A grouped, selectively weighted false discovery rate procedure

TL;DR

This paper introduces a new grouped, selectively weighted FDR procedure called sGBH that improves power in structured hypotheses testing by focusing on relevant groups and selectively weighting p-values, without relying on a groupwise mixture model.

Contribution

The paper proposes the sGBH method, which enhances FDR control by selectively identifying and weighting hypothesis groups, outperforming existing grouped weighted FDR procedures.

Findings

sGBH demonstrates significantly improved power over GBH.

The method maintains FDR control under simple conditions.

Application to gene expression data shows practical effectiveness.

Abstract

False discovery rate (FDR) control in structured hypotheses testing is an important topic in simultaneous inference. Most existing methods that aim to utilize group structure among hypotheses either employ the groupwise mixture model or weight all p-values or hypotheses. Thus, their powers can be improved when the groupwise mixture model is inappropriate or when most groups contain only true null hypotheses. Motivated by this, we propose a grouped, selectively weighted FDR procedure, which we refer to as "sGBH". Specifically, without employing the groupwise mixture model, sGBH identifies groups of hypotheses of interest, weights p-values in each such group only, and tests only the selected hypotheses using the weighted p-values. The sGBH subsumes a standard grouped, weighted FDR procedure which we refer to as "GBH". We provide simple conditions to ensure the conservativeness of sGBH, together with empirical evidence on its much improved power over GBH. The new procedure is applied to a gene expression study.

Equations

$${\pi }_0=m^{-1}{\sum }_{j=1}^l{\pi }_{j0}{n}_j.$$

$${\tilde{\pi }}_0={\sum }_{j\in S}{\pi }_{j0}{n}_j{\left({\sum }_{j\in S}{n}_j\right)}^{-1},$$

$$v_j=\frac{{\pi }_{j0}\left(1-{\tilde{\pi }}_0\right)}{1-{\pi }_{j0}}\text{ for each }j\in S,$$

$${\hat{v}}_j=\frac{\left(n_j-R_j\left(\lambda \right)+1\right)\left(R\left(\lambda \right)+l-1\right)}{m\left(1-\lambda \right)R_j\left(\lambda \right)},$$

$${\hat{\pi }}_{0,\hat{S}}=\left({\sum }_{j\in \hat{S}}{\hat{\pi }}_{j0}{n}_j\right){\left({\sum }_{j\in \hat{S}}{n}_j\right)}^{-1}$$

$$\mathbb{E}\left[1/{\hat{\pi }}_{0,k}^{†}\right]\le 1/{\pi }_0^{\prime }\text{for each }k\in I_0^{\prime }.$$

$${\hat{\pi }}_0^{♯}=\frac{1}{m^{\prime }\left(1-\lambda \right)}+\frac{1}{m^{\prime }}\sum _{i=1}^{m^{\prime }}\frac{1_{\left\{p_i^{\prime }>\lambda \right\}}}{1-\lambda }$$

$${\hat{\alpha }}_m\le \frac{\alpha }{m_S}\frac{1}{1-{\check{\pi }}_0}{\sum }_{j\in S}{n}_j\left(1-{\pi }_{j0}\right)+\mathrm{Pr}\left(\hat{S}\ne S\right)+\mathrm{Pr}\left({\hat{\pi }}_{0,\hat{S}}>{\check{\pi }}_0\right).$$

$$\underset{m\to \infty }{\lim }\underset{j\in S}{\inf }\frac{1}{\sqrt{n_j}\left(1-{\pi }_{j0}\right)}\underset{t\in \left[0,1\right]}{\sup }\left|{\sum }_{i\in G_j\setminus S_{j0}}\left(1_{\left\{p_i\le t\right\}}-t\right)\right|=\infty$$

$$\underset{m\to \infty }{\mathrm{limsup}}\underset{j\in S}{\sup }{\pi }_{j0}=q\text{ for some }q\in [0,1)\text{,}$$

$${\varphi }_{\mu ,\sigma }\left(x\right)={\left(\sqrt{2\pi }\sigma \right)}^{-1}\exp \left(-2^{-1}{\sigma }^{-2}{\left(x-\mu \right)}^2\right)$$

$${\kappa }_{\sigma }\left(t;x\right)={\int }_{\left(-1,1\right)}\omega \left(\zeta \right)\exp \left(2^{-1}{t}^2{\zeta }^2{\sigma }^2\right)\cos \left(t\zeta x\right)d\zeta \text{,}$$

$${\phi }_n\left(t;\mathbf{y}\right)=\frac{1}{n}\sum _{i=1}^n\left(1-{\kappa }_{\sqrt{s_{ii}}}\left(t;y_i\right)\right).$$

$$n^{-2}{\Vert \mathbf{S}\Vert }_1=O(n^{-\delta })\text{for some }\delta >0,$$

$${\hat{\pi }}^{\ast }={\phi }_n\left(\sqrt{2\gamma \log n};\mathbf{y}\right)\text{ for some }\gamma \in (0,2^{-1}\delta ].$$

$$\underset{n\to \infty }{\lim }\sqrt{2\gamma \ln n}\min \left\{\left|u_j\right|:u_j\ne 0,1\le j\le n\right\}=\infty .$$

