma = 0$.\n\nHowever, there are commonalities between causal inference and mediation analysis in terms of constructing statistics—both methodologies aim to derive meaningful statistics, with causal inference concentrating on effect estimation and mediation analysis focused on hypothesis testing. \n\nIn the distributed framework, methods for constructing statistics generally fall into two categories: one-shot and multi-shot approaches. To estimate the Average Treatment Effect (ATE), $\\tau(e)$, where $e$ is the nuisance propensity model, the simplest approach is to take the weighted average of estimates from each site, $\\sum_{k=1}^{K} \\eta_k \\hat{\\tau_k}(\\hat{e_k})$. An improvement proposed by Guo et al. (2024) is to use the weighted average of the propensity estimates, $\\sum_{k=1}^{K} \\hat{\\tau_k}(\\sum_{r=1}^{K} \\eta_r \\hat{e_r})$. Khellaf et al. (2024) also studied one-shot and multi-shot methods in the context of linear outcome models. We have added the discussion in Introduction in revision.\n\nTo construct the Sobel test statistic $T_{Sobel} = T_{Sobel}(T_{\\gamma}, T_{\\beta})$ in the distributed setting, the simplest method is the weighted average of each site’s estimates: $\\sum_{k=1}^{K} \\xi_k T_{Sobel}^{(k)}(T_{\\gamma}^{(k)}, T_{\\beta}^{(k)})$. \nWe propose an improved approach $T_{Sobel}(\\bar{T_\\gamma}, \\bar{T_\\beta})$,\nwhere $ \\bar{T_\\gamma} $ and $ \\bar{T_\\beta} $ are the weighted averages of $ \\bar{T_\\gamma}^{(k)} $ and $ \\bar{T_\\beta}^{(k)} $ across all sites $ k = 1, \\dots, K $. This method performs significantly better than the simple averaging approach.\n Theoretically, our test statistic has the same asymptotic distribution as the global test statistic, and our simulations confirm that the performance of our method is nearly identical to that of the global approach.\n\n\n**Weakness 3:** No discussion is made about the distribution of the data across the centers - what happens if there is heterogeneity?\n\n**Answer:** Thank you very much for your valuable suggestion. We truly appreciate your attention to this important aspect. In response to your comment, we have included an additional simulation in the revised manuscript to explore the scenario of heterogeneity. In this case, we consider a situation where the distribution of $X$ varies across different centers, and the parameters $\\gamma$ and $\\beta$ differ for each machine, while maintaining the consistency of their signs across all centers.\n\nSpecifically, we define $X^{(k)} \\sim N(\\mu_k 1_p, \\Sigma_X)$, where $\\mu_1, ..., \\mu_K \\sim_{i.i.d} U(-3, 3)$, and $1_p = (1, ..., 1)^T$ is a $p$-dimensional vector. The true parameter values for machine $k$ are given by:\n\n$$\\beta_k = 0.05 + e_{\\beta,1}^{(k)}, 0.1 + e_{\\beta,2}^{(k)}, 0.2 + e_{\\beta,3}^{(k)};$$\n$$\\gamma_k = 0.05 + e_{\\gamma,1}^{(k)}, 0.1 + e_{\\gamma,2}^{(k)}, 0.2 + e_{\\gamma,3}^{(k)},$$\n\nwhere $e_{\\beta,1}^{(k)}, e_{\\gamma,1}^{(k)} \\sim_{i.i.d} U(-0.02, 0.02)$, $e_{\\beta,2}^{(k)}, e_{\\gamma,2}^{(k)} \\sim_{i.i.d} U(-0.04, 0.04)$, and $e_{\\beta,3}^{(k)}, e_{\\gamma,3}^{(k)} \\sim_{i.i.d} U(-0.08, 0.08)$.\n\nThe results from this simulation are consistent with those obtained under the homogeneous case (as shown in Figure 4 of the manuscript). Specifically, our method performs similarly to the global test, with noticeable differences only when $K$ is particularly large ($K = 64$). This suggests that our method is well-suited to handle heterogeneity across centers.\n\n\n**Question 1:** How do your methods relate to meta analysis techniques?\n\n**Answer:**  Thank you for your question. We have already addressed this issue in \\textbf{Weakness 1}.\n\n**Question 2:** How do they technically and methodologically relate to existing federated causal inference works? What are the new techniques introduced here?\n\n**Answer:**  Thank you for your question. We have already addressed this issue in \\textbf{Weakness 2}.\n\n**Question 3:** How would data heterogeneity affect the results and the framework?