AMDP: An Adaptive Detection Procedure for False Discovery Rate Control in High-Dimensional Mediation Analysis
Authors: Jiarong Ding
Xuehu Zhu
Venue: NeurIPS 2023 spotlight (2023)
Keywords: Mediation analysis
Composite null hypothesis
Local false discovery rate
Optimal ranking rule
High-dimensional

Abstract: High-dimensional mediation analysis is often associated with a multiple testing problem for detecting significant mediators. Assessing the uncertainty of this detecting process via false discovery rate (FDR) has garnered great interest. To control the FDR in multiple testing, two essential steps are involved: ranking and selection. Existing approaches either construct p-values without calibration or disregard the joint information across tests, leading to conservation in FDR control or non-optimal ranking rules for multiple hypotheses. In this paper, we develop an adaptive mediation detection procedure (referred to as "AMDP") to identify relevant mediators while asymptotically controlling the FDR in high-dimensional mediation analysis. AMDP produces the optimal rule for ranking hypotheses and proposes a data-driven strategy to determine the threshold for mediator selection. This novel method captures information from the proportions of composite null hypotheses and the distribution of p-values, which turns the high dimensionality into an advantage instead of a limitation. The numerical studies on synthetic and real data sets illustrate the performances of AMDP compared with existing approaches.

AMDP: An Adaptive Detection Procedure for False Discovery Rate Control in High-Dimensional Mediation Analysis
Jiarong Ding School of Mathematics and Statistics
Xuehu Zhu' School of Mathematics and Statistics Xi'a an Jiaotong University zhuxuehuOxj tu. edu. en
an Jiaotong University ajr99010stu. xjtu. edu. cn
Abstract
nigh-dimeensioal mediation analysis is often associated with multiple testing problem for detecting significant mediators. Assessing the uncertainty of this de- tecting process via false discovery control the FDR in multiple testing, two essential steps are involved: ranking and selection. Existing approaches either construct p-values without calibration dis- regard the joint information across tests, leading to conservation in FDR control non-optimal ranking rules for multiple hypotheses. In this paper, we develop adaptive mediation detection procedure (referred to "A "AMDP") to identify relevant mediators while asymptotically controlling the FDR in high-dimensional mediation analysis. AMDP produces the optimal rule for ranking hypotheses and proposes a data-driven strategy determine the threshold for mediator selection. This novel method captures information from the proportions of composite null hypotheses and the distribution of p-values, which turns the high dimensionality into advantage instead of a limitation. The numerical studies on synthetic and real data sets illustrate the performances of AMDP compared with existing ap- proaches.
(FDR) has garnered great interest. To
Introduction
Mediation analysis is regarded as a prevalent tool to dissect a mediation relationship between expo- sures and outcomes, and it has been widely applied in different fields, such epidemiology [37]] public health [21]]. policy evaluation [M]] social sciences [25]], neuroscience [[C]], and many oth- ers. Baron et al. [BI] provided the basis for the advance of mediation analysis. They proposed conventional regression- based approach, commonly referred to as the causal steps method, to exam- ine the logical relationships among exposure, mediator, and outcome variables linking in causal chain. While the causal steps method established necessary conditions for causal inference, did not provide joint test of the indirect effect of exposure the outcome through mediator, as recommended by MacKinnon et al. [32]. Thus, MacKinnon et al. [32] investigated the normality- based Sobel's test [[88] means detect mediation effects under the framework of the product- coefficient method, the product the exposure-mediatte and mediator-outcome effects. However, highlighted by MacKinnon et al. [[I]], the distribution of product coefficients is not normally dis- tributed, leading the Sobel's test being overly conservative. Hence, MacKinnon et al. [62] proposed the joint significance test (also name as the MaxP test) alleviate this conservatism. Nonetheless, the joint significance test still suffers from low statistical power, as it overlooks the impact of the composite null structure in mediation analysis [(33, 82).
Corresponding author
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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To address the aforementioned issues, Taylor et al. [ recommended the utilization of the distri- bution of the product method or bootstrapping as alternative procedures. These methods had been shown to exhibit higher power while still maintaining reasonable control over the Type I error rate. In the study of Kamukama et al. [27]], a z-value based Sobel's test was introduced to investigate the mediation effect of competitive advantage on the relationship between intellectual capital and finan- cial performance. Nuijten et al. [G6]] proposed an approach based on Bayesian models for testing the presence of an indirect effect. More recently, Zhang [H6]] developed two data-adaptive tests that outperform both Sobel's 's test and joint significance tests.
Despite these advances, a further challenge is that the methods mentioned above typically assume a -owddimensional mediator, whereas the development of high-throughhn technologies promotes a growing need for research dealing with high- gigh-dimensional data. The multiple testing problem arising from high-dimension mediation analysis is aimed at identifying the relevant mediators that explain the effect of exposure on an outcome. Differ from a single test, multiple testing approaches are aimed to handle simultaneous testing, and often use the false discovery rate (FDR) to measure the uncertainty of detecting process [[]], that is,
# {j: Do.JE E }}
FDP
FDR E(FDP),
#(jEβJVI
where j represents the index of mediator, Au is the index set of the null mediators, Â is the index set of selected mediators.
The control of FDR in multiple hypothesis testing consists of two primary steps: ranking and se- lection. In the initial step, a ranking statistic is calculated to evaluate the significance of each test, resulting in the ranking of hypotheses. The subsequent step is to maximize the selection set based on the established ranking order from the first step, while simultaneously maintaining the FDR at the target level. Although the p-value generated from the single hypothesis testing method can be con- sidered as a ranking statistic and further combined with the BH procedure [[]] for FDR control, there are still some limitations inherent to this approach. Firstly, the single p-value without calibration may be conservative and lead to excessive conservation in FDR control. Secondly, in the absence of joint information across multiple hypotheses, the ranking statisties based on the p-value may not be optimal.
To overcome these limitations, recent advancements in mediation analysis attempt construct cali- brated test statistics that account for the composite nature of the null hypothesis, and further achieve FDR control in multiple testing. For example, Dai et al. [I22] developed a JS-mixture procedure, which utilizes the maxP statistics [62]] as the ranking statistics and corrects the conservatism in the joint significance test by estimating the mixture distribution of p-values. Although JS-mixture sharply achieves FDR control, it is still underpowered for the reason that the maxP statistics does not account for the distribution information of two-dimension p-values under different hypotheses. Furthermore, Liu et al. [GM] proposed a procedure called DACT focusing on constructing calibrated p-values for each single test by combining information across large-scale tests. Specially, DACT estimates the proportions of sub-null hypotheses and generates weighted p-values accordingly. How- ever, DACT may suffer from underpowered performance in certain situations due to the following reasons. Intuitively, it ignores the alternative hypothesis information in its weighting scheme. On a deeper level, it does not fully consider the distributed information of wwo-dimensional p-values, limiting its ability to leverage valuable insights for improved power.
In this paper, we propose an adaptive mediation detection procedure (AMDP) for identifying rele- vant signals in high-dimensional mediation analysis. Our main contributions are summarized below:
AMDP utilizes a two-dimensional p-value based local FDR as a test statistic, allowing for the comprehensive utilization of structural information of large-scale tests. Additionally, it determines the optimal rule for the order of selecting mediators.
We establish theoretical results showing that AMDP enables asymptotic control of the FDR for selected mediators using the estimated local FDR.
We reveal the critical importance of information retention in the ranking step for achiev- ing optimal statistical power by discussing the limitations of the ranking statistics used in existing methods and conducting comparisons with AMDP.
We empirically demonstrate the effectiveness of AMDP on synthetic and real data sets. Simulation results confirm the validity of our approach, and an application to prostate

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cancer dataset illustrates its satisfactory performance in identifying CpG methylation sites that mediate between risk SNPs and gene expression.
The remainder of this paper is organized as follows: ing rule based on AMDP along with an estimator local FDR. We theoretically prove the ability of AMDP in controlling the FDR while mimicking the optimal power. Section & presents the simula- tion studies to evaluate the performance of AMDP. In Section EL we demonstrate the practical utility of AMDP by applying it to the prostate cancer dataset in TCGA 2015. We conclude the paper in Section B The technical proofs and additional discussion are postponed to Appendix.
Section / formally introduces an optimal rank-
2 AMDP: an optimal multiple testing procedure for FDR control
Let X be the exposure (independent variable), {M,..... MM be candidate mediators, and Y be the outcome (dependent variable). In the context of mediation analysis in the genome-wide association studies (GWASs), often refers the single nucleotide polymorphisms (SNPs), M corresponds to DNA methylation, and Y pertains to gene expression or risk of disease. As stated by Baron et al. [ the mediation relationship can be expressed by the following models:
E (M, |X) cuj + axxX, E (Y Mj = Po5 + 3,M, + 31
where a denotes the effect of on Mj. 33 represents the effect of M, on Y, totally the product of aj and B, represents indirect effect of X on Y. 31 is the direct effect of X on Y with the M, being fixed. We assume that there are no unmeasured confounding variables, also known as the sequential ignorability assumption [25, [Σ]. Any confounders can be adjusted by additional covariates [[2)]
and such an adjustment omitted in model (2) for simplification.
Testing whether [M....., Mr} plays an intermediary role in the causal path from X to Y in (L) can be transformed into a multiple testing problem:
Hoj ay 0 versus Hıj 0j,3 +
The above composite hypothesis can be decomposed into four disjoint cases follows:
Case 1, Hoo. Og = 0 and
3, = 0,
Case 2, Hot
ag = 0 and
B, X O,
Case 3, H1o
ag * O and
B, = O,
Case 4, Hig:: ag # 0 and
B, * O,
where Case 1-3 represents the composite null hypothesis, and Case 4 is the alternative hypothesis. rejection of Hoj indicates the presence of a mediation effect by Mj. In this paper, the p-values for testing aj
and By 0 are respectively denoted Pıj = 2(1 Φ(1 å 1/0a,) }} and
P2j = 2(1 (1 3, 1/āg,)] where aj. 8y.. da, and og, are the least squares estimators based on the models (2)) Under the sequential ignorability assumption for mediation analysis, Pış and P2j independent [30].
2.1 Optimal rejection region under the four-group model
To provide an optimal ranking guideline, consider a pair of p-values p, (P1j. P2j) under the empirical null inference framework [] Let Hoo.; = 1 Case holds, H0l,; = 1 if Case holds, H1o, = 1 if Case 3 holds, and H11. otherwise. Assume the conditional density of p, follows
Hoo.J ~ Bernoulli }m
foo (p)
if Hoo.; = 1,
Hor,5 Bernoulli (mo1))
So1 (p)
if HoL = 1,
H, =
P5 Hy
(4)
Hooo Bernoulli {m10),
f1o (p)
if H10, = 1,
Hug Bernoulli (mn),
In (p)
if Hug
with (v (2) E R22 Too. Tor, moo, and represents the proportions of Hoo,j. Hor.j. Hioj and H1,5, respectively. It follows that Top + Toi + 3 + æ1 due to the disjoint nature of the composite hypothesis.
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Then, the density function of pj follows the following four-grop group model, which can be considered as variant version the random mixture model [[7]].
~ fp) = Too + Fon (p) + (p) + (p))
(5)
Under the four-group model (5), the local FDR [[N [1] is defined as
fdr(p) = P (Hoo.) U Hor,j U H1o, Pj = p) Too + mon fon (p) + mio/1o(p)
(6)
f(p)
It refers to the posterior probability that a hypothesis is null, given its corresponding p-value.
Before delving into the optimal ranking guidelines, we introduce several key definitions relevant to this objective. For any rejection region S E [o, 1)2, we define the global FDR as
gFDR(S) = Poo U Hoi U H10 = Pj E S);
where Hoo, Ho1, and H1 composite null hypothesis. The power is defined as Power(S) (p E Hn =).
In the ranking step, the primary objective is to establish an optimal ranking rule that accurately reflects the significance order of the tests, while adhering to the optimality goal set in the selection step. Under the Neyman-Peane Pearson framework [[5]] this optimality goal entails maximizing power while simultaneously controlling the global FDR at targeted level of a. This process can be formulated as a constrained optimization problem, i.e.
max (S) subject to gFDR(S) <
(9)
The optimal rule under the two-group model has been extensively studied in the literature. Re- searchers have proposed various methods for optimal decision-making based on different frame- works [S. g]. Our optimality goal shares similarities with the work of Lei et al. [28]. They had demonstrated that, under Bayes rule, the optimal rejection thresholds are the level surfaces of local FDR. We extend this insight p-values in two dimensions, and define the form of the rejection region as S(C) = {p= fdr(p) S C). A detailed and comprehensive explanation of this concept is provided in Theorem MI Theorem 1. Assume that
(i) foo(p), (p), fiolp), and fú (p) are continuous; (ii) v(p fdr(p) =
for any t (0, 1}, where v is a Lebesgue-Stielyg measure on the two two-dimensional Borel space
Then, for any given global FDR level ax there exists unique value Ç such that S((*) is the solution of the constrained optimization problem in (2). And the local FDR involved in S(C*) corresponds to the optimal ranking rule.
Remark 1. Genovese et al. (20) have shown that under weak conditions, gF EF = FDR +0 + where J represents the number of mediators. Hence, controlling gFDR and FDR are asymptol- ically equivalent as the number of mediators J tends to infinity. A similar result supporting this equivalence was also obtained by Storey [80].
2.2 The estimator of local FDR
From Theorem we have established that the optimal ranking rule under the Neyman-Pearsee framework is the local FDR. However, it is worth noting that the discussions in Section 2.1 are based on the assumption that the distribution of p-values and proportions of the composite hypothesis are known. In the following, we emphasize that our results still hold if the local FDR can be consistently estimated. Assuming that foo (p-values follow uniform distribution Case 1). The estimation of fdr(p) can be divided into three parts: (i) The proportions of the composite null hypothesis
Too, Tol, Tio; (ii) The mixture density fp) (iii) The densities of the composite null hypothesis for (p). (p)
Motivated from Storey et al.
Ro1, 110. and Too can be estimated follows:
(pi A) in (A) = J(1 - X)
o = Σ (p2j A) J(1 - 2)
#oo E((pi > A2p > A)
J(1 - 2)2
(10)

