SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Therefore, the local FDR estimator is derived as:
$$
\widehat {\operatorname {f d r}} (p) = \frac {\hat {\pi} _ {0 0} f _ {0 0} (p) + \hat {\pi} _ {0 1} \hat {f} _ {0 1} (p) + \hat {\pi} _ {1 0} \hat {f} _ {1 0} (p)}{\hat {f} (p)}. \tag {17}
$$
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  $\zeta^{\star}$  of the optimization problem (9). However, determining such an optimal threshold  $\zeta^{\star}$  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
$$
\operatorname {F D P} (\zeta) = \frac {\# \left\{j : j \in \Delta_ {0 0} \cup \Delta_ {0 1} \cup \Delta_ {1 0} , j \in \widehat {\Delta} \right\}}{\# \{j : j \in \widehat {\Delta} \} \vee 1} \text {a n d} \operatorname {F D R} (\zeta) = \operatorname {E} (\operatorname {F D P} (\zeta)), \tag {18}
$$
where  $\widehat{\Delta}$  is the index set of selection in rejection region  $\widehat{S} (\zeta) = \{p:\widehat{\mathrm{fdr}} (p)\leq \zeta \}$ , i.e.,  $j\in \widehat{\Delta}$  when  $p_j\in \widehat{S} (\zeta)$ , the denominator of  $\mathrm{FDP}(\zeta)$  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 (18) poses a challenge, since the rejection region  $\widehat{S}(\zeta)$  comprises a mixture of four distinct types of hypotheses. However, we can draw inspiration from Assumptions 1 and 2 to leverage the symmetry property of p-values under the composite null hypothesis to estimate the number of false positives. We define the symmetric regions of  $\widehat{S}$  as  $\tilde{S}_{01} = \{(1 - p^{(1)}, p^{(2)}): \widehat{\mathrm{fdr}}(p) \leq \zeta\}$ ,  $\tilde{S}_{10} = \{(p^{(1)}, 1 - p^{(2)}): \widehat{\mathrm{fdr}}(p) \leq \zeta\}$ ,  $\tilde{S}_{00} = \{(1 - p^{(1)}, 1 - p^{(2)}): \widehat{\mathrm{fdr}}(p) \leq \zeta\}$ . It's noteworthy that the rejection region  $\widehat{S}$  is a subset of the region  $D$  defined in Section 2.2, and its symmetric regions,  $\tilde{S}_{01} \subseteq \tilde{D}_{01}$ ,  $\tilde{S}_{10} \subseteq \tilde{D}_{10}$ , and  $\tilde{S}_{00} \subseteq \tilde{D}_{00}$ . Indeed, Assumptions 1 and 2 provide us with an approximation of the number of false positives in (18):
$$
\begin{array}{l} \# \left\{j \in \Delta_ {0 1} \cup \Delta_ {0 0}: p _ {j} \in \widehat {S} (\zeta) \right\} \approx \# \left\{j \in \Delta_ {0 1} \cup \Delta_ {0 0}: p _ {j} \in \tilde {S} _ {0 1} (\zeta) \right\} \approx \# \left\{j: p _ {j} \in \tilde {S} _ {0 1} (\zeta) \right\}, \\ \# \left\{j \in \Delta_ {1 0} \cup \Delta_ {0 0}: p _ {j} \in \widehat {S} (\zeta) \right\} \approx \# \left\{j \in \Delta_ {1 0} \cup \Delta_ {0 0}: p _ {j} \in \tilde {S} _ {1 0} (\zeta) \right\} \approx \# \left\{j: p _ {j} \in \tilde {S} _ {1 0} (\zeta) \right\}, \\ \# \left\{j \in \Delta_ {0 0}: p _ {j} \in \widehat {S} (\zeta) \right\} \approx \# \left\{j \in \Delta_ {0 0}: p _ {j} \in \tilde {S} _ {0 0} (\zeta) \right\} \approx \# \left\{j: p _ {j} \in \tilde {S} _ {0 0} (\zeta) \right\}. \\ \end{array}
$$