\n\n**Answer:**  Thank you for your question. We have already addressed this issue in \\textbf{Weakness 3}.\n\n**Reference**\n \n[1] Guo T, Karimireddy S P, Jordan M I. Collaborative heterogeneous causal inference beyond meta-analysis[J]. arXiv preprint arXiv:2404.15746, 2024.\n\n[2] Khellaf R, Bellet A, Josse J. Federated causal inference: Multi-study ate estimation beyond meta-analysis[J]. arXiv preprint arXiv:2410.16870, 2024.\n\n[3] Lee H, Hübscher M, Moseley G L, et al. How does pain lead to disability? A systematic review and meta-analysis of mediation studies in people with back and neck pain[J]. Pain, 2015, 156(6): 988-997.\n\n[4] Saunders K R K, Landau S, Howard L M, et al. Past-year intimate partner violence perpetration among people with and without depression: an individual participant data (IPD) meta-mediation analysis[J]. Social psychiatry and psychiatric epidemiology, 2023, 58(12): 1735-1747.\n\nWe thank the reviewer again for the helpful comments and suggestions for our work. We are more than happy to address any further questions that you may have."}, {"review_id": "aU9MymCtID", "reviewer": "Reviewer_XAfo", "summary": "The 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.", "questions": "1. 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?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "The paper provides a novel and relevant method for mediation analysis when data is distributed across silos. Overall, the reviews were positive with some minor concerns/questions, similar to my own. The authors have addressed my main concern regarding the homogeneity assumption by incorporating a simulation, analysis, and discussion into the paper. My suggestion is to accept the work after reading the peer reviews and the author's rebuttals.", "rebuttal": "We thank the reviewer for the careful reading and thoughtful comments. We address the reviewer's questions in the following. We hope the responses below address the reviewer's concerns.\n\n**Weakness 1:** 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\n**Answer:** Thank you for your thoughtful question. You've raised an important point regarding the assumption of homogeneity in our method. In the manuscript, we initially assume that the $\\beta$ and $\\gamma$ effects are homogeneous across all data clusters. However, as highlighted by your comment, this assumption is indeed critical and may not always hold in practice.\n\nTo address this concern, we have included an additional simulation in the revised manuscript. This simulation explores a scenario where the distribution of $X$ varies across different centers, and where the parameters $\\beta$ and $\\gamma$ differ for each machine, while ensuring that the signs of $\\beta$ and $\\gamma$ are consistent across all centers.\n\nIn particular, we define $X^{(k)} \\sim N(\\mu_k 1_p, \\Sigma_X)$, where $\\mu_1, ..., \\mu_K \\sim_{i.i.d} U(-3, 3)$, and $1_p = (1, ..., 1)^T$ is a $p$-dimensional vector. The true parameter values for machine $k$ are given by:\n\n$$\\beta_k = 0.05 + e_{\\beta,1}^{(k)}, 0.1 + e_{\\beta,2}^{(k)}, 0.2 + e_{\\beta,3}^{(k)}$$\n$$\\gamma_k = 0.05 + e_{\\gamma,1}^{(k)}, 0.1 + e_{\\gamma,2}^{(k)}, 0.2 + e_{\\gamma,3}^{(k)}$$\n\nwhere $e_{\\beta,1}^{(k)}, e_{\\gamma,1}^{(k)} \\sim_{i.i.d} U(-0.02, 0.02)$, $e_{\\beta,2}^{(k)}, e_{\\gamma,2}^{(k)} \\sim_{i.i.d} U(-0.04, 0.04)$, and $e_{\\beta,3}^{(k)}, e_{\\gamma,3}^{(k)} \\sim_{i.i.d} U(-0.08, 0.08)$.\n\nThe results from this simulation are consistent with those obtained under the homogeneous case (as shown in Figure 4 of the manuscript). Specifically, our method performs similarly to the global test, with noticeable differences only when $K$ is particularly large ($K = 64$). This suggests that our method is well-suited to handle heterogeneity across centers.\nWe acknowledge that more rigorous theoretical results are needed to fully address heterogeneity, and we plan to explore this further in future research.