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where no. (A) denotes the estimator of the proportion of null P1j (A) denotes the estimator of the proportion of null P2j. I(-) is an indicator function, and [0, 1) is tuning parameter. In practice, there is a bias versus variance tradeoff for choosing suitable Å. Further research on selecting an appropriate value of À is detailed in the Appendix A. Following that
#o (A) = To. (A) -A), (A) R.o(A) - Too (A).
(11)
Next, turn to the estimation of fp). For this purpose, employed adaptation of the beta
kernel function proposed by Chen [[]]]. This choice is made considering the fact that p (p , p(2) falls within [0, 1)2. The beta kernel function allows for a flexible and smooth estimation of f(p).
providing suitable estimation approach for our analysis. Our beta kernel estimator is:
= - 160 (2)) J-2
Σ K Kon,, (koossorss)
Σ K (hosssops))
(12)
where K2,6 is boundary beta kernel defined
Kp/0,(1-p//000) (1) = K Kolob),((-)//001)
if E (2b, 1 - 26), if E [0,26], if 2b, 1],
(n/b.o(1-p)(t)
Ku,o be the density function of Beta(u, v) random variable, b is smoothing parameter, and p(p, b) 2p² 2.5 /4p* + 6p² 2.25 pp² p/b.
In the context of mediation analysis, the density of p-values follows a mixture distribution, as indi- cated in (5) This mixture distribution involves three distinet types of null hypotheses: Ho1, Hio- and Hoo- Distinguishing between Hor and H10. as well as obtaining accurate estimators for foi (p) (corresponding to Hot) and (p)(p) (corresponding to H1)) is indeed challenging task. Motivated by the knockoff method [2], we consider leveraging the symmetry property of p-values under the composite null hypothesis to tackle this issue. Before diving into the details of utilization of the symmetry property for estimating for (p) and fIo(p), we introduce some essential notations and assumptions.
Denote Auo. Ao1. and A10 as the index set of the null mediators under the composite null hypothesis Hoo. Hoi. and 1o,, respectively. We define the region D as [0, 0.5) with its symmetric regions as follows: Do [0.5, 1] [0,0.5). Dio = [0, 0.5) [0.5, 1]. and Doo = [0.5, 1|2 The assumptions are given as follows.
Assumption 1. For j E Aoo. the sampling distribution of Pj is symmetric about p = 0.5 and
p(2)) 0.5; For j E Do1. the sampling distribution of Pj is symmetric about p(1) = 0.5; For
j E A1v. the sampling distribution of P3 is symmetric about p 0.5.
Assumption 2. The symmetric regions of D satisfy: (i) For p
Doo. limn fu (p) O. 0, lim,-+oo f1o (p) 0: (ii) For p Doz. limn-soo fú1 (p) 0 and
lim, foi (p) lim, f10 (p) 0; (iii) For D10. lim,- f11 (p) and lim, Foi (p) 0.
Assumption 01 is only required for the null mediators. It indicates that at least one of pı; and p2; follows a uniform distribution under the composite null hypothesis. Assumption Q holds for any reasonable p-value. Since a non-null p-value should fall within [o, 0.5), we can infer that as the sample size tends to infinity, the probability of p- p-values under alternatives falling within [0.5, 1] approaches zero. Additional explanations on Assumptions M-D are detailed in the Appendix D.2.
Remarkably, under Assumption II we can decompose fio(P) and for (p) as follows:
f10(P) = fi.(p fo(v (1)) for
()) ()())))) (z
(13)
where (1)( fo)(p(2)) 1. This decomposition allows to transform the problem into estimating (1) and (2)),,,, representing the marginal probability density of P1, pnd P2ý under alternatives, respectively. Assumption provides the inspiration to utilize the symmetric regions about D to address this estimation task, that is,
11 (2) Too + 11051.((())
Too + ( (2(p)2))=
(14)
Too T10
Too + Tul
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where h1 () denotes the density of Pıj under H10 and Hoo, and k2 (p(2)) denotes the density of P2j under Hot and Hoo.
We apply the beta kernel function to Pıù in region Dio and Doo, as well as to Paj region Don and Doo. to get the estimation of k1 (p())) and (2) as
J1 (D( = J Σ S0, (p1s), k2(1 = J2
J Ke (P25),
Σ j=1
(15)
where J1 is the number of p-values in region Dig and Doo. J2 is the number of p-values in region Doi and Doo Combining the estimators in (II)) (T3) (I5), we obtain the estimation of fi. (p!))
and f.1 (p(2)) as:
(p (#00 + 11o) - #00
fa(p(2)) (ooo + mos (p2(2)) At0
(16)
110
A01
Therefore, the local FDR estimator derived as:
får(p) foo + non for (p) + fnofiol))
(17)
f(p)
2.3 Asymptotic FDR control
In this section, we present a selection strategy for the second step. The primary goal of our proposed selection strategy is to maximize power while simultaneously controlling the FDR based on the established ranking order from the first step, i.e., finding the optimal threshold X* of the optimization problem (9)). However, determining such an optimal threshold (* is a challenging task, as it involves decision-making based on the estimation of global FDR. To address this challenge effectively, we propose a data-driven strategy. Based on the notations in Section 2.2, the form of the FDP and FDR are given by
(DP = # {j:jE Doo U Aoi U A1o.J E A
and FDR(() = E(FDP(()), (18)
#U:jEEJVi
where Å is the index set of selection in rejection region S(() = {p: får(p) - {). i.e., j E â when Py S()), the denominator of FDP(() represents the total number of rejections and the numerator represents the number of false positives.
In mediation analysis, accurately estimating the number of false discoveries in (L) poses chal- lenge, since the rejection region S(() comprises mixture of four distinet types of hypotheses. However, we can draw inspiration from Assumptions I and Ø to leverage the symmetry property of p-values under the composite null hypothesis estimate the number of false positives. We define
the symmetric regions of S as So1 = ((1-p),,(C)
(dr(p) S (). Sio = ((D(), p(2))= fdr(p) S (). Soo ((1 (1) 1 1 (2)) ffr(p)) S 5}. It's noteworthy that the rejection region $ is a subset of the region D defined in Section 2.2, and its symmetric regions, So C. Dov S10 S Dio. and Soo C. Don- Indeed, Assumptions II and @ provide us with an approximation of the number of false positives in (I))
# #fi E Ao U Apo E $(c) # {i E Aon U Auo:py E Sur(C)) = # (i e Son(S)}.
# j E AIO Aon:; E 8()) # {i
J Aio U Aoo: E ((o(()) ~ # {i:pz e S1o(3)).
f E 8(() # {ie Ao0: E ( # ji:p; E S
The number of the selection # {i [:: ps E Son(5) # {i pi S1o(3) - # {i p; E Suv(() +1 can be considered as an overestimation of # {i ; E Aoo U Aoi Ud E }.. and FDP(() is given by
# #fD:Ds e Sol(5) + # {i :p E Su((s) - # (i pipps E Suo(C) +1
FDP(()
(19)
#(i : P3 E ((())v