The number of the selection  $\# \left\{j:p_j\in \tilde{S}_{01}(\zeta)\right\} +\# \left\{j:p_j\in \tilde{S}_{10}(\zeta)\right\} -\# \left\{j:p_j\in \tilde{S}_{00}(\zeta)\right\} +1$  can be considered as an overestimation of  $\# \left\{j:j\in \Delta_{00}\cup \Delta_{01}\cup \Delta_{10},j\in \widehat{\Delta}\right\}$ , and  $\widehat{\mathrm{FDP}} (\zeta)$  is given by
$$
\widehat {\mathrm {F D P}} (\zeta) = \frac {\# \left\{j : p _ {j} \in \tilde {S} _ {0 1} (\zeta) \right\} + \# \left\{j : p _ {j} \in \tilde {S} _ {1 0} (\zeta) \right\} - \# \left\{j : p _ {j} \in \tilde {S} _ {0 0} (\zeta) \right\} + 1}{\# \{j : p _ {j} \in \widehat {S} (\zeta) \} \vee 1}. \tag {19}
$$
Then the data-driven cutoff  $\zeta^{\star}$  can be determined as follows:
$$
\zeta^ {\star} = \sup  \left\{\zeta > 0: \widehat {\mathrm {F D P}} (\zeta) \leq \alpha \right\}, \tag {20}
$$
and the final selection is  $\widehat{\Delta}_{\zeta^{\star}} = \left\{j:p_j\in \widehat{S} (\zeta^{\star})\right\}$
Finally, we summarize our proposed FDR control procedure in Algorithm 1.
Algorithm 1 A data-driven algorithm for FDR control.
1: Calculate a pair of p-values  $p_j = (p_{1j}, p_{2j})$  following model (2), where  $j = 1, \dots, J$ .
2: Estimate the proportions of the composite null hypothesis  $\hat{\pi}_{00},\hat{\pi}_{01},\hat{\pi}_{10}$
3: Estimate the null densities  $\hat{f}_{01}(p),\hat{f}_{10}(p)$  and the mixture density  $\hat{f} (p)$  using the adaptation of the beta kernel estimator.
4: Estimate the  $\mathrm{fdr}(p)$  following (17).
5: For a nominal FDR level  $\alpha \in (0,1)$ , select the mediators  $\left\{j:p_j\in \widehat{S} (\zeta^{\star})\right\}$  where  $\widehat{S} (\zeta) = \{p:\widehat{\mathrm{fdr}} (p)\leq \zeta \}$  and the cutoff  $\zeta^{\star}$  is
$$
\zeta^ {\star} = \sup  \{\zeta > 0: \widehat {\mathrm {F D P}} (\zeta) = \frac {\# \left\{j : p _ {j} \in \tilde {S} _ {0 1} (\zeta) \right\} + \# \left\{j : p _ {j} \in \tilde {S} _ {1 0} (\zeta) \right\} - \# \left\{j : p _ {j} \in \tilde {S} _ {0 0} (\zeta) \right\} + 1}{\# \left\{j : p _ {j} \in \widehat {S} (\zeta) \right\} \vee 1} \leq \alpha \}.
$$
Remark 2. In a recent work of Deng et al. [13], a procedure called JM was introduced for detecting 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 partially 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 information from large-scale testing, AMDP can accurately estimate the local FDR. Additionally, motivated by the symmetric property of the composite null hypothesis, we proposed a data-driven algorithm to determine the optimal rejection region. As shown in (19), the number of the selection  $\# \{j:p_j\in \tilde{S}_{01}(\zeta)\} +\# \{j:p_j\in \tilde{S}_{10}(\zeta)\} -\# \{j:p_j\in \tilde{S}_{00}(\zeta)\} +1$ , provides a less conservative estimation of  $\# \{j:j\in \Delta_{00}\cup \Delta_{01}\cup \Delta_{10},j\in \hat{\Delta}\}$  than JM, which relies on conditional mirror conservation to estimate FDP in masked region.