\n\nOnce again, we appreciate your insightful feedback and hope that the additional simulation clarifies this concern. Thank you for helping us improve the manuscript.\n\n**Weakness 2:** 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\n**Answer:**  Thank you for your feedback. The Sobel and MaxP tests are indeed classical methods, and there are several improvements based on them. In the revision, we discuss the implementation of two such improvements within a distributed framework: the HDMT method proposed by Dai et al. (2022) and the JT-comp method from Huang (2019). Both of these methods are well-suited for high-dimensional mediators.\n\nLet there be $J$ candidate mediator variables $M_1, \\dots, M_j$, leading to $J$ hypothesis tests $H_{0j}: \\beta_j \\gamma_j = 0$. Denote $T_{\\beta,j}$ and $T_{\\gamma,j}$ as the test statistics for the $j$-th test.\n\nHuang (2019) constructs a p-value as follows:\n\n$$\n\\widehat{p_{comp,j}} = F\\left( \\frac{T_{\\gamma,j} T_{\\beta,j}}{\\text{Var}(T_{\\gamma})} \\right) + F\\left( \\frac{T_{\\gamma,j} T_{\\beta,j}}{\\text{Var}(T_{\\beta})} \\right) - F\\left( \\frac{T_{\\gamma,j} T_{\\beta,j}}{\\text{Var}(T_{\\gamma}) + \\text{Var}(T_{\\beta})} \\right),\n$$\n\nwhere $F(z) = 2 \\int_{|z|}^{\\infty} \\frac{K_0(z)}{\\pi} \\, dz$, and $K_0(z)$ is the Bessel function of order $0$. The sample variances of $T_{\\gamma,j}$ and $T_{\\beta,j}$ across all tests are used to estimate $\\text{Var}(T_{\\gamma})$ and $\\text{Var}(T_{\\beta})$.\n\nIn the distributed framework, we replace $T_{\\gamma,j}$ and $T_{\\beta,j}$ with the averages $\\overline{T_{\\gamma,j}} = \\sum_{k=1}^{K} \\sqrt{\\dfrac{n_k}{N}} T_{\\gamma,j}^{(k)}$ and $\\overline{T_{\\beta,j}} = \\sum_{k=1}^{K}\\sqrt{\\dfrac{n_k}{N}} T_{\\beta,j}^{(k)}$, respectively, as described in equation (4) of the paper.\n\nAdditionally, Dai et al. (2022) propose the HDMT method, which constructs the test statistic:\n\n$$\nT_{HDMT,j} = (\\widehat{\\pi_{01}} + \\widehat{\\pi_{10}}) T_{MaxP,j} + \\widehat{\\pi_{00}} T_{MaxP,j}^2,\n$$\n\nwhere $\\widehat{\\pi_{01}}$, $\\widehat{\\pi_{10}}$, and $\\widehat{\\pi_{00}}$ are the estimated proportions of $(\\beta_j = 0, \\gamma_j \\neq 0)$, $(\\beta_j \\neq 0, \\gamma_j = 0)$, and $(\\beta_j = 0, \\gamma_j = 0)$ across all $J$ tests, respectively. These proportions can be estimated using non-parametric methods (e.g., Storey, 2002).\n\nIn the distributed framework, $T_{MaxP,j}$ is replaced by the distributed test statistic $T_{MaxP,j}^{Dis}$, and $\\widehat{\\pi_{index}}$ is approximated by the simple average $\\frac{1}{K} \\sum_{k=1}^{K} \\widehat{\\pi_{index}}^{(k)}$, $index=00,01,10.$\n\nThe performance and statistical properties of these distributed test statistics will be further studied, and we plan to address these aspects in future work.\n\n\n**Question 1:** 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?\n\n**Answer:**  Thank you for your question. We have already addressed this issue in \\textbf{Weakness 1}.\n\n\n**Question 2:** 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\n**Answer:** Thank you for your thoughtful comment. With condition $K = o(\\sqrt{N})$, we can derive the limiting distribution of the proposed distributed statistics under the null hypothesis and the power function under the alternative hypothesis. When $K = o(\\sqrt{N})$ is not satisfied, aside from the issue of reduced power, a more serious concern is that the type I error might become uncontrollable. In practice, determining this boundary is indeed challenging. From our simulations, we observed that when $N = 2^{11}$ and $K = 32$ (which is less than $\\sqrt{N} \\approx 45$), our method produces results quite similar to the global method. In the revision, we further tested the case when $K = 64$ and found that the difference between our method and the global method increased. This suggests that our method is somewhat tolerant of larger values of $K$, so we recommend ensuring that $K < \\sqrt{N}/2$ in practice, which should be sufficient.