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Then the data-driven cutoff 5* can be determined as follows:
5 sup(c > 0: FDP(() s a))
(20)
and the final selection Aç. = {i pp E }
Finally, we summarize our proposed FDR control procedure in Algorithm M1
Algorithm A data-driven algorithm for FDR control.
1: Calculate pair of p-values = (P1j. P25) following model (L),, where j = I.....JJ
2: Estimate the proportions of the composite null hypothesis #oo, for, $10-
3: Estimate the null densities (p) fio(p) and the mixture density f(p) using the adaptation of the beta kernel estimator.
4: Estimate the fdr(p) following (L
5: For nominal FDR level E (0, 1), select the mediators {5:p1 {((5)) where S(C) {p-
fdr(p)= S {} and the cutoff 5* is
# # E + # # E Bu((1) # #hip E Suo(C) +1 S a}.
<' =s=sup(< > 0 FDP(()
=
# Py E S(C))
Remark 2. In recent work of Deng et al. [U33, a procedure called JM was introduced for de- tecting simultaneous signals across multiple independent experiments. The core idea behind JM is to partition the region of p-values into masked and unmasked areas, and then utilize p-values from each of these regions to estimate FDR and local FDR, respectively. By leveraging the par- tially revealed information from the unmasked area, JM updates the rejection region in the masked area iteratively until it reaches the desired FDR level. In contrast to the stepwise updates in the JM procedure, the AMDP does not require such iterative adjustment. By leveraging informa- tion from large-scale testing, AMDP can accurately estimate the local FDR. Additionally, moti- vated by the symmetric property of the composite null hypothesis,
proposed data-driven al- gorithm to determine the optimal rejection region. As shown in (T9), the number of the selection {s::; B1o(S)] } { :p Suo(C) 1, provides less conservative estima- tion of # {j j E Aao U Aoi u A1o.j E 2} than JM, which relies on conditional mirror conservation to estimate FDP masked region.
# {1= [j:P) Ps (or(S))
Theorem Q below shows that for any nominal FDR level a E (0. 1), both FDP((") and FDR((*) under control using Algorithm M. as the sample size and the number of mediators J tend to infinity.
Theorem 2. Assume that
(i) E fdr(p,) fdr(p;)) P. 0 n, n, oc;
(ii) For Ç (0, 1). P (fdr(p)) < slij E Aoo U Doi U (10) is continuous;
(iii) For any FDR level of E (0, there exists constant Sa (0, 1] such that P (FDP (Sa) ( a) + 1 as J oc.
When Assumptions 0ε holds, we have FDP (C*) S a + (1) and lim (C*) S a. n,J-+00
Proofs of Theorems MH are given in Appendix E.
Remark 3. To demonstrate the effectiveness of our proposed method, we provide examples un- der two scenarios that highlight how the loss of associated information across tests result in decreased power. We compare our method, AMDP, with the JS-mixture test[U2] and the DACT test/B0], which provides further evidence for the utility of AMDP. Due to space limitations, this section is postponed to the Appendix B.
Simulation Study
In this section, we conduct thorough set of simulations to assess the performance of our proposed method AMDP. For a comprehensive comparison, we evaluate two competing methods, the JS- mixture a and the DACT [30]. The DACT method consists of two variants: DACT (Efron) and
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DACT (IC). The R implementations of JS-mixture and DACT can be found at https://github Com/ cran/HDMT and https:/ com/ /ooaggaaaaaaa respectively. The exposure X is simulated from Ber(0.5), then the mediator Mj and the outcome Y, are generated as follows:
Y, = 3,M, t, 10,
M, - a,x + eg. ((, 1).
We respectively calculate the FDP and true discovery proportion (TDP) follows:
FDP = # #(i:jEAo.JEA #
TDP = # # Do.j E }
(21)
where Ao is the index set of the composite null mediators, Â is the index set of selected mediators. The FDR and power are measured by averaging FDP and TDP over 200 replications, respectively. Let T be the mediation effect size parameter. We utilize the following six examples to conduct a comprehensive comparison of the FDR and power for the four procedures.
Example 1. We fix (n, J) = (1000, 10000), (TrO. Toi. T10 111) = (0.85, 0.05, 0.05, 0.05), and vary T from 0.6 to 1.2. Under Hoo. aj 0 and 33 O: under H01. 0g = 0 and 3 = 0.37; under H10. 02 = 0.2T and 3, = 0; under H11. aj = 0.27 and B, = 0.3r:
Example 2. We fix (n, J) = (1000, 10000), (Too. Tol, Ti0, T11) = (0.4, 0.2, 0.2, 0.2). and vary from 0.6 to 1.2. Under Hou. aj
and 8 = under Ho1. 0j = and 8, 0.3T; under H10. aj = 0.27 and 33 = O; under H1,, 0j 0.2r and 53 = 0.3r:
Example 3. We fix (n, (1000, 1). (Ton, Toi, Ti0, T1)
(0.85, 0.05, 0.05, 0.05). and vary
J E (5000, 8000, 10000, 20000}. Under Hoo. aj = 0 and 8j = 0; under HOI. og = 0 and 8, = 0.37: under H10. a, = 0.27 and B, 0; under H11, aj 0.27 and 3, = 0.37:
Example 4. We fix (n, T) (1000, 1). (Foo, T0 T10, 111) = (0.4, 0.2,0.2.0.2), and vary J E (5000, 8000, 10000, 20000}. Under Hoo. a; 0 and B3 0; under Hot. aj 0 and B, 0.37;
under H1.. 23 = 0.2r and A3 = O; under H11 02 = 0.2r and 8, = 0.37;
Example 5. We fix (J, = (10000, 1), (Tou, Toi, Tuu T11) = (0.85, 0.05, 0.05, 0.05), and vary
{600, 800, 1000, 1200). Under Hog- aj 0 and 8 0: under HØT. aj and 3, 0.37;
under Hio. 0i- = 0.2r and 33 O: under H11, 0j- 0.2r and 8j = 0..r.
0.24 and vary n {600, 800, 1000, 1200). Under Hoo. aj = 0 and 3, = 0: under H01. ag = 0 and By = 0.3T; under
Example 6. We fix (J, 7) = (10000, 1). (Too; o0 100 111)
H10, ay 0.27 and B₁ 0: under H11, aj 0.27 and B, 0.3r.
Mathodi
00-
1.5
0.50
Figure A1: The average FDR and power performance of four methods under Examples 1-2. The sample size n 1000 and the number of mediators J is 10000. The effect size parameter T varies from 0.6 to 1.2. The nominal FDR level is 0.1.
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To assess how the four methods are affected by effect size under sparse alternatives (T11 0.05) and dense alternatives [ 0.2), we apply the four methods in Examples 1-2. The effect size, T, is varied from 0.6 to 1.2 in both examples. The results of estimated FDR and power are summa- rized in Figure AT. For sparse alternatives in Example 1, AMDP and JS-mixture maintain stable FDR control at the nominal level across various effect sizes. DACT (Efron) consistently controls the FDR but can be overly conservative, leading to potential under-identification of significant signals. Moreover, the FDR level of DACT (JC) exhibits inflation under weak effects. In terms of power analysis, AMDP emerges as the top performer, consistently outperforming the other three methods in Example 1. JS-mixture ranks second when the effect is strong, while DACT (Efron) lags behind due to its conservative behavior. For dense alternatives in Example 2, DACT (JC) fails to effec- tively control the FDR, leading substantially higher FDR than the nominal level. While DACT (Efron) still exhibits overly conservative behavior. However, AMDP and JS-mixture maintain stable FDR levels across different effect sizes in Example 2, highlighting their robustness in controlling FDR. Regarding power analysis in Example 2, AMDP consistently outperforms the other methods, demonstrating its ability handle scenarios with substantial proportion of Hor and 110 while still achieving high power. JS-mixture ranks second in terms of power. It is noteworthy that in certain settings of Example 2. DACT (JC) may exhibit higher power than AMDP. Nevertheless, this higher power is often associated with severely inflated FDR levels. The conservation of DACT (Efron) results in lower power compared the other three methods.
Mechod
DAST
Mettod
DiGhithorr
0.2
0.-
13
1060
1853
(a)
(b)
Figure A2: The average FDR and power performance of four methods under the targeted FDR level of 0.1. (a) The comparison under Examples 3-4 for varying the number of mediators J E (5000, 8000, 10000, 20000} with n = 1000, T 1; (b) The comparison under Examples 5-6 for varying the sample size E {600, 800, 1000, 1200} with J = 10000, T 1.
Next, we move on to investigate whether the four methods are sensitive to changes in the large me- diator size J under both sparse and dense alternative scenarios. Panel (a) of Figure A2 displays the FDR and power performance of the four methods under Examples 3-4. In the sparse alternatives scenario of Example 3, AMDP and JS-mixture demonstrate remarkable stability in controlling FDR at the nominal level across different values of J. DACT (Efron) exhibits conservative FDR control. While DACT (JC) is less conservative than DACT (Efron), it remains underpowered. In terms of power analysis, AMDP and JS-mixture are the leading methods. DACT (JC) demonstrates higher power when J is not very large, but its power decreases as J grows. DACT (Efron) consistently displays lower power in all settings due to its conservative behavior. In the dense alternative see- nario of Example 4, DACT (JC) suffers from inflated FDR. In contrast, the FDRs of the other three methods are under control with varying mediator sizes, though DACT (Efron) continues to be overly conservative. Moreover, AMDP consistently delivers the highest power among all methods in the dense alternative scenario. We note that JS-mixture performs competitively in terms of power. On the other hand, the power of DACT (JC) decreases with the growth of J. which raises concerns about its ability to detect true positives accurately in scenarios with larger mediator sizes. DACT (Efron) consistently displays lower power in all settings.
In Examples 5-6, we explore the influence of sample size on FDR and power performance under sparse and dense alternatives, respectively. The results of these analyses are presented in panel (b) of
9

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Figure A In Example 5, where a small proportion of alternative hypotheses is considered. AMDP and JS-mixture stand out as more accurate and stable in controlling the FDR among all methods. When the sample size is small, DACT (JC) exhibits slightly higher FDR compared to AMDP and JS-mixture DACT (Efron) remains overly conservative. The power of the four approaches initially decreases and then increases with the growth of n. Specially, for all four methods, the lowest power is achieved at n = 800 among all the tested sample sizes. Moreover, AMDP consistently delivers reasonably higher power compared to the other three methods. In the dense alternatives of Example 6, DACT (IC) encounters challenges in maintaining FDR control, particularly when 1 is small. In contrast, AMDP, JS-mixture, and DACT (Efron) effectively control the FDR across different settings. Regarding power performance, AMDP, JS 1S-mixture, and DACT (Efron) demonstrate a consistent increase in power Nevertheless, this seemingly higher power of DACT (JC) is a result of the severely inflated FDR levels. We note that the consistent superiority of AMDP in both FDR control and power, as observed in Examples 5 and 6, aligns with the theoretical results presented in Theorem II.
grows. In some settings, DACT (JC) appears to perform better than AMDP.
4 Data Analysis
Prostate cancer is a prevalent disease among men, with a multifactorial etiology involving genetic, environmental, and lifestyle factors. There is a growing recognition that DNA methylation plays an important role in regulating gene expression [[ Additionally, the number of GWASs-identified risk SNP that influence DNA methylation levels in prostate cancer has reached a total of 167 [[]. Despite significant progress in understanding the role of DNA methylation in gene expression regu- lation and identifying prostate cancer risk SNPs, further research is strongly encouraged to uncover the specific CpG sites that contribute to the regulatory effects of risk SNPs on their target genes.
We apply our proposed AMDP, JS JS-mixture [D and DACT [BO]. including DACT (Efron) and DACT (JC), to analyze the TCGA prostate cancer dataset. The dataset is freely available at https: I.portal.dcc cancer gov, Our analysis focuses on 495 primary prostate tumor samples with information on 147 prostate cancer risk SNPs, DNA methylation, and gene expression. In total, we consider 69.602 CpG methylation probes (M) as potential mediators. The risk SNPs are the exposure variable (X), and gene expression is the outcome variable of interest (Y). The primary objective of our analysis is to explore the potential causal role of CpG methylation in the association between prostate cancer risk SNPs and gene expression. We estimate the null proportions as froo- 0.52, -10- 0.03, and मंं 0.42, respectively. Figure A2 of the Appendix C displays the number of significant triplets (X- - M - Y) detected by AMDP, JS-mixture, DACT (Efron), and DACT (JC) at different nominal FDR levels ranging from 0.01 to 0.1. It can be seen that, in the majority of cases, AMDP outperforms the other three methods by identifying more triplets at the same FDR level. On average, the discoveries made by AMDP are approximately 20. 20.2% higher than those of JS- mS-mixture, 86.6% higher than those of DACT (Efron), and 42.6% higher than those of DACT (JC), across the range of values from 0.01 to 0.1. This substantial improvement in performance highlights the effectiveness of AMDP in identifying non-zero mediation effects in the prostate cancer dataset.
Additional results of Section g are postponed to the Appendix C.
5 Discussion
In this paper, we develop a novel adaptive mediation detection procedure (AMDP) to identify signif- icant mediators in digensiimnsional mediation analysis. The novel approach determines the optimal ranking for hypotheses, and then employs a data-driven strategy to select the threshold for mediator identification. We demonstrate the effectiveness of our proposed method through theoretical analy- sis and simulation results. There is a potential avenue for future research. We discuss the mediation effect based on the marginal model in this paper, where the p-values independent. How to further study relevant mediators from two aspects of theory and application is an interesting topic.
Limitation Our approach effectively handles high-dimensional mediators but may not perform opti- mally when confronted with low-dimensional mediators. This distinction is attributed to the nature of method, wherein the two-dimensional p-values linked to each exposure- cerssuuremediitoo-uuuuoooo relationship effectively serve as "samples" for the estimation of local FDR and FDP. Consequently, the reduction in dimensionality can lead to less precise estimates of local FDR and FDP.
10

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Acknowledgements
The authors thank the editor and the anonymous reviewers for their constructive suggestions that significantly improved an early manuscript. The research described herewith was supported by grant from the National Key R&D Program of China (2022YFA 1003803), a grant from the National Social Science Foundation of China (2IBTJ048), a grant from the National Scientific Foundation of China (12371276, 12131006), Basic Science Research Foundation of Shaanxi (22JSY038) and the Zhongying Young Scholar Program.
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AMDP:
An Adaptive Detection Procedure for False Discovery Rate Control in High-Dimensional Mediation Analysis
Appendix
A Selection of an appropriate X
In Section 2.2, we estimate Auo (A), and fo1 (A) using the method proposed by Storey et al. [I with fixed parameter A. The estimators are calculated in (10)-(11). Theoretical considerations suggest that as À approaches 1, the estimators of the composite null hypothesis become more accu- rate asymptotically. However, in finite samples scenarios, with a larger value of the chance of these null p-values falling within (A, 1] gets smaller, resulting in less accurate estimates. Conversely, when À becomes smaller, the bias of the null estimators increases while the variance decreases [[]]]. Consequently, there exists inherent bias-variance trade-off in the selection of X.
To strike reasonable balance between bias and variance, we aim to determine À by minimizing the mean-squared error (MSE) of the estimators. The MSE is defined as E[( Too) (A)o(A) ] 2 + f (A) Tol 2. For achieving this goal, we consider a range of cutpoints for À (e.g.. A 0.1. 0.9) and calculate the MSE for each value of A. As highlighted by Barfield et al. [E)] a substantial proportion of null hypotheses may exhibit both 0 and 3 in genome-wide study involving high-dimensional mediation hypotheses. To investigate the choice of À in such scenarios, we consider the following settings, as shown Table A1
Table A1: The composite null proportions under different scenarios.
Hypothesis Configuration
Too
T10
To1
Scenario 1
0.2
0.3
0.3
0.2
Scenario 2
0.4
0.2
0.2
0.2
Scenario 3
0.5
0.2
0.2
0.1
Scenario 4
0.6
0.15
0.15
0.1
Scenario 5
0.75
0.1
0.1
0.05
Scenario 6 0.85
0.05
0.05
0.05
Table A2 shows means and MSE of the estimated null proportions under the six scenarios with 1=1000, J=10000, a,= 0.2, and 3,=0.3. In each scenario, identified the top three smallest MSES among the estimated null proportions. Notably, we observed that the optimal value of varied across the different scenarios. However, it is noteworthy that À 0.5 consistently appeared among the top three MSE values in all the simulated scenarios. As mentioned earlier, smaller values of Á tend to result in larger biases the null estimate, while excessively large values of À may yield inaccurate estimates in finite sample scenarios. Therefore, the consistent appearance of X = 0.5 among the perforrfonn MSE values suggests that provides a reasonable trade-off between bias and variance, leading to accurate estimates across a wide range of scenarios. Considering its stability and computational efficiency, we believe that X 0.5 is a suitable choice for estimating the null proportions in high-dimensional mediation analysis.
B Comparison with existing methods
In this section, we demonstrate the loss of information during the ranking step can result in decreased statistical power, despite controlling the FDR at the desired level.