Theorem 2 below shows that for any nominal FDR level  $\alpha \in (0,1)$ , both  $\mathrm{FDP}(\zeta^{\star})$  and  $\mathrm{FDR}(\zeta^{\star})$  are under control using Algorithm 1, as the sample size  $n$  and the number of mediators  $J$  tend to infinity.
Theorem 2. Assume that
(i)  $\frac{1}{J}\sum_{j = 1}^{J}\left|\widehat{\mathrm{fdr}} (p_j) - \mathrm{fdr}(p_j)\right|\xrightarrow{P}0\text{as} n,J\to \infty;$
(ii) For  $\zeta \in (0,1]$ ,  $\mathbb{P}(\mathrm{fdr}(p_j)\leq \zeta \mid j\in \Delta_{00}\cup \Delta_{01}\cup \Delta_{10})$  is continuous;
(iii) For any FDR level of  $\alpha \in (0,1)$ , there exists a constant  $\zeta_{\alpha} \in (0,1]$  such that  $\mathbb{P}(\mathrm{FDP}(\zeta_{\alpha}) \leq \alpha) \to 1$  as  $J \to \infty$ .
When Assumptions 1-2 hold, we have
$$
\operatorname {FDP} \left(\zeta^ {\star}\right) \leq \alpha + o _ {p} (1) \text {a n d} \lim  _ {n, J \rightarrow \infty} \operatorname {FDR} \left(\zeta^ {\star}\right) \leq \alpha .
$$
Proofs of Theorems 1-2 are given in Appendix E.
Remark 3. To demonstrate the effectiveness of our proposed method, we provide examples under two scenarios that highlight how the loss of associated information across tests can result in decreased power. We compare our method, AMDP, with the JS-mixture test[12] and the DACT test[30], which provides further evidence for the utility of AMDP. Due to space limitations, this section is postponed to the Appendix B.
3 Simulation Study
In this section, we conduct a 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 [12] and the DACT [30]. The DACT method consists of two variants: DACT (Efron) and
DACT (JC). The R implementations of JS-mixture and DACT can be found at https://github.com/cran/HDMT and https://github.com/zhonghualiu/DACT, respectively. The exposure  $X$  is simulated from  $\mathrm{Ber}(0.5)$ , then the mediator  $M_{j}$  and the outcome  $Y_{j}$  are generated as follows:
$$
Y _ {j} = \beta_ {j} M _ {j} + \epsilon_ {j}, \epsilon_ {j} \sim N (0, 1),
$$
$$
M _ {j} = \alpha_ {j} X + e _ {j}, e _ {j} \sim N (0, 1).