\n\nNevertheless, when $K = o(\\sqrt{N})$ does not hold, or when $K$ is very large and the data size in each cluster is relatively small, what are the asymptotic properties of the proposed distributed statistics, is an important research question worth exploring. We consider this as a direction for future work.\n\n**Reference**\n\n\n[1] Dai J Y, Stanford J L, LeBlanc M. A multiple-testing procedure for high-dimensional mediation hypotheses[J]. Journal of the American Statistical Association, 2022, 117(537): 198-213.\n\n[2] Huang Y T. Genome-wide analyses of sparse mediation effects under composite null hypotheses[J]. The Annals of Applied Statistics, 2019, 13(1): 60-84.\n\n[3] Storey J D. A direct approach to false discovery rates[J]. Journal of the Royal Statistical Society Series B: Statistical Methodology, 2002, 64(3): 479-498.\n\nWe thank the reviewer again for the helpful comments and suggestions for our work. We are more than happy to address any further questions that you may have."}, {"review_id": "ckJCBmZ7S8", "reviewer": "Reviewer_aebb", "summary": "This 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).", "questions": "Please 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?", "limitations": "Yes, in section 6.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "The authors addressed my concern and now I have more confidence to support this paper", "rebuttal": "We thank the reviewer for the careful reading and thoughtful comments. We address the reviewer's questions in the following. We hope the responses below address the reviewer's concerns.\n\n**Weakness 1:** 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.\n\n**Answer:** Thank you for your insightful comment.\nThe condition $K = o(\\sqrt{N})$ is not as restrictive as it may seem at first glance. It is equivalent to $K / n_k \\to 0$ as $n_k \\to \\infty$, meaning the number of sites $K$ is much smaller compared to the sample size within each site. This setup is quite common in practice, especially in meta-analysis, where researchers often aggregate conclusions from dozens of studies, each with hundreds or even thousands of data points. \n\nFurthermore, the condition $K = o(\\sqrt{N})$ is widely used in distributed statistical inference and federated learning.\n In distributed frameworks, common estimation methods are categorized as either one-shot or multi-shot methods. The one-shot method specifically requires this assumption to ensure that the cumulative bias across machines can be neglected. Without this assumption, it would be challenging to guarantee the consistency of the estimates.\n\nWe acknowledge the concern regarding the applicability of this assumption and will consider how to relax or generalize it in future work.\n\n**Weakness 2:** 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.\n\n**Answer:** Thank you for your thoughtful comment.\nRegarding the consideration of multiple continuous mediators, we have addressed this issue in the final section of the revised manuscript. In genomic settings, where multiple candidate mediators $M_j, j = 1, \\ldots, J$ may exist, a common approach is to test each mediator $M_i$ individually within the model to check for mediation effects. Afterward, testing continues with the next mediator $M_{i+1}$, and this process continues until all mediators have been evaluated. By controlling the false discovery rate (FDR), one can identify which mediators exhibit significant mediation effects. An example of this approach can be found in Liu et al. (2022) in the Normative Aging Genome-Wide Epigenetic Study. This clearly represents a multiple hypothesis testing problem, and our method can be extended to handle such scenarios directly. Additionally, the power of our approach can be enhanced by incorporating techniques such as HDMT (Dai et al., 2022) and the JT-comp method (Huang, 2019).\n\nHowever, there is also a more complex scenario where multiple mediators $M_j, j = 1, \\ldots, J$ are included in a structural equation model (SEM), and interactions may exist among these mediators. In this case, the structural equations are given by:\n\n$$\nY = \\beta_0 + \\beta_A A + \\boldsymbol{\\beta}^T M + \\boldsymbol{\\beta_X}^T \\mathbf{X} + \\epsilon_Y,\n$$\n$$\nM_j = \\gamma_{0j} + \\gamma_j A + \\boldsymbol{\\gamma_{Xj}}^T \\mathbf{X} + \\epsilon_{Mj}, \\quad j=1, \\ldots, J.