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Table A2: The performance of the estimated proportions of the composite null hypothesis (mean and MSE) under the six scenarios. The tuning parameter À varies from 0.1 0.9.
Scenario
Moo
T10
MSE
Scenario 2
Too
110
fo1
MSE
A=0.1
0.222
0.278
0.314
1.17e-3
λ=0.1
0.415
0.185
0.215
6.4le-4
A=02
0.211
0.289
0.308
3.10e-4
A=
0.408
0.192
0.208
1.88e-4
A=0.3
0.207
0.293
0.306
1.20e-4
A=0.3
0.406
0.194
0.205
9.14e-5
A=0.4
0.205
0.295
0.305
6.69e-5
λ=0.4
0.404
0.196
0.204
4.6le-5
A=0.5
0.203
0.296
0.303
3.03e-5
A=0.5
0.402
0.196
0.203
2.56e-5
0=066
0.203
0.296
0.303
3.25e-5
λ=0.6
0.403
0.197
0.202
2.46e-5
X=0.7
0.202
0.297
0.302
1.96e-5
X=0.7
0.406
0.194
0.198
7.62e-5
0=08
0.201
0.299
0.302
5.94e-6
A=0.8
0.405
0.195
0.199
5.96e-5
0=9
0.206
0.295
0.296
8.54e-5
X=0.9
0.405
0.192
0.198
8.69e-5
Scenario 3
foo
110
MSE
Scenario 4
$oo
#10
fto1
MSE
A=0.1
0.515
0.185
0,207
4.84c-4
λ=0.1
0.611
0.139
0.157
2.94c-4
A=02
0.507
0.192
0.204
1.38e-4
A=0.2
0.606
0.144
0.154
8.07e-5
A=0.3
0.505
0.194
0.203
6.82e-5
A=0.3
0.604
0.146
0.152
3.79e-5
A=0.4
0.504
0.196
0.201
3.81e-5
λ=0.4
0.602
0.148
0.152
1.18e-5
A=55
0.503
0.197
0.201
1.91e-5
X=0.5
0.602
0.148
0.151
7.87e-6
A=066
0,502
0.196
0.202
2.24e-5
λ=0.6
0.603
0.147
0.150
1.79e-5
0=0.7
0.503
0.195
0.199
3.15e-5
X=
0.604
0.148
0.148
2.12e-5
A=08
0.501
0.197
0,202
1.36e-5
X=0.8
0.604
0.147
0.147
2.92e-5
A=09
0.501
0.198
0.201
6.69e-6
A=
0.602
0.150
0.150
3.19e-6
Scenario 5
foo
f1o
#oi
MSE
Scenario 6
$oo
fro
MSE
A=0.1
0.757
0.093
0.104
1.22e-4
λ=0.1
0.854
0.046
0.053
4.12e-5
A=02
0.754
0.096
0.103
3.69e-5
A=
0.853
0.048
0.052
1.25e-5
A=0.3
0.753
0.097
0,102
2.22c-5
X=0.3
0.853
0.048
0.050
1.19e-5
A=044
0.751
0.099
0.102
6.94e-6
A=0.4
0.852
0.049
0.050
3.79c-6
A=55
0.752
0.099
0.101
3.38e-6
A=
0.850
0.049
0.051
9.69e-7
A=0.6
0.754
0.097
0.099
2.35e-5
λ=0.6
0.851
0.050
0.051
1.40e-6
A=0.7
0.755
0.096
0.098
4.53e-5
A=0.7
0.849
0.052
0.052
1.16e-5
A=0.8
0.751
0.098
0.101
5.44c-6
X=0.8
0.848
0.054
0.056
6.07e-5
X=0
0.765
0.091
0.090
4.09e-4
X=0.9
0.857
0.056
0.055
1.1le4

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To conduct this investigation, we consider three different approaches for comparison: our proposed AMDP, along with two existing methods, the JS-mixture [[22] and the DACT [30]. During the selec- tion step, it is assumed that the information about the proportions of the composite null hypothesis and the distributions of p-values under alternatives are known. This provided knowledge allows for effectively controlling the FDR of all three procedures at the predefined level of in the selection step. With the FDR under control, we then proceed to investigate how different ranking strategies impact the power performance of the three methods. To achieve this, compare the rejection re- gions of each method under various scenarios. Formally, the ranking statistic for each method is as follows:
8-5mmiature
Pmans
8DACT = -wip() + w2p( + WyPmens
8AMDP- = fdr(p),
where Pmoz (1) p(2), V denotes the maximum of the two p-values. W1, w2 and wa are normal- ized relative proportions of the composite null. We consider two scenarios to compare the ranking statistic of the three methods:
Scenario 1 Balanced null proportions of Hor and H1o:
-0. 004 -0.7 f( p1(2) = 0.49 0.21 x (2) + 0.21 0.3y 0.7 + 0.09 x 0.18p(1) (1) p
where the density functions of p- p-values under alternatives are f(p(1) H1a) Beta 1) and f()12) Hot) ~ Beta(0.3, 1). The proportions composite hypothesis are Too0 = 0.49, n01 = TiO = 0.21, and Iा = 0.09.
Scenario 2 Unbalanced null proportions of Ho1 and H1o:
)()) 0.4 + 0.1 x (4) -0.6 + 0.4 0.6p 0.4 + 0.1 0.24p()) -0.6 p(2) -0.4
where the density functions of p-values under alternatives f(p(") H10) Beta (0.4. 1) and fp! Hoi) Beta(0.6, 1). The proportions of composite hypothesis are Too 0.4, moi 0.4, Tio = 0.1, and 111 = 0.1.
Meshod
500
100
(a)
(b)
Figure A3: (a) The rejection regions the three methods under Scenario 1; (b) The rejection regions of the three methods under Scenario 2.
Theorem 1 has proved that the ranking statistic of AMDP is optimal among those methods that effectively control the FDR the nominal level. Therefore, we refer to the rejection region of AMDP as the oracle rejection region in the sense that it achieves the highest power under FDR control. In Scenarios 1-2, we compare the rejection regions of JS-mixture and DACT with the oracle rejection region when the FDR level can be precisely controlled to the specified level. This comparison provides deep insights into the impact of information loss during the ranking step on power.
3

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Figure A visually presents the local FDR for the well as the rejection region for the three procedures. The color intensity in the figure represents the level of local FDR, with darker colors indicating lower local FDR, thus the corresponding hypothesis more likely to rejected. While the rejection regions of all three methods are all located in areas with lower local FDR, we emphasize that the AMDP is superior since it simultaneously considers information about proportions of the composite null hypothesis and the distributions of p-values un- der alternatives, leading more accurate and reliable selection of mediators. In contrast, the other two methods only take into account partial information during the ranking step, which results in decreased power. The insensitivity of the JS-mi IS-mixture method changes in proportions and distribu- tions is noteworthy. As described in panels (a)-(b), the shape its rejection region remains square regardless of the scenarios. On the other hand, the DACT method only considers partial information about proportions of the null, since its rejection domain remains symmetrical under Scenario (Bal- anced proportion of Ho and 11)), and shifts towards the larger distribution side under Scenario (Unbalanced proportion of Ho and H10). In contrast, the AMDP method fully captures all relevant information above, the oracle rejection region is sensitive to the change in both proportions and distributions. This allows the AMDP method to adapt and adjust its rejection region accordingly, making it more effective in identifying significant signals.
model (5) in both Scenarios 1-2, as
C Additional results of Section 4
In this section, we demonstrate additional results related to the data analysis of the prostate cancer dataset in Section 4, including Table AK and Figures A4HA
400
DACT (Ehron)
304
210.
Number of discovered triplets
DACT
.23
300
217
28p
356
227
200-
100
33.
242
120
200-
:20
0-
0.02
0.03
0.04
0.05
0.08
0.09
0.1
Figure A4: The number of triplets discovered by four methods. The nominal FDR level varies from 0.01 to 0.1. The blue, green. orange, and pink bars represent the numbers of triplets identified by JS-mixture, DACT (Efron). DACT (JC), and AMDP, respectively.
Figure A provides an overview observation of the prostate cancer dataset as well as the rejection regions of JS-mixture, DACT (Efron), DACT (JC), and AMDP. Panel (a) shows the dispersion of p-values. However, the high density of the p-values makes it difficult to observe carefully. Thus, we depict the details of the TCGA dataset in different aspects in panels (b), (c), and (d), respectively, for providing clearer insight. In panels (e)-(h), and (i)-(), we compare the rejection regions of four methods: JS-min JS-mixture, DACT (Efron), DACT (JC), and AMDP FDR levels of 0.05 and 0.1, respectively.
From panels (b)-(d), can be seen that the distribution of p-values is inffuenced by information related to the composite null hypothesis. In panel (b), there is a slightly denser concentration of p-values near the p(1) = axis. This occurrence can be attributed to the presence of Å = 0.03, mentioned earlier. On the other hand, panel (c) exhibits notable concentration of p-values near the p!2) 0 axis. This pattern is influenced by the presence of significant number of cases falling under Ho1. which affects the distribution of p-values. As a result, we observe an accumulation of p-values near the p axis in the plot. Panel (d) demonstrates a seemingly uniform distribution of valuus.. This uniformity can be attributed to the theoretical expectation that only p-values under

---

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101
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0003
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01011
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3.00-
8.00
11
161
3.006
0.00-
0.04
3.004
0.031
3.00
3.50
0005
8.50
3.504
0.56
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3.54
2003
0.03
0.244
$.500
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Figure A5: (a) An overview of p-values obtained from TCGA prostate cancer dataset; (b) The dispersion of p-values in the region [0, 0.2) [0.8, 1}: (c) The dispersion of p-values in the region [0.8, 1] x [0, (d)) The dispersion p-values in the region [0.8, 1] x [0.8, 1}; (e)-(h) The rejection domains of JS-m IS-mixture, DACT (Efron), DACT (JC), and AMDP the targeted FDR level 0.05, respectively; (i)-(1) The rejection domains of JS-mixture, DACT (Efron), DACT (JC), and AMDP the targeted FDR level 0.1, respectively; The black dots represent the p-values all triplets. The blue, green, orange, and pink dots represent p-values of triplets identified by JS-mixture, DACT (Efron), DACT (JC), and AMDP, respectively.
Hoo exist in the region [0.8, 1] x [0.8, 1]. At the FDR level of 0.05, the rejection region of JS- mixture in panel (e) corresponds any information related to the distribution of p-values or the proportions of the composite null. In contrast, DACT (Efron) considers the proportion of null hypotheses and demonstrates preference for rejecting fewer hypotheses with p-values close 22) 0 minimize false discoveries, as shown in panel (f). However, the number of triplets identified by DACT (Efron) is the least among all methods, resulting an overly conservative behavior. The conservatism observed in DACT (Efron) is alleviated by DACT (JC), as panel (g) reveals that DACT (JC) identifies more significant triplets efficient approach by adjusting the threshold of the rejection region to achieve a higher sensitivity. In panel (h), we observe that the rejection region of AMDP is adaptive. AMDP estimates the number of false discoveries based on symmetric regions of the rejection region, allowing for more effective and accurate control of false discoveries, and well-calibrated adjustments to the rejection region. AMDP strikes a better performance on detecting significant triplets among all procedures. Next, we turn to investigate the rejection regions of these four methods the FDR level 0.1. In panel (i), it is observed that the rejection region of JS-mixture remains insensitive to changes in FDR levels, maintaining its square shape. As shown panels (i)-(k,, both DACT (Efron) and DACT (JC) exhibit increased identification of triplets compared to the FDR level of 0.05. Nevertheless, they still appear to be somewhat underpowered in efficiently detecting significant triplets. In contrast, AMDP outperforms all the other methods the same FDR level, as demonstrated in panel (I). By leveraging information on the proportions of null
square shape. However, this symmetric shape does reflect
compared to DACT (Efron). DACT (JC) offers
5