$$4^{-2}{m}^{-2}{\Vert {\mathbf{\Sigma }}_k\Vert }_1=\frac{1}{16}\frac{1+{\rho }_k}{1-{\rho }_k}{m}^{-1}+O\left(m^{-2}\right).$$

$${\mathcal{H}}_k=\left\{H_i:i=4^{-1}m\left(k-1\right)+1,\dots ,4^{-1}mk\right\}$$

$$\frac{\left(1-{\tilde{\pi }}_0\right)m_S}{\left(1-{\pi }_0\right)m}=1.$$

$$p_{j_k}^{\ast }=\frac{p_{j_k}{\pi }_{j0}}{1-{\pi }_{j0}}\left(1-{\pi }_0\right)\text{ and }{\tilde{p}}_{j_k}=\frac{p_{j_k}{\pi }_{j0}}{1-{\pi }_{j0}}\left(1-{\tilde{\pi }}_0\right)$$

$$p_{j_k}^{\ast }\ge {\tilde{p}}_{j_k}\text{ for each }j\in S\text{ and }k\in G_j.$$

$${\hat{\pi }}_{0,S}=\left({\sum }_{j\in S}{\hat{\pi }}_{j0}{n}_j\right){\left({\sum }_{j\in S}{n}_j\right)}^{-1},$$

$${\hat{p}}_{j_k}^{\ast }=\frac{p_{j_k}{\hat{\pi }}_{j0}}{1-{\hat{\pi }}_{j0}}\left(1-{\hat{\pi }}_0\right)\text{ and }{\hat{\tilde{p}}}_{j_k}=\frac{p_{j_k}{\hat{\pi }}_{j0}}{1-{\hat{\pi }}_{j0}}\left(1-{\hat{\pi }}_{0,S}\right).$$

$$D_r=\left\{{\mathbf{p}}_S\in {\left[0,1\right]}^{m_S}:R<r\right\}$$

$$t\mapsto \mathrm{Pr}\left(R<r|p_{i^{\prime }}\le t\right)$$

$$1>{\pi }_0\ge {\tilde{\pi }}_0\text{ and }m\ge m_S.$$

$$p_{j_k}^{\ast }=\frac{p_{j_k}{\pi }_{j0}}{1-{\pi }_{j0}}\left(1-{\pi }_0\right)\text{ and }{\tilde{p}}_{j_k}=\frac{p_{j_k}{\pi }_{j0}}{1-{\pi }_{j0}}\left(1-{\tilde{\pi }}_0\right)$$

$$\kappa =\frac{{\tilde{p}}_{j_k}}{p_{j_k}^{\ast }}=\frac{1-{\tilde{\pi }}_0}{1-{\pi }_0}\ge 1\text{ for each }{j}_k\in G_S.$$

$$p_{\left(R^{\ast }\right)}^{\ast }\le \frac{R^{\ast }\alpha }{m}\text{ and }{p}_{\left(r\right)}^{\ast }>\frac{r\alpha }{m}\text{ for all }{m}_S\ge r>R^{\ast }.$$

$$\frac{\left(1-{\tilde{\pi }}_0\right)m_S}{\left(1-{\pi }_0\right)m}=1$$

Full text

1 Introduction

Multiple testing aiming at false discovery rate (FDR) control has been routinely applied in genomics, genetics, neuroimaging, drug safety study and other fields. In many multiple testing scenarios, there is prior information on certain characteristics of hypotheses or statistics. For example, groups of hypotheses may have different proportions of true nulls, or statistics associated with a group of hypotheses may possess the same type of dependency structure or have similar powers. To utilize such information, methods based on hypotheses grouping and weighting (Liu et al., 2016; Basu et al., 2018) or p-value grouping and weighting (Cai and Sun, 2009; Hu et al., 2010; Chen et al., 2017; Nandi and Sarkar, 2018) have been developed.

Even though these methods can often be more powerful than the procedures of Benjamini and Hochberg (1995), Benjamini and Hochberg (2000) and Sun and Cai (2007), they have some limitations. For example, those in the Bayesian paradigm, e.g., “TLTA” in Liu et al. (2016), employ the groupwise mixture model where component densities are assumed to be continuous, whereas those in the frequentist paradigm, e.g., the “GBH” procedures of Hu et al. (2010); Nandi and Sarkar (2018), weight each p-value by treating each group of hypotheses equally importantly.

In contrast, there are many multiple testing settings with the following two features: (i) the number of false nulls is relatively small and some groups of hypotheses may contain no false nulls, which happens in, e.g., differential gene expression studies based on microarrays, and (ii) the null distributions of test statistics or p-values are different from each other and the groupwise mixture model is inappropriate, as in, e.g., differential gene expression studies based on discrete RNA-seq data. Even though “TLTA” can accommodate (i) by taking into account false discoveries both within and between groups and GBH accommodates (ii) for sub-uniform p-values with heterogeneous null distributions, neither of them is able to accommodate both (i) and (ii).

1.1 Main contribution

We propose “sGBH”, a grouped and selectively weighted multiple testing procedure as a refinement and extension of GBH