$$
We respectively calculate the FDP and true discovery proportion (TDP) as follows:
$$
\mathrm {F D P} = \frac {\# \left\{j : j \in \Delta_ {0} , j \in \widehat {\Delta} \right\}}{\# \{j \in \widehat {\Delta} \} \vee 1}, \quad \mathrm {T D P} = \frac {\# \left\{j : j \notin \Delta_ {0} , j \in \widehat {\Delta} \right\}}{\# \{j \notin \Delta_ {0} \} \vee 1}. \tag {21}
$$
where  $\Delta_0$  is the index set of the composite null mediators,  $\widehat{\Delta}$  is the index set of selected mediators. The FDR and power are measured by averaging FDP and TDP over 200 replications, respectively. Let  $\tau$  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  $(n,J) = (1000,10000)$ ,  $(\pi_{00},\pi_{01},\pi_{10},\pi_{11}) = (0.85,0.05,0.05,0.05)$ , and vary  $\tau$  from 0.6 to 1.2. Under  $\mathrm{H}_{00}$ ,  $\alpha_{j} = 0$  and  $\beta_{j} = 0$ ; under  $\mathrm{H}_{01}$ ,  $\alpha_{j} = 0$  and  $\beta_{j} = 0.3\tau$ ; under  $\mathrm{H}_{10}$ ,  $\alpha_{j} = 0.2\tau$  and  $\beta_{j} = 0$ ; under  $\mathrm{H}_{11}$ ,  $\alpha_{j} = 0.2\tau$  and  $\beta_{j} = 0.3\tau$ ;
Example 2. We fix  $(n,J) = (1000,10000)$ ,  $(\pi_{00},\pi_{01},\pi_{10},\pi_{11}) = (0.4,0.2,0.2,0.2)$ , and vary  $\tau$  from 0.6 to 1.2. Under  $\mathrm{H}_{00}$ ,  $\alpha_{j} = 0$  and  $\beta_{j} = 0$ ; under  $\mathrm{H}_{01}$ ,  $\alpha_{j} = 0$  and  $\beta_{j} = 0.3\tau$ ; under  $\mathrm{H}_{10}$ ,  $\alpha_{j} = 0.2\tau$  and  $\beta_{j} = 0$ ; under  $\mathrm{H}_{11}$ ,  $\alpha_{j} = 0.2\tau$  and  $\beta_{j} = 0.3\tau$ ;
Example 3. We fix  $(n,\tau) = (1000,1)$ ,  $(\pi_{00},\pi_{01},\pi_{10},\pi_{11}) = (0.85,0.05,0.05,0.05)$ , and vary  $J\in \{5000,8000,10000,20000\}$ . Under  $\mathrm{H}_{00}$ ,  $\alpha_{j} = 0$  and  $\beta_{j} = 0$ ; under  $\mathrm{H}_{01}$ ,  $\alpha_{j} = 0$  and  $\beta_{j} = 0.3\tau$ ; under  $\mathrm{H}_{10}$ ,  $\alpha_{j} = 0.2\tau$  and  $\beta_{j} = 0$ ; under  $\mathrm{H}_{11}$ ,  $\alpha_{j} = 0.2\tau$  and  $\beta_{j} = 0.3\tau$ ;
Example 4. We fix  $(n,\tau) = (1000,1)$ ,  $(\pi_{00},\pi_{01},\pi_{10},\pi_{11}) = (0.4,0.2,0.2,0.2)$ , and vary  $J \in \{5000,8000,10000,20000\}$ . Under  $\mathrm{H}_{00}$ ,  $\alpha_{j} = 0$  and  $\beta_{j} = 0$ ; under  $\mathrm{H}_{01}$ ,  $\alpha_{j} = 0$  and  $\beta_{j} = 0.3\tau$ ; under  $\mathrm{H}_{10}$ ,  $\alpha_{j} = 0.2\tau$  and  $\beta_{j} = 0$ ; under  $\mathrm{H}_{11}$ ,  $\alpha_{j} = 0.2\tau$  and  $\beta_{j} = 0.3\tau$ ;
Example 5. We fix  $(J,\tau) = (10000,1)$ ,  $(\pi_{00},\pi_{01},\pi_{10},\pi_{11}) = (0.85,0.05,0.05,0.05)$ , and vary  $n\in \{600,800,1000,1200\}$ . Under  $\mathrm{H}_{00}$ ,  $\alpha_{j} = 0$  and  $\beta_{j} = 0$ ; under  $\mathrm{H}_{01}$ ,  $\alpha_{j} = 0$  and  $\beta_{j} = 0.3\tau$ ; under  $\mathrm{H}_{10}$ ,  $\alpha_{j} = 0.2\tau$  and  $\beta_{j} = 0$ ; under  $\mathrm{H}_{11}$ ,  $\alpha_{j} = 0.2\tau$  and  $\beta_{j} = 0.3\tau$ .