---

and calibrating its rejection region dynamically, AMDP achieves better power to identify significant triplets.
Table A3: Top ten triplets identified by AMDP at the FDR level 0.1.
SNP ID
CpG Name
Gene
Chromosome
P1
P2
rs12653946
cg00626856
IRX4
5
6.60e-56
2.66e-20
rs rs12653946
cg03587843
IRX4
5
1.95e-51
1.03e-19
rs rs12653946
cg06161964
IRX4
5
1.99e-53
2.02e-22
rs 16639446
cg09672187
IRX4
5
4.01e-65
2.13e-33
rs 26533594
cgl cgl 1279838
IRX4
5
3,97e-64
3.61e-27
rs rs12653946
cg14051264
IRX4
5
7.62c-67
8.86e-26
rs 1s12653946
cg26195178
IRX4
5
2.52c-61
2.43e-26
rs5945619
cg16065628
NUDTII
X
6.75e-32
2.79e-42
rs 191933488
cg23651356
RGS17
6
7.25e-20
2.81e-16
rs12653946
cg14823763
IRX4
5
8.13e-47
1.27e-15
The top ten triplets identified by AMDP are summaried in Table AS These ten triplets consist of ten CpG sites and three genes. The CpG sites involved in these triplets are located in close proximity to the transcription starting sites, and their DNA methylation level are closely related to the expression of the corresponding genes [[2]] Among the identified triplets, the three genes, IRX4, NUDTII, and RGS17, have been shown to be associated with altered CpG methylation. IRX4 is a causative gene of the prostate cancer susceptibility locus [А5] The corresponding SNP rs 1653949, a variant previously confirmed be associated with prostate cancer, is significantly associated with IRX4 expression [6]] The increased expression of NUDTII I has been confirmed to be associated with the risk variant rs5945619 [22, 29). RGS17 is a commonly induced gene in prostate tumors, and has been found crucial for the maintenance of the proliferative potential of tumor cells [8]]
D Discussions on the parameter choice and assumptions
D.1 Parameter choice
In Figures 1-2 in Section 3, we assess how the four methods (JS-mixture, DACT (Efron). DACT (JC), and AMDP) are influenced by effect size, the large mediator size and sample size. To ensure the realism of our experiments, we carefully selected our simulation parameters. Motivated from several real-world datasets including the TCGA lung cancer cohort dataset [[ the Multi-EStee Ethnic Study of Atherosclerosis [[.5], and the TCGA prostate cancer dataset [02]], we adopt similar parameter settings as those used [12] to construct the simulation examples in Section
Regarding the choice of nominal FDR level, we initially used an FDR level of 0.1, which is a widely accepted standard in the field 2
Another common FDR level 0.05 [23] 39]. To provide
a comprehensive analysis, we conducted experiments at the FDR threshold of 0.05 across a wide
range of sample sizes (200, 500, 1000, and 5000). We present the experimental results under sparse alternatives scenario and dense alternatives scenario in Tables A4 It's noteworthy that the results are similar with those obtained using the FDR level of 0.1 in Section 3.
D.2 Discussion Assumptions 1-2
Our method extracts a pair (p()),((2) for each exposure-meddiatorootttom relationship and em- ploys these pairs to estimate the FDP a two-dimensional plane [0, 1] [0, 1]. The theoretical basis supporting FDP estimation is the assumption that p-values are uniformly distributed under the null hypothesis, which is widely recognized principle [M] [z]] Due the presence of composite null hypothesis in the mediation effect, we elaborate on Assumptions 1-2 to illustrate the properties of the p-value distribution under composite null hypothesis.
For Assumption 1, under Hon. both Pıj and p2j obey the uniform distribution, resulting in (P1j, P25) also following the uniform distribution on the two-dimensional plane [0, 1|x (0, 1]. Consequently, the

---

Table A4: The FDR and power performance of the four methods with effect size o,= 83=0.3 under sparse alternatives scenario. The nominal FDR level 0.05, and the number of mediators is 15000.
Method
FDR
Power
AMDP
0.0530
0.0474
JS-mixture
0.0284
0.0187
n=200
DACT (Efron)
0.0091
0.0055
DACT (JC)
0.0859
0.0960
AMDP
0.0454
0,4665
IS-mixture
0.0438
0.3714
n=500
DACT (Efron)
0.0140
0.2254
DACT (JC)
0.0793
0,4960
AMDP
0.0488
0.8698
IS-mixture
0.0500
0. 0.7931
n=1000
DACT (Efron)
0.0114
0.6101
DACT (JC)
0.0299
0.7460
AMDP
0.0498
0.9999
JS-mixture
0.0529
1
n=5000
DACT (Efron)
8.00e-05
0.9986
DACT (JC)
0.0935
Table A5: The FDR and power performance of the four methods with effect size a,=0.2, 3,= under dense alternatives scenario. The nominal FDR level is 0.05, and the number of mediators is 15000.
= 0.3
Method
FDR
Power
AMDP
0.0471
0.0331
JS-mixture
0.0348
0.0206
n=200
DACT (Efron)
0.0449
0.0379
DACT (JC)
0.1156
0.1673
AMDP
0.0460
0.4768
IS-mixture
0.0504
0.4168
n=500
DACT (Efron)
0.0333
0.3488
DACT (JC)
0.1640
0.6839
AMDP
0.0487
0,8734
IS-mixture
0.0541
0.8208
n=1000
DACT (Efron)
0.0315
0.7636
DACT (JC)
0.0818
0.8674
AMDP
0.0501
JS-mixture
0.0544
n=5000
DACT (Efron)
0.0116
DACT (JC)
0.1085
sampling distribution of (PLjs P2;) symmetrical around (1 0.5 and p(2) 0.5. Under Hol Pi; still obeys the uniform distribution, but P2j does not, leading to (Pıj. P2j) being only symmetrical about () = 0.5 on [0, 1] x [o, 1]. Similarly, under H10- P25 obeys the uniform distribution, but P1, does not, resulting in (Pij- P2;) being only symmetrical about (2)) 0.5 on [0. 1] x [0. 1). It is essential to emphasize that Assumption specifically applies to the null mediators.
For Assumption 2, a non-null p-value theoretically lies within [0, 0.5). Therefore, as the sample size tends infinity, the probability of p-values under alternative hypotheses falling within [0.5. 1] approaches zero. For example, as n goes to infinity. p-values under H11 and H10 are not expected to fall within the region Do1 [0.5, 1] x [0, 0.5) because non-null Pıj not lies within [0.5, 1] theoretically, therefore the region Doi only contains p-values under Hoo and Hol- Similarly, the region D1o thcoretically only includes p-values under Hoo and H

---

E
Proofs
E.1
Proof of Theorem 1
For any rejection region S E [0, 1J2, the global FDR in mediation analysis defined as follows
gFDR(S) = Hoo U Hon HI0 = 1 py E 5)
Tool E Hoe 1) mmm E Ho es Top ES Hon 1] P(p) E Hon -1
1)
Too foop)d S. Too Ig foo foo noi Is fon (p)d 110 fno f1o
Jo
fn (p)dp
(A.1)
We introduce notations
Doo(5) = -sg (po(p) (oi for (p)dp, D1o(S) flo())dp. D(((s)= = f f1(())p.
Thus, gFDR(S) transformed into
T00 Doo + noi Doi (5) + T10
gFDR(S) Toa (o0( (S) Toi Doi (5) 1 110 (1(
(A.2)
1 (200 No Do no Proi(())))
where
7000
7oo
701
For any threshold Ç E (0, 1]. define the rejection region
71
S(C) as
S(C) = pp: p: Too tol moofol() fn)
Top foo non + miof1o(p)
<c}
(A.3)
I+ (fan)))()( No. for (oo S sc
Here we prove that gFDR(S(()) is non-decreasing function ç. Suppose (2 <1. considering two cases:
Case v(S((c)) S(G)) 0. We derive that gFDR(S(G)) gFDR()((2))
Case 2 v(S((a) S(G)) 0. We can prove that gFDR(S(()) is non-decreasing function of if
Di(()(()) S(C))
"oo S(G,) + You (S((2) S(G)) - no Dro(((()))) -S(())
(A.4)
(s((((())
(ooDDo(S(G)) 70u Dor (oi(((())) (o0D
holds, the reason is as follows. Let
fu1(p)
(oofoo()) + fui + mwfiv()) (pE
mi sup
702 inf
Nuoouo(p) fur (p) hofilpp) :PES(a)
fu(())
By the definition of region S(C), have ma mi obviously. Therefore, we have (S(G))
"onDon + Non Do Don (o
m2 Don (Soon(S(( (S(()) non (o)))))))) no mo
Doo (S((1)) No1 Doi Doi (S((i) no Dio
(A.S)
700 Doo ( (S(C) S((1)) + + Do1
5((1) + no Dio(((()) S((1)) You Yoo Dao (S (S(Cz) S(G1)) + Yo1 Doi (S((((
Diu S(G1))
( S(G))
Don (S((2) S((1) + 701 Do Do (S((2)
Pho -((1))

---

Furthermore, we decompose the region S(22) as follows
(n((S(a)) 7oo (S(D)) Doi (1o
(A.6)
+701 Doi
Combined with (A.4), we obtain
Dnt((())
(s1()(
(o) +ro
too (1 (C()))
(A.7)
Moreover, by the definition of gFDR(S), it holds that
gFDR(S((1)) < gFDR(S((2)).
Under the Assumption (ii) in Theorem 1, for given a E (0, 1), there exists threshold Ç* > 0, s.t. gFDR(S((")) a. For the ease of presentation, denote S((*) as S In the following, will prove that S* is the optimal rejection region.
Considering any set T that satisfies Dn((T) > Dn((S). Let Pr = - S and Rs S - T. We can derive that
Dn(T) (o(TTSS+++ (n((r),
(A.8)
(n(S) = Dn Dn + Dn (Rs).
Then, we have D1I(RT) > (Rs( By the definition of s*, we have
inf Sou + nux for (p) (10/1(pp))
> sup { Swm(p) + nun Joz (p) 770 p E Rr
p E Rs
f((
(p)
(A.9)
Therefore,
700 Dook Ri) ( 7on Don RT) + Dro
Doo (Rs)) yo Don (Rs) Dio( Do
(A.10)
Dii (RT)
D1i(R))
In similar way. we can derive that
Doo + Don (Ro)) Pon +om Do (oo
(RT)
Du((TnS))
(A.11)
Finally, we have
700 (T) + 701 tio
D1.(T)
(T ha
(T'
Dn (((TTSS))) (S
777 http:///aaaaaaaaaaoo//////oommaa Don (Rs) Di
700 (100 D11(S*)
(A.12)
The second inequality holds because (nr(R)
(ns))). implying gFDR(T) >
GFDR(S((")) aa Therefore, we can conclude that the rejection region (C*) is optimal.
E.2 Proof of Theorem 2
E.2.1
The consistent estimator local FDR
To justify Assumption (i) for the corresponding local FDR estimator in Theorem 2, we first prove the consistency of local FDR estimator under Loo norm in Proposition MI.