\n$$\n\nIn this setting, one would need to test the hypothesis $H_0: \\boldsymbol{\\gamma}^T \\boldsymbol{\\beta} = 0$, where $\\boldsymbol{\\gamma}^T = (\\gamma_1, \\ldots, \\gamma_J)^T$. This scenario is more complex because even when $\\boldsymbol{\\gamma} \\neq 0$ and $\\boldsymbol{\\beta} \\neq 0$, their inner product may still be zero. Moreover, the distribution of $\\hat{\\boldsymbol{\\gamma}}^T \\hat{\\boldsymbol{\\beta}}$ is challenging to compute (Preacher et al., 2008). While this is outside the scope of the current paper, it presents an interesting direction for future research. \n\n**Weakness 3:** Privacy is still a major concern in distributed settings, but this paper did not provide privacy guarantees, and information leakage was not discussed.\n\n**Answer:** Thank you for your important comment.\nRegarding privacy concerns in distributed settings, we would like to clarify that our approach inherently offers privacy protection. This is because the individual sites do not need to share raw data. Instead, they only need to transmit summary statistics, which allows our method to achieve the same results as if all data were shared. As demonstrated in the real data example section, each class does not need to provide all the student scores, but rather just two summary statistics, $T_{\\beta}^{(k)}$ and $T_{\\gamma}^{(k)}$. Therefore, our method ensures privacy protection by not requiring the transfer of sensitive individual data.\n\nWe appreciate your concern, and while we did not explicitly discuss information leakage in the paper, we believe that the use of summary statistics mitigates such risks.\n\n\n**Question 1:** Does the author have any idea or predictions on how the results would be if $K$ is larger than $o(\\sqrt{N})$?\n\n**Answer:** Thank you for your insightful question.\nThe condition $K = o(\\sqrt{N})$ is crucial for deriving the limiting distribution of the proposed distributed statistics under the null hypothesis and for determining the power function under the alternative hypothesis. When $K$ exceeds this threshold, the cumulative bias from each machine's estimated $T_{\\beta}^{(k)}$ and $T_{\\gamma}^{(k)}$ can no longer be ignored. This could potentially lead to reduced power, and the Type I error rate may become uncontrollable. \n\nIn the revision, we examined the case where $K = 64$ and observed that the difference between our method and the global method became more pronounced. This further underscores the importance of the $K = o(\\sqrt{N})$ condition.\n\n**Question 2:** Was the privacy discussion omitted due to technical difficulties or was it taken care of inside the framework already?\n\n**Answer:**  Thank you for your question. We have already addressed this issue in \\textbf{Weakness 3}.\n\n**Reference**\n\n\n[1] Dai J Y, Stanford J L, LeBlanc M. A multiple-testing procedure for high-dimensional mediation hypotheses[J]. Journal of the American Statistical Association, 2022, 117(537): 198-213.\n\n[2] Huang Y T. Genome-wide analyses of sparse mediation effects under composite null hypotheses[J]. The Annals of Applied Statistics, 2019, 13(1): 60-84.\n\n[3] Liu Z, Shen J, Barfield R, et al. Large-scale hypothesis testing for causal mediation effects with applications in genome-wide epigenetic studies[J]. Journal of the American Statistical Association, 2022, 117(537): 67-81.\n\nWe thank the reviewer again for the helpful comments and suggestions for our work. We are more than happy to address any further questions that you may have."}, {"review_id": "eqFeToBVsL", "reviewer": "Reviewer_rUFx", "summary": "This 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.", "questions": "1. 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.", "limitations": "Please see section questions.