Example 6. We fix  $(J,\tau) = (10000,1)$ ,  $(\pi_{00},\pi_{01},\pi_{10},\pi_{11}) = (0.4,0.2,0.2,0.2)$ , and vary  $n \in \{600,800,1000,1200\}$ . Under  $\mathrm{H}_{00}$ ,  $\alpha_{j} = 0$  and  $\beta_{j} = 0$ ; under  $\mathrm{H}_{01}$ ,  $\alpha_{j} = 0$  and  $\beta_{j} = 0.3\tau$ ; under  $\mathrm{H}_{10}$ ,  $\alpha_{j} = 0.2\tau$  and  $\beta_{j} = 0$ ; under  $\mathrm{H}_{11}$ ,  $\alpha_{j} = 0.2\tau$  and  $\beta_{j} = 0.3\tau$ .
Figure A1: The average FDR and power performance of four methods under Examples 1-2. The sample size  $n$  is 1000 and the number of mediators  $J$  is 10000. The effect size parameter  $\tau$  varies from 0.6 to 1.2. The nominal FDR level is 0.1.
To assess how the four methods are affected by effect size under sparse alternatives  $(\pi_{11} = 0.05)$  and dense alternatives  $(\pi_{11} = 0.2)$ , we apply the four methods in Examples 1-2. The effect size,  $\tau$ , is varied from 0.6 to 1.2 in both examples. The results of estimated FDR and power are summarized in Figure A1. 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 effectively control the FDR, leading to 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 to handle scenarios with a substantial proportion of  $H_{01}$  and  $H_{10}$  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 to the other three methods.
(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  $J \in \{5000, 8000, 10000, 20000\}$  with  $n = 1000$ ,  $\tau = 1$ ; (b) The comparison under Examples 5-6 for varying the sample size  $n \in \{600, 800, 1000, 1200\}$  with  $J = 10000$ ,  $\tau = 1$ .
Next, we move on to investigate whether the four methods are sensitive to changes in the large mediator 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 scenario 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
Figure A2. 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 a 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 (JC) encounters challenges in maintaining FDR control, particularly when  $n$  is small. In contrast, AMDP, JS-mixture, and DACT (Efron) effectively control the FDR across different settings. Regarding power performance, AMDP, JS-mixture, and DACT (Efron) demonstrate a consistent increase in power as  $n$  grows. In some settings, DACT (JC) appears to perform better than AMDP. 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 1.
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 [44]. Additionally, the number of GWASs-identified risk SNP that influence DNA methylation levels in prostate cancer has reached a total of 167 [6]. Despite significant progress in understanding the role of DNA methylation in gene expression regulation 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-mixture [12], and DACT [30], including DACT (Efron) and DACT (JC), to analyze the TCGA prostate cancer dataset. The dataset is freely available at https://portal.gdc.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  $\hat{\pi}_{00} = 0.52$ ,  $\hat{\pi}_{10} = 0.03$ , and  $\hat{\pi}_{01} = 0.42$ , 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  $\alpha$  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  $20.2\%$  higher than those of JS-mixture,  $86.6\%$  higher than those of DACT (Efron), and  $42.6\%$  higher than those of DACT (JC), across the range of  $\alpha$  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 4 are postponed to the Appendix C.
In this paper, we develop a novel adaptive mediation detection procedure (AMDP) to identify significant mediators in high-dimensional 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 analysis 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 are 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 optimally when confronted with low-dimensional mediators. This distinction is attributed to the nature of our method, wherein the two-dimensional p-values linked to each exposure-mediator-outcome 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.
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 a grant from the National Key R&D Program of China (2022YFA1003803), a grant from the National Social Science Foundation of China (21BTJ048), a grant from the National Scientific Foundation of China (12371276, 12131006), Basic Science Research Foundation of Shaanxi (22JSY038) and the Zhongying Young Scholar Program.