---

Proposition 1. Assume that the smoothing parameter satisfies
lim b b= 0 and lim Jb = +00.
J-00
J-000
Then, we have
f får(p,)
Ja
sup Py 10)
Let be probability density on [0, 11. and be the beta kernel estimator:
o(D = J- Σ K (popp(pr)) i 1,
To prove the consistency of the beta kernel estimator J. i.e
((p - g(p(o) F J
sup E
(A.13)
we first need to establish the uniform convergence of its bias on the interval [0. 1]. Lemma 1. Let be the probability density on [0, 1), and ğ be the beta kernel estimator. We have
E ( - a b O, 1,2.
sup g
Proof of Lemma 1. Without loss of generality, we replace p(0), 1,2 with p for simplifying the proof steps, and discuss three cases in the following.
Case 1 p E (2b, - -b) Denote 14 and of are mean and variance of P, variable following Beta(p/b, (1 /)). According to Johnson al. [26], there exists constant C' such that
P1 P.
(A.14)
= bp(1 p) + R2(p),
(A.15)
where R2(p) < Cb Because probability density on [0. 1], for 0, there exists 8 such that
lo(t) m()) < for lp t| <
(A.16)
for all p (2b, 1 2b); According 44), have
111 pl 8/2 for all p (2b, 1 26).
(A.17)
Therefore, we can derive that
26
P 1-p (1.5:
E (((D)) - m(p)1
(o(1) - g(())K
26
(1 1-p
(0(1)) k(x)
|r-m1l8/2
(1,2) 1 1-
lg(t) - s(p][K
dt
1-11/26/2
b
1, P 1-p p
(0(1) g(1) K
dt.
/K-p1/66/2
<(t,)
p 1 -
+ 2 sup
lg(t)|
K t,
dt
1E(2b,1-2))
10-01126/2
b' b
=M1 -M2.
According (A. 16 and
7), obtain
Mi Se.
(A.18)
10

---

Combining the Chebyshev's 's inequality and (A 15), and there also exists be such that
} 18 S for all b S be. lo(()] (b (b+
M2 S
sup (6(26,1-26)
ls(()oo}
182
2 sup (6(2),1-20)
(A.19)
Thus, from (A. 1 and (A. 99, we conclude that
sup p-(26.1-26)
E (D(p)} - g(p)l < 2s
for all S be.
Case 2 p E [0, 26]
Based on the notations of Case 1, we have
P2 = p + E(p, b),
(A.20)
o R2(p),
(A.21)
where μ2 and of are mean and variance of of P, variable following Beta (p(p, b), (1
p)/b).
E(p, b) = (1 - (p(p, b) p/b / + bp(p. 6) p}. and (p( S Cbb For e > 0, there exists
ò > 0 such that
lo(t) m()) < e for Ip t <
(A.22)
for all p [0, 26); According (A.. since E(p, b) is bounded function for p [0, 26), there also exists by such that
I/2 < 6/2 for b < bs for all p E [0,2b). 26).
(A.23)
Therefore, we can derive that
I.- (a(t) - g(o) (L.p(D.)), 1 1-D
26
E (o(p)) g(p)
dt
lg(t) - g(p)k (s.pes.)) b),
1-
dt
10-wal<6/2
b
19(t) s(p)i (t,p(p.)) b),
1- p
dt
10-02128/2
(g(1) - A(DlK (Lpl,
1-p
dl
18 18-wal66/2
+ 2 sup 1g(()) /
t, p() 1-p
K
dt
1E[0,26]
1t-02128/2
b
=M1 M2.
According to (22 and (A there exists b(2) such that
M1 S E for all b < b(t).
(A.24)
Combining the Chebyshev's ' inequality and AA and there also exists ((2)) such that
M2 - 8 sup 1o(t)lo? re(0,26]
182 <
8 sup <e(0..0)
ls(t)(Cb | 18 s for all b - b!?)
(A.25)
Thus, from A and
we conclude that
E {ğ(p)} - g(p) < 2s for all $ min (B8)) (169,690)
sup
rE [0,2b)
Case 3 E [[-[[]6. 26, 1] Case can be proven similar procedure. We note that b)b p))),))),
Na = p b. b-E(1 03 = R2(p),
(A.26)
(A.27)
where μa and oz mean and variance of P. variable following Beta(p/b, p(1 p,b)), and R2(p) & CD2
This completes the proof of Lemma MI

---

Proof of Proposition 1. To prove the consistency of the beta kernel estimator, we use the inequality:
[p() s(p) v sup [p(p) - E ( p
sup
E [}(p))} ( pE(o.1]
sup
(A.28)
[0
From Lemma II the second term converges to zero. In the following, we prove that
sup [Ā(p) (((()) P 0 as J 0 pE
(A.29)
We also consider three cases:
Case 1 p E (26, 1 10-2b) 26)
The beta kernel estimator f(p) is expressed
1-26
11 P 1-P ((),
((p) =
K
(A.30)
26
where Fn is the empirical distribution. The expectation of the beta kernel estimator is
16
P
1
E E(G(p))
K
t
dF(t).
(A.31)
2b
b b
Thus, for p (2b, 1 26), can derive that
lØ(p) - [G()))
K
1
P
1-1
1
((Fr(()) - F(O)
d
(A.32)
(1--3)
11-20
1-p)
sup rE(b,1-2h
(t) - F(t)
dk
t,
6' b
26
Note that the integral
(A
is bounded above by
2b
1 1-P
1 1 -1
1 b
K
t
1,
K
t,
dt
S
(A.33)
b
b
2h
Therefore,
E
52-
1-b
sup 1=(26,1-2h)
[Fa(t) F(t)).
(A.34)
From Dvoretzky et al. [16]. we can obtain
b
}
P sup 12(2)11-2)
lØ(p) - -E (b()) 2 SPP sup
[Fa(t) - F(O) >
1-b b
LE(26,1-22)
(A.35)
822
< 2exp
-J
2 (1-b)²
By utilizing the Borel-Cantelli Lemma, it is shown that under the beta kernel estimator is consistent.
Case 2p E [0, 2b]
Case 2 can proven a similar procedure of Case 1. Note that, for all E [0, 26].
2b K (1.ple.)) b), 1-P)
lØ(p) E (0(p))
d ((F.()) F(O))
(A.36)
< sup [F) - F(L) ( 26
dK (1, b), 1.,1--p) 1 b
t6 (0,2))
Since P(p, b) is monotonic increasing [0. 2b}, ((0,b) b) = 1, p22,, b) = 2. For p E (0,2 the
integral in (A 3.000 bounded above by 2 1+ For 0. it is bounded above by
b
(p(p,b) b) + 1-p - 1)
2b
plple, b). 1-p -
-
-1) dt = 1+b
K
(A.37)
b
12

---

Thus, we can obtain
b
P sup pE[U,26]
lØ(p) E (o(p))] 2
P sup (E|0,26]
[F,(t)
F())
1+b]
(A.38)
82
82
$ 2 exp
(1 + b)2
which concludes the proof of the consistency of beta kernel estimator in Case 2.
Case 3 3p [1 -2,, 1] Case 3 can be proven similar procedure of Case 1. Note that for all p E [1 -b, 2b. 1].
lØ(p) - (0())) Lr K (117.0011- -P.b)
d ((Fa(()) ((pbb)) FFO)
(A.39)
[ dK -r,bi
sup 1E[1-26.1]
[F((1) P(O)
1
For [1 - 2b, 1), the integral in (A 39) is bounded above by b And for p it is bounded
above by
(0(1 p,b) + 1)
K (1, P -1,110 1 - p..) di
+ b
(A.40)
b
1-2b
Thus for all p E [1 2b, 1).
b+b
lő(p) F (i(p)) s
sup 1E(1-b6.1]
(F((1) F(())-
(A.41)
Similarly, we obtain
b
sup pE[1- 26,1]
(((p))
S sup [6,1]
[F,(t) F(O) 2 1+b
(A.42)
2 82
< 2 exp
(1 + b)2
which concludes the proof of the consistency of beta kernel estimator in Case 3. From Dai et al. [122] for fixed J and X, the biases of tooo aun, and Noo go to zero oo:
lim Āoo = Too+ n-000
lim for = Tol n
lim u1o = T20- n00
(A.43)
And derive
f(p) = (() (2)) = (()()) ((2(2)).
(A.44)
By combining equations
and
according continuous mapping theorem [
we have
fdr(p) fdr(p) P. 0 as n, J + co.
sup (0.11
(A.45)
According to equation
A 45 verify the rationality of Assumption (i) in Theorem 2:
fdr(p,) - fdr(p;))
P 0 oo.
(A.46)
j
13

---

E.2.2 Proof of Theorem 2
To begin with, we introduce some notations. For Ç E (0, 11, denote
(C) =
1 Σ
I (p;3E((c)
E S(()). (c) = Jon
Σ
(p; S(())
Joo JEDoo
jE
(C) = J10
Σ
(ps E B(()).
T)(()
1
Σ
(p)
Pj E E())
Jn
jEAio
jEAn1
(C) = Joa
1
Σ
(ps E =S(())
P
GT = Jol
Σ
P
(p, E S(()
JEAva
JEAo1
(C) = J10
1
Σ
P
P(ps E =S(C))
jEA10
(9)
1
E
I
(ps E Sools)),
(C)
Σ
I E Sor(3))
Juo JEDoo
Jol Joo jedoiUAoo
(J9(S)= J10 Joo
Σ SED
(pp E sio(c))
where Joo = A Jol = 1oi1/ J10 = Aiol, Jn = Anl) Denote r Joo/
J11
Jo1/J11, J10/ 8J
And
Joo + Jo1 + 110"
R¹(() Us (r (0
k}(C) Us (r ())))))
R9(S) os u)((19) 100 (-+0)
FDP (C) ,90 (5) + (5) ((+
K9(()/os
FDP](()) T Eg0 (5) + 9 69 (5) +r)
K((S
PDP = G (C) + r 69 (C) + (()))((((++++))) + C))
Before proceeding with the proof of Theorem 2, we prove Lemma 0 first Lemma 2. Under Assumption (i)-(ii) in Theorem 2, if Joo + co, Joi + 0o, 310 - 00 as J + and n > oc, we have in probability,
2 (c) - GO0 (C)
G9 (C) GG)(C) 0,
sup
0.
sup (E(0.1)
(E(D.1)
sup (E(D0.)
(C) G3) O.
sup (E(0.1)
(C((C) - {2(
0.
Proof of Lemma 2.
We consider three cases under composite null hypothesis.
Case 1
Under Hoo: We can derive that
(C)
1
Σ
(Ts E E((()) -T(p E s(()) +I(p; E E(()))
=
Juo
A00
(A.47)
G
1
Σ
P
E ()pE)(())))))))))){{))) - (Ppss E S(()) + +p(ps e =(()))
Joo
AE
14

---

Thus, we have
-9) (c)
Σ
(ps E 3(() [p; E S(()))
|1
sup cE(a.3)
S sup
<e(u.1] Jou
je
pPps E =(()) (ps E S(())]
<E(0,1 Joo jE
sup
(A.48)
Σ
[ps E(()) (p) ES(1))
+ sup CE(U,1 Jue
To deal with the first term, we have
1
Σ
I (p, E S(() - (p, E S(O))
Joo
jEA,o
I fdr (p²) s {c [fdr (p;) s 5)
Joo
jEdmo
fdr (pz) S Ss fdr (p,) > {) + 1 fdrr
S f, > c}] (p;) (pj)
Juo jEAio
1
I fdr (pj) s 56 fir (00) > 5) + I {s- - < fdr (pj) S (, fdr (p,) > c]
Joo jEA,0
[ fdr (dr (pj) - (pr >+ + I { f får (pz) S 5--556660 c, fär (p,) > c}]
+ Joo je
1
I[C fdr (p/) S < +
[rr (pj) fdr (p.)
Joo
Jone
jeAm
jEAon
(A.49)
Combine with the Gilivenko-Cantlll theorem and Assumption (i), we can derive
Σ
I fdr (py) s c) - I [får (pz) S sc
= sup CE(O,1] Joo
< sup
1
Σ
11c - E < fdr (p;) S < + o) + Jooe jEDoo
Σ
Fdr (p,) fdr (p.)
(E(0,1]
Joo
jE
1
IP(S << for (p,) s s
< sup [E(0.1]
Joo
GE
+ sup (E(0.1)
Σ I(ldr (p;) S
1
Σ (c)
Joo
Joo
JEAm
JEAno
1
Rdr (pz) - fdr fdr( (r3)
Juoe
< sup CE(0.1]
IP(S - < for (pz) 5 + op(1).
Juo
JE
Since e can arbitrarily small, suPçe(0.1) Jam Ejesn IP(CS - e < Idr (p,) S Ç +e) can be small due to Assumption (ii). Consequently, we have Q = (1) and thus the first term holds. Before addressing the second term, we obtain that
fdr (p,) c)
fdr (p,) S S, fdr (p,) s(+c) + (fár (p;) S (d > >(+c)
(A.50)
<P (pj) S ((+)) P [Tr (p,) - rdr (wy) > z).
15