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 2, "strengths_and_weaknesses": "### **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.", "was_revised": true, "final_justification": "I have reviewed the authors' response to my initial comments and am satisfied with their revisions. They have addressed my concerns thoroughly. As a result, I will raise my overall score to reflect the improvements made.", "rebuttal": "We thank the reviewer for the careful reading and thoughtful comments. We address the reviewer's questions in the following. We hope the responses below address the reviewer's concerns.\n\n**Weakness 1:** The underlying split-and-conquer framework is not new. Prior work (e.g., Battey et al.,2018, Zhao et al., 2019) has extensively studied distributed statistical testing, and this paper appears to be a direct extension of existing methods to mediation models.\n\n**Answer:** Thank you for your insightful comment. We would like to clarify that this paper is not a direct extension of existing methods. \n\nFirst, the hypothesis testing problems addressed in the referenced papers are different from our approach. \nBattey et al. (2018) focus on testing whether a specific coefficient $\\beta_v$ in a high-dimensional generalized linear model is equal to a given value, such as testing $H_0: \\beta_v = 0$. Zhao et al. (2019) test whether the regression function $E(y|X)$ equals a given parametric model, i.e., $H_0: E(y|X=x) = g(x; \\theta_0)$ for some $\\theta_0 \\in \\Theta$. In contrast, our paper addresses the testing of the product of two coefficients in a structural equation model, i.e., testing $H_0: \\beta \\gamma = 0$. \nFor different testing problems, one must construct specific test statistics and carefully examine their statistical properties to derive rejection regions and power functions.\n\nAdditionally, Battey et al. (2018) and Zhao et al. (2019) construct distributed test statistics by taking a weighted average of local test statistics, a commonly used method, but not necessarily the optimal one. In contrast, our study develops distributed Sobel and MaxP test statistics without relying on simple weighted averaging. In our revised simulations, we added comparisons with the weighted average method and found that our approach significantly outperforms it. When $N$ is fixed, the power of the weighted average method decreases sharply as $K$ increases, while our method remains almost unaffected, achieving power nearly equivalent to that of the global method. This demonstrates the innovative nature of our approach.\n\nWe hope this clarifies the novelty of our method. We will update the manuscript to highlight these distinctions more clearly.\n\n**Weakness 2:** 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\n**Answer:** Thank you for your thoughtful comment. Indeed, there are several improvements that can be made to our method, such as the HDMT method proposed by Dai et al. (2022) and the JT-comp method from Huang (2019), both of which are well-suited for high-dimensional mediators.\nSuppose there are $J$ candidate mediator variables $M_1, \\dots, M_j$, leading to $J$ hypothesis tests $H_{0j}: \\beta_j\\gamma_j=0$. Let $T_{\\beta,j}$ and $T_{\\gamma,j}$ represent the test statistics for the $j$-th test. \n\nHuang (2019) constructs a p-value as follows:\n\n$$\n\\widehat{p_{comp,j}} = F\\left( \\frac{T_{\\gamma,j}T_{\\beta,j}}{\\text{Var}(T_{\\gamma})} \\right) + F\\left( \\frac{T_{\\gamma,j}T_{\\beta,j}}{\\text{Var}(T_{\\beta})} \\right) - F\\left( \\frac{T_{\\gamma,j}T_{\\beta,j}}{\\text{Var}(T_{\\gamma}) + \\text{Var}(T_{\\beta})} \\right),\n$$\n\nwhere $F(z) = 2 \\int_{|z|}^{\\infty} \\frac{K_0(z)}{\\pi} \\, dz$, and $K_0(z)$ is the Bessel function with order $0$. The sample variances of $T_{\\gamma,j}$ and $T_{\\beta,j}$ across all tests are used to estimate $\\text{Var}(T_{\\gamma})$ and $\\text{Var}(T_{\\beta})$.\n\nIn the distributed framework, we can replace $T_{\\gamma,j}$ and $T_{\\beta,j}$ with the averages $\\overline{T_{\\gamma,j}} = \\sum_{k=1}^{K} \\sqrt{\\dfrac{n_k}{N}}T_{\\gamma,j}^{(k)}$ and $\\overline{T_{\\beta,j}} = \\sum_{k=1}^{K}\\sqrt{\\dfrac{n_k}{N}} T_{\\beta,j}^{(k)}$,  respectively. This is precisely the formula in equation (4) of the article.\n\nAdditionally, Dai et al. (2022) propose the HDMT method, which constructs the test statistic:\n\n$$\nT_{HDMT,j} = (\\widehat{\\pi_{01}} + \\widehat{\\pi_{10}}) T_{MaxP,j} + \\widehat{\\pi_{00}} T_{MaxP,j}^2,\n$$\n\nwhere $\\widehat{\\pi_{01}}$, $\\widehat{\\pi_{10}}$, and $\\widehat{\\pi_{00}}$ are the estimated proportions of $(\\beta_j = 0, \\gamma_j \\neq 0)$, $(\\beta_j \\neq 0, \\gamma_j = 0)$, and $(\\beta_j = 0, \\gamma_j = 0)$ across all $J$ tests, respectively. These proportions can be estimated using non-parametric methods (e.g., Storey, 2002). \n\nIn the distributed framework, $T_{MaxP,j}$ can be replaced by the distributed test statistic $T_{MaxP,j}^{Dis}$, and $\\widehat{\\pi_{index}} $ can be approximated by the simple average $\\frac{1}{K} \\sum_{k=1}^{K} \\widehat{\\pi_{index}}^{(k)}$, for $index=00,01,10$.\n\nThe performance and statistical properties of the distributed test statistics constructed in this way require further study, which will be addressed in future work.\n\n**Weakness 3:** 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). \n\n**Answer:**  Thank you for your insightful comment. In genomic settings, where multiple candidate mediators $M_j, j=1,\\ldots,J$ may exist, a common approach is to test each mediator $M_i$ individually in the model to check for mediation effects, then move to the next mediator $M_{i+1}$ and continue the testing process until all mediators have been evaluated. By controlling the false discovery rate (FDR), one can identify which mediators exhibit significant mediation effects. An example of this approach can be found in Liu et al. (2022), in the Normative Aging Genome-Wide Epigenetic Study. Clearly, this represents a multiple hypothesis testing problem, and our method can be directly extended to handle such scenarios. Furthermore, we can improve the power of our approach by incorporating methods like DACT from Dai et al. (2022) and JT-comp method from Huang (2019), as detailed in the response to Weakness 2.\n\nHowever, there is also another situation where multiple mediators $M_j, j=1,\\ldots,J$ are included in a structural equation model (SEM), and interactions may exist among these mediators. In this case, the structural equations are given by:\n\n$$\nY = \\beta_0 + \\beta_A A + \\boldsymbol{\\beta}^T M + \\boldsymbol{\\beta_X}^T \\mathbf{X} + \\epsilon_Y,\n$$\n$$\nM_j = \\gamma_{0j} + \\gamma_j A + \\boldsymbol{\\gamma_{Xj}}^T \\mathbf{X} + \\epsilon_{Mj}, \\quad j=1,\\ldots,J.\n$$\n\nIn this setting, one would need to test the hypothesis $H_0: \\boldsymbol{\\gamma}^T \\boldsymbol{\\beta} = 0$, where $\\boldsymbol{\\gamma}^T = (\\gamma_1, \\ldots, \\gamma_J)^T$. This scenario is more complex because, even when $\\boldsymbol{\\gamma} \\neq 0$ and $\\boldsymbol{\\beta} \\neq 0$, their inner product may still be zero. Moreover, the distribution of $\\hat{\\boldsymbol{\\gamma}}^T \\hat{\\boldsymbol{\\beta}}$ is challenging to compute (Preacher et al., 2008). This falls outside the scope of the current paper, but it is an interesting direction for future research. We have included this discussion in the final section of the revised manuscript.\n\n\n**Questions 1-3** have been already addressed  in the response to Weakness 1-3.\n\n**Question 4:** Please add comparisons with naive distributed implementations (e.g., averaging p-values) in simulation studies to emphasize the advantage of the proposed methods.\n\n**Answer:** We truly appreciate your recommendation to compare our approach with naive distributed implementations in the simulation studies. In the revised version of our study, we included comparisons with two weighted averaging methods:\n\n$$T_{Sobel}^{ave} = \\sum_{k=1}^{K} \\sqrt{\\frac{n_k}{N}} T_{Sobel}^{(k)},$$\n$$T_{MaxP}^{ave} = \\sum_{k=1}^{K} \\sqrt{\\frac{n_k}{N}} T_{MaxP}^{(k)}.