---

Combine with Assumption (i). can derive that IP ( fdr (p,) fdr (ps) > 0.
Then, we have
Σ JEDa0
(ms
P
(ng E =$(S) IP (D) ES(C))
sup CE(0.1)
PP fdr (p3) s ss) P (fdr [
sup
CE(O.1) Joo
jE
(A.51)
PP (fdr (pz) + P (fdr (pj) S
sup
(E(U.1) Joo jE
( fdr fdr ( Sc+
sup (E(O.1)
Joo
jE
As can be arbitrarily small, suP(e(U.1) Joo Eje IP fdr (pj) S < +
The third term can be proved using the Glivenko-Cantelli theorem. Thus, we have shown the proof of the first claim in Lemma A
Under Hor and i Following the similar procedure in Case 1, we can conclude the proof of the second and third claims in Lemma M
Cases 2-3
According the symmetric property of p-values pj for j Doo U Doi 1io. follow the similar steps in Case I, thus, we have
Case 4
sup
(9(()-('(0 0.
(A.52)
CE(O.1)
This concludes the proof of the fourth claim.
Proof of Theorem 2. For any e E (0, or), suppose there exists Sa-c > 0, then
PP (FDP (a
By Lemma M for any constant e > 0, we have
sup o<c5e
FDP -DP)(S)-FDP,(C) B0.
(A.53)
By the definition of (*, i.e., 5* = sup { (0, FD) s a). we have
P (5 6a-c) PF P FDP (Sa-c) sa)
2 [DP) (Sa_e) ( FDP, (Ga-c) S F, (Sa-c) (5a
(FDP (Ca-e) S -P |DP1 (C.P)((G-.)) (Co-c) - FDP, (Sa-e)
> E, FDP (
2 (FDP (a--)) < a -P
FDP] (Sa-c) FDP, (Sa-e) >e)
1-
(A.54)
for J large enough. Thus, we have
PP (C 2 Ca-e) 2 1 e.
(A.55)
Conditioning on the event 5* 2 Sa-e, we have
16

---

limsup E FDP, (c*) < lim sup E [DP,, (5 C (a-c)++
R,J-+00
n,J
< lim E
FDPJ (C)) - (PDD)(C))-DDPPC))112222 (5* 2 Sa-r)
n
limsup E n,J-+00
[DP] (5*) FDF) (5 2 (2 P((* 2 La-c)
lim E n,J-+00
FDP] (C*) 15* 2 Ca-e P 2 Sa-e) +
lim limsup E n,J=+00
sup CElCo-r.1]
FDP - (1)
limssup E n,J-
sup
[E(Sa-e11]
lim sup E
FDP] (5T) +
n,J-000
(A.56)
The first two terms are based on Lemma and the dominated convergence theorem. the third term, we have FDP] (C*) S a by the definition (*, This concludes the proof of Theorem 2. 0
17


Reviews:
[{"review_id": "u8CHoQnlOn", "reviewer": "Reviewer_oiJi", "summary": "UPDATE \n\nIn light of the revisions the authors made I have raised my score and now support acceptance of this work. Thank you very much.\n\n-----------------------------------------\nThe authors propose a procedure to increase statistical power to identify mediators while controlling the FDR in high-dimensional data sets. Their method leverages the proportions of composite null hypotheses and the distribution of p-values under the alternative to derive an algorithm to estimate p-values for mediator variables while controlling the FDR. They perform a theoretical analysis suggesting that their method is asymptotically optimal and showcase that it controls FDR more consistently than other methods (DACT, JC, JS-mixture) in a simulation study and identifies more mediators than JS-mixture in an empirical study suggesting higher statistical power.", "questions": "This is very interesting work with broad applications to many empirical studies. I am willing to support acceptance of this paper, if the following concerns/questions are addressed appropriately. I hope you find my comments helpful and constructive.\n\n**Major points**\n- Can you provide a motivation for the choice of simulation parameters (i.e., sample sizes, number of mediators, FDR alpha and effect sizes). Current genetic studies tend to have larger sample sizes ( in the order of 20.000) while older studies had smaller sample sizes so including a wider range of samples e.g. 100, 500, 1000, 10000, 20000 may better cover empirical scenarios. Moreover, I don’t understand why you picked alpha = 0.1 as reference, since most studies choose alpha=0.05, for genetic studies this might even be smaller.\n- In Figure 1, it appears that AMDP deviates more from the 0.1 FDR line when the effect sizes get smaller. Could you speculate on why that is? On that note, the effect sizes chosen are rather large. What happens for smaller effect sizes (up to .2 or .3). Does this deviation get larger, is this a breakpoint for the method?\n- Similarly, in the empirical analysis (Figure 3), JS-mixtures identifies more triplets for alpha=0.01. Could you explain why that is, does this trend hold for alphas that are even smaller like 0.001?\n- What are the conditions under which the method is not appropriate? When does it break? I am missing a clear limitation section.\n\n\n**Minor point**\n- p.8 ll. 288 ff. essentially contains a figure caption of a figure that is presented in the supplement, which is not very helpful for the reader. I would suggest to either summarize the take-aways from this supplementary analysis here and move the caption to the figure where it belongs or spend this space on discussion the implications including break points and limitations of the analyses you present in the main manuscript. \n- Just out of curiosity, can you elaborate on how this method could be used for image analysis? Can this be applied to neuroimaging (e.g., fMRI) as well?\t\n\n", "limitations": "- The paper does not have a clear limitation section (see Questions). ", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The problem is very relevant as high-dimensional data are frequent not only in genetics, but also in other domains like neuroscience and user data, which all potentially suffer from similar problems.\n- The work is very extensive comprising theoretical analysis, simulations and an empirical application of the methods.\n- Related methods are extensively discussed in the introduction.\n", "weaknesses": "- There is no real discussion of the limitations/breakpoints of this method.\n- The effect sizes chosen for the simulation analysis seem to be unreasonably large (at least showing how the method behaves for smaller effect sizes, which are more commonly observed empirically, would be informative).\n- The paper is occasionally quite dense and would benefit from focusing on some key aspects.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the thoughtful comments. \n\n**Weaknesses 1：** Thanks for your reminder. Our method is designed to high-dimensional mediation analysis, but its performance may fail in the low-dimensional designs. We will rewrite our discussion section and incorporate the limitation in revised version to provide a more comprehensive analysis.\n\n**Weaknesses 2：** We adopt the similar settings as these in Dai et al. (2023).  More details about the parameter settings can be found in our reply to Question 1.\n\n**Weaknesses 3：** Your suggestion will lead to a significant improvement of an early manuscript. The high-dimensional mediation analysis is often associated with a multiple testing problem for detecting significant mediators, which has attracted significant interest. _**To control the FDR, our article will focus on two key steps: ranking and selection.**_  We will emphasize the two key aspects in revised version to enhance the readability of the article.\n\n\n**Question 1:**  \n\n* The reviewer's inquiry regarding the motivation for the choice of simulation parameter.  \n\n**Response:** \n\n(1) In Figures 1-2, we assess how the four methods (JS-mixture, DACT (Efron), DACT (JC), and AMDP) are influenced by effect size, the large mediator size and sample size.  \n\n(2) We aimed to closely simulate real-world data scenarios. (a) We referred to several real datasets including the TCGA lung cancer cohort dataset (Zhang et al., 2021), the Multi-Ethnic Study of Atherosclerosis (Du et al., 2023), and the TCGA prostate cancer dataset (Dai et al., 2023), and then construct the simulation examples. (b) We adopt the similar parameter settings as these in Dai et al. (2023). \n\n(3) We follow widely used FDR level 0.05 (Song et al., 2020; Guo et al., 2023) and FDR level 0.1 (Dai et al., 2023; Mosig et al., 2021).\n\n* The reviewer asked if we could cover empirical scenarios with a wider range of sample size, and the reason why we pick alpha=0.1 as reference.\n\n**Response:** As the response to Question 1, we will adopt the commonly used FDR levels 0.05 and 0.1. Our method is specifically designed for genetic data, as suggested by Mosig et al. (2001), we adopted FDR level of 0.1 in the original manuscript. Following your suggestion, we have integrated experiments at the FDR threshold of 0.05 with a wide range of sample sizes (200, 500, 1000, 5000).  We present the experimental results under sparse alternatives scenario and dense alternatives scenario in PDF format. From Tables 1-2 in PDF format, we can observe the similarly results as these with the FDR level of 0.1. Note that when the sample size reaches 5000, the power of all four methods converges close to 1, so we do not undertake experiments with larger sample sizes of 10000 and 20000.\n\n\n**Question 2:** \n\n* The reviewer asked the reason why AMDP deviates more from the 0.1 FDR line when the effect sizes get smaller in Figure 1.\n\n**Response:** There is the missing \"+1\" in Equation (10). To enhance the robustness of FDR control, our method estimates the numerator of Equation (10) as (the number of false discoveries + 1), which was used in Du et al. (2023) and Guo et al. (2022). When the effect size is smaller, it yields a small denominator for the FDP estimation. Thus, the \"+1\" term becomes non-negligible, leading to slightly conservative FDR controls. To mitigate this problem, the \"+0.5\" adjustment can be employed effectively.  As depicted in Figure 1, the JS-mixture method also exhibits a similar phenomenon. The difficulty to detect significant variables with minor effect sizes results in rare cases of no rejection in some experiments, which subsequently reduces the average FDR.\n\n* The reviewer shows concern about the effect sizes chosen in our simulation study.\n\n**Response:** Our simulation settings encompass a wide range of effect sizes, including 0.2, 0.3, and even smaller values such as 0.12. More details can be found in the response to Question 1. \n\n**Question 3:** \n\n* The reviewer’ question about the trend observed in Figure 3 for different alpha levels, including alphas smaller than 0.01.\n\n**Response:** We agree that JS-mixture identifies more triplets than our method when alpha is smaller. As the response in Question 2, our method employs a +1 adjustment in Equation (10) to enhance FDR control. As the alpha level decreases, the number of rejections is significantly reduced, and the +1 term becomes non-negligible, leading to fewer identified triplets. It's essential to point out that people usually care about the results of widely used FDR levels, such as 0.05 and 0.1.\n\n**Question 4:** \n\n* The reviewer asked a clear limitation section.\n\n**Response:** Although the performance of AMDP suffers in low-dimensional designs, as stated in Weaknesses 1, our method does not exhibit a well-defined breakpoint.\n\n**Minor question:** \n\n* **Response:** Thanks for your valuable suggestion. We will incorporate the necessary modifications in the revised version.\n\n* **Response:** Thanks for the interesting question. The application of our method to neuroimaging is a plausible avenue. In voxel-based analyses, our method may be used to capture active voxels in various brain regions, including the visual, auditory, and motor regions. For further inspiration, see Chang et al. (2023).\n\n**Reference:** \n\nChang, J., He, J., Kang, J., & Wu, M. (2023), …, JASA, 1-14.\n\nDai, J. Y., Stanford, J. L., & LeBlanc, M. (2022), …, JASA, 117, 198-213.\n\nDu, J., Zhou, X., …, & Mukherjee, B. (2023), …, Genet Epidemiol, 47, 167-184.\n\nDu, L., Guo, X., Sun, W., & Zou, C. (2023), …, JASA, 118, 607-621.\n\nEfron, B. (2004), …, JASA, 99, 96-104.\n\nGuo, X., Ren, H., Zou, C., & Li, R. (2022), …, JASA, 1-13.\n\nGuo, X., Li, R., Liu, J., & Zeng, M. (2023), …, JBES, 1-14.\n\nMosig, M. O., Lipkin, E., ..., & Friedmann, A. (2001), …, Genetics, 157, 1683-1698.\n\nSong, Y., Zhou, X., ..., & Mukherjee, B. (2020), …, Biometrics, 76, 700-710.\n\nZhang, H., Zheng, Y., Hou, L., Zheng, C., & Liu, L. (2021), …, Bioinformatics, 37, 3815-3821."}, {"review_id": "BzAykqNNTn", "reviewer": "Reviewer_9J5U", "summary": "The paper proposes and analyses a new method for FDR control.", "questions": "Please see the weakness section. ", "limitations": "No. ", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The paper is clear on the method, the assumptions and the proofs. The empirical section is promising and adequate. \n", "weaknesses": "Some of the assumptions and the writing are not clear, or seem too strong to me. For example the assumption about \"no confounding\" which is not listed as a formal assumption but is mentioned in the text, and attributed to another paper. It would be helpful is the authors could provide an explanation about why this is not a limiting assumption. \n\nThe assumption 2 seems reasonable, but assumption 1 again seems limiting, with a few distributions that could satisfy it. It is not a paper breaking assumption, but it would be useful if the authors are upfront about it, and discuss which possible distributions satisfy it. Similarly, about assumption 2 it would be helpful if the authors could write a couple of sentences discussing what the assumption means and implies. \n\nThe abstract and the intro makes a big deal about the curse of dimensionality becoming a blessing, but this was not obvious to me from the paper's text. could the authors explain what they mean by this exactly ? could you point out where this is discussed in the paper? \n\nI thought the \"limitations\" section was required this year for neurips submissions. but this is missing, and the discussions section at the end is not comprehensive about this. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer 9J5U for your comments and for citing your concerns. 9J5U’s main concern is that some assumptions in our paper seem too strong. We can provide some explanations.\n\nMajor questions:\n\n**Question 1:** \n\n* The reviewer concern about the assumption about \"no confounding\". \n\n**Response:** To be more precise, this assumption is well known as the sequential ignorability assumption, as established in the modern causal inference framework (Imai et al., 2010; Valeri et al., 2013). Specifically, this assumption comprises two components: (1) \"no unmeasured confounding\" of the exposure-outcome relationship and (2) \"no unmeasured confounding\" of the mediator-outcome relationship. Under the sequential ignorability assumption, the Natural Indirect Effect (NIE) can be identified and is equivalent to the product of $\\alpha$ and $\\beta$ in our model (Imai et al., 2010). In the revised version, we have taken the reviewer's feedback into account and further clarified the language by switching \"no confounding\" to \"no unmeasured confounding\" in Section 2.1. This change helps ensure a clearer and more precise description of our assumptions and their implications for the mediation analysis.\n\nAlthough the sequential ignorability assumption is a common assumption in traditional mediation analysis, it's worth noting that certain studies have delved into the intricacies of the hidden confounder issue, as mentioned in Song et al. (2020) and Song et al. (2021). Given this perspective, we will further consider extending our current work to settings where hidden confounders are present.\n\n**Question 2:** \n\n* The reviewer asked for explanations of Assumptions 1-2. \n\n**Response:** Our method extracts a pair $(p_1, p_2)$ for each exposure-mediator-outcome relationship and employs these pairs to estimate the FDP on a two-dimensional plane $[0,1] \\times [0,1]$. The theoretical basis supporting FDP estimation is the assumption that p-values are uniformly distributed under the null hypothesis, a widely recognized principle (Benjamini et al., 1995; Hung et al., 1997). Due to the presence of a composite null hypothesis in the mediation effect, we elaborate on Assumptions 1-2 to illustrate the properties of the p-value distribution under composite null hypothesis. \n\n(1) For Assumption 1, under $H_{00}$, both $p_1$ and $p_2$ obey the uniform distribution, resulting in $(p_1, p_2)$ also following the uniform distribution on the two-dimensional plane $[0,1]\\times[0,1]$. Consequently, the sampling distribution of $(p_1, p_2)$ is symmetrical around $p_1=0.5$ and $p_2=0.5$. Under $H_{01}$, $p_1$ still obeys the uniform distribution, but $p_2$ does not, leading to $(p_1, p_2)$ being only symmetrical about $p_1=0.5$ on $[0,1]\\times[0,1]$. Similarly, under $H_{10}$, $p_2$ obeys the uniform distribution, resulting in $(p_1, p_2)$ being only symmetrical about $p_2=0.5$ on $[0,1]\\times[0,1]$. It is essential to emphasize that Assumption 1 specifically applies to the null mediators.\n\n(2) For Assumption 2, since a non-null p-value theoretically lies within $[0, 0.5)$, it implies that the rejection region $S$ is a subset of $[0, 0.5) \\times[0, 0.5)$ without requiring additional information. Consequently, $S$ and its symmetric regions do not overlap. Moreover, as the sample size $n$ tends to infinity, the probability of p-values under alternative hypotheses falling within $[0.5, 1]$ approaches zero. For example, as $n$ goes to infinity, p-values under $H_{11}$ and $H_{10}$ are not expected to fall within the region $\\tilde{S_{01}}$ because the region $\\tilde{S_{01}}$ \ntheoretically only contains p-values under $H_{00}$ and $H_{01}$. Similarly, the region $\\tilde{S_{10}}$ \ntheoretically only includes p-values under $H_{00}$ and $H_{10}$.\n\n\n**Question 3:** \n\n* The reviewer asked for explanation of “the curse of dimensionality becoming a blessing”. \n\n**Response:** The core idea behind our method is to extract a set of $(p_1, p_2)$ for each exposure-mediator-outcome relationship and constructing a p-value-based statistic, which we refer to as the local FDR. We can estimate the local FDR by treating each set of p-values as a sample point. It effectively converts the high-dimensional problem into a large sample scenario. The increasing dimensionality yields more accurate estimates of the local FDR involving the nonparametric density estimation. Consequently, as the dimension increases, we can better control the FDR, making our method more effective in practical applications.\n\n**Question 4:** \n* The reviewer asked for \"limitations\" section.\n\n**Response:** As previously discussed in Question 3, our article focuses on addressing high-dimensional mediation effects. Consequently, the estimation of local FDR and FDP might not be as effective for low-dimensional mediation problems. We will rewrite our discussion section and incorporate the limitation in revised version to provide a more comprehensive analysis.\n\n**Reference:** \n\nBenjamini, Y., & Hochberg, Y. (1995), ..., Journal of the Royal statistical society: series B (Methodological), 57(1), 289-300.\n\nHung, H. J., O'Neill, R. T., Bauer, P., & Kohne, K. (1997), ..., Biometrics, 11-22.\n\nImai, K., Keele, L., & Tingley, D. (2010), ..., Psychological methods, 15(4), 309.\n\nSong, Y., Zhou, X., Zhang, M., Zhao, W., Liu, Y., Kardia, S. L., ... & Mukherjee, B. (2020), ..., Biometrics, 76(3), 700-710.\n\nSong, Y., Zhou, X., Kang, J., Aung, M. T., Zhang, M., Zhao, W., ... & Mukherjee, B. (2021), ..., Statistics in medicine, 40(27), 6038-6056.\n\nValeri, L., & VanderWeele, T. J. (2013), ..., Psychological methods, 18(2), 137."}, {"review_id": "aRQHx8DqKi", "reviewer": "Reviewer_ZQdD", "summary": "This paper describes a method to control for false discoveries in high-dimensional mediation analysis. This is the problem of inferring whether any of a large set of possible mediator variables act as a mediator between an exposure variable and an outcome variable. This work is particularly interested in the application where the exposure is a single-nucleotide polymorphism (SNP), the exposure is the expression level of a gene, and the mediators are the methylation levels at a set of CpGs. It is known that methylation is involved in the regulation of gene expression, so in this setting one wishes to discover whether the SNP is affecting the methylation at a CpG and in turn this is affecting the level of gene expression of a gene. This is an interesting problem and one of interest when seeking to uncover the functional effect of genetic variants. In this setting, there will typically be many CpGs as possible mediators, so we are in the realm of high-dimensional statistics and need to account for the fact that we are running a large number of tests in parallel. Previous methods for this problem are deemed to be too conversatve, sacrificing power for the suppression of false discoveries, and the authors seek to identify an approach which has improved power.\n\nThe method is verified on simulated data and illustrated on a prostate cancer data-set, where it is shown to identify more (SNP, CpG, gene) triples than existing methods", "questions": "No questions", "limitations": "I do not believe that the method has any obvious negative societal impacts. ", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The paper tackles a problem of practical interest\n- The method is justified theoretically (I did not check the proofs)\n- The method is justified empirically on simulated data and illustrated on a real-world data set", "weaknesses": "The presentation felt a bit compressed with many relevant details deferred to the appendix, such as the examples described in Remark 2 and many of the plots relating to the prostate cancer data-set.\n\nThe authors provide limited information on the context, assuming a reader will be familiar with genetic and epigenetic analysis.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Question 1:** \n\n* The presentation felt a bit compressed with many relevant details deferred to the appendix, such as the examples described in Remark 2 and many of the plots relating to the prostate cancer dataset.\n\n**Response:** Thanks for you carefully to review our paper. We will make adjustments to the typesetting of both the main text and appendix, as well as revise and go through the content in the updated version.\n\n**Question 2:** \n\n* The authors provide limited information on the context, assuming a reader will be familiar with genetic and epigenetic analysis.\n\n**Response:** Thanks for your insightful comments to improve the quality and depth of our work. In the revised version, we will expand on the information in the field of genetics and epigenetics, offering a more detailed and comprehensive explanation.\n"}, {"review_id": "BMNkaVgizX", "reviewer": "Reviewer_qu3t", "summary": "This paper proposes a high-dimensional mediation analysis detection procedure with false discovery control. Classic FDR control for multiple testing does not use information across tests making it overly conservative. The authors propose a local FDR-based procedure for identifying mediators and a data-driven approach for estimating the null densities and determining thresholds. Experiments show the proposed approach enjoys high power while still maintaining good FDR control.   ", "questions": "None. ", "limitations": "Yes", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The application of multiple testing on high-dimensional mediation analysis is interesting. \n\nThe local FDR uses across-dimensional information, and the test statistics and thresholds can be estimated from data. \n\nThe proposed algorithm maximises power while maintaining FDR control. \n\nApplying a local FDR-based method mediation analysis problem is not trivial. The data-driven approach proposed in Section 2.2 could be useful in other FDR control applications (e.g. how the null densities are obtained). ", "weaknesses": "This paper is very technical. I would appreciate some more introduction and motivation: \n\nWhat is local FDR? \nWhy do we want to use local FDR in mediation analysis? \nWhat may be the obstacles when using local FDR in mediation analysis? \n\nIf the authors could present answers to these questions at the beginning, it would make the paper feel more motivated. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your valuable suggestions. We will add more introduction and motivation in the revised version. Now we respond the comments raised by you point-by-point.\n\n**Question 1:** \n\n* What is local FDR? \n\n**Response:** In the context of multiple hypothesis testing, the false discovery rate (FDR) is a measure that quantifies the proportion of false positives among the total number of rejected null hypotheses. The local false discovery rate takes this concept one step further and aims to estimate the false discovery rate for each individual test or hypothesis separately (Efron et al., 2001). It represents the posterior probability that a hypothesis is null, given its corresponding p-value. The local FDR plays an important role in multiple hypothesis testing as it considers the varying degrees of uncertainty and statistical evidence supporting or refuting each individual hypothesis, and it allows for a more nuanced and accurate evaluation of the significance of each hypothesis by leveraging information across large-scale tests.\n\n**Question 2:** \n\n* Why do we want to use local FDR in mediation analysis? \n\n**Response:** High-dimensional mediation analysis is often associated with a multiple testing problem for detecting significant mediators. Assessing the uncertainty of this detecting process via false discovery rate (FDR) has garnered great interest. To control the FDR in multiple testing, _**two essential steps are involved: ranking and selection.**_ Existing approaches either construct p-values without calibration or disregard the joint information across tests, leading to conservation in FDR control or non-optimal ranking rules for multiple hypotheses. In contrast, local FDR allows for assessing the significance of each individual mediation path separately while considering the multiple testing aspect. Moreover, it produces the optimal rule for ranking hypotheses, as demonstrated in Theorem 1 of the manuscript. In summary, the proposed local FDR improves the accuracy and reliability of identifying significant mediation paths between variables in multiple hypothesis testing.\n\n\n**Question 3:** \n\n* What may be the obstacles when using local FDR in mediation analysis?\n\n**Response:** The main challenge we encounter pertains to estimate the local FDR within the framework of mediation analysis. In this context, the density of p-values follows a mixture distribution, as indicated in Equation (4). This mixture distribution involves two distinct types of null hypotheses: $H_{01}$ (corresponding to $f_{01}(p)$) and $H_{10}$ (corresponding to $f_{10}(p)$). Effectively distinguishing between these two components and obtaining precise estimations for $f_{10}(p)$ and $f_{01}(p)$ present a formidable obstacle. Motivated by the knockoff method (Barber et al., 2015), we explore a strategy rooted in the symmetry property exhibited by p-values under the composite null hypothesis, see lines 183-194 for more details.\n\nWe will rewrite our introduction section and integrate the motivation above in revised version to enrich the readability of the article.\n\n**Reference:** \n\nBarber, R. F., & Candès, E. J. (2015). Controlling the false discovery rate via knockoffs. The Annals of Statistics, 43(5), 2055 – 2085.\n\nEfron, B., Tibshirani, R., Storey, J. D., & Tusher, V. (2001). Empirical Bayes analysis of a microarray experiment. Journal of the American statistical association, 96(456), 1151-1160.\n"}]

Author rebuttal:
Thanks again for the comments. We present additional experimental results under sparse alternatives scenario and dense alternatives scenario with the FDR level of 0.05 in the PDF format.

Meta-review: The paper discusses an important problem dealing with high-dimensional data in many domains.  Overall, we found the method to be theoretically sound and empirically validated; we think the paper is worthy of being accepted.  However there are several aspects that can be improved.  Please reflect reviewers' comments and the discussions in making a final version.  More specifically, it is desired to include discussions on the parameter choice and assumptions.  Also, be sure to include the limitation as you comment in the discussion.
Decision: Accept (spotlight)