$$\n\nWe found that both of these naive methods showed much poorer performance compared to our proposed method. For a fixed $N$, as the value of $K$ increased, the power of both averaging methods noticeably declined. In fact, their performance only slightly surpassed the results obtained from using the $T_{Sobel}^{(1)}$ and $T_{MaxP}^{(1)}$ statistics from just the first machine. When $K = 8$, the power of the averaging methods dropped to below 0.1, while the power of our proposed method remained identical to the global test.\n\nThese results further emphasize the robustness and efficiency of our method, showcasing how it maintains strong performance even as $K$ increases. We hope that this comparison adds more clarity and context to the advantages of our approach.\n\n**Reference**\n\n[1] Battey H, Fan J, Liu H, et al. Distributed testing and estimation under sparse high dimensional models[J]. Annals of statistics, 2018, 46(3): 1352.\n\n[2] Dai J Y, Stanford J L, LeBlanc M. A multiple-testing procedure for high-dimensional mediation hypotheses[J]. Journal of the American Statistical Association, 2022, 117(537): 198-213.\n\n[3] Huang Y T. Genome-wide analyses of sparse mediation effects under composite null hypotheses[J]. The Annals of Applied Statistics, 2019, 13(1): 60-84.\n\n[4] Liu Z, Shen J, Barfield R, et al. Large-scale hypothesis testing for causal mediation effects with applications in genome-wide epigenetic studies[J]. Journal of the American Statistical Association, 2022, 117(537): 67-81.\n\n[5] Preacher K J, Hayes A F. Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models[J]. Behavior research methods, 2008, 40(3): 879-891.\n\n[6] Storey J D. A direct approach to false discovery rates[J]. Journal of the Royal Statistical Society Series B: Statistical Methodology, 2002, 64(3): 479-498.\n\n[7] Zhao Y, Zou C, Wang Z. A scalable nonparametric specification testing for massive data[J]. Journal of Statistical Planning and Inference, 2019, 200: 161-175.\n\nWe thank the reviewer again for the helpful comments and suggestions for our work. We are more than happy to address any further questions that you may have."}]

Author rebuttal:

Meta-review: This paper considers the task of performing causal mediation analysis in the federated setting where data are stored across multiple machines due to either storage limitations or privacy concerns. To the best of mine and all reviewers' knowledge, this problem setup is interesting, important, and quite novel, with no previous paper considering it. One reviewer described it as "highly understudied yet critical".

The paper builds proposes distributed versions of two classic mediation tests and proves that the proposed distributed variants achieve the same power as the non-distributed versions asymptotically, and are moreover highly communication efficient, requiring only a single round of communication. The reviewers all agreed that the proposed tests were "simple and practical" and "principled" and that the work as a whole showed "strong novelty".

In addition to synthetic studies that corroborate the theoretical results, the paper also includes an interesting case study on real-world education data, asking whether the effect of high school mathematics education on performance in college-level statistics courses is mediated by knowledge of calculus. Although the reviewers did not say much about this case study, I think it really strengthens the paper and illustrates the method's applicability.

The reviewers posed several possible weaknesses, including possibly unjustified assumptions and relation to previous work that was underdiscussed. However, the authors offered highly detailed responses which resolved these concerns in all cases.

I think this is a clear accept. I am recommending for a poster (rather than spotlight/oral) for the sole reason that the setting, while novel, seems quite niche. That said, by average score it's among the higher ones in my batch, so I would not at all be opposed to bumping it up.
Decision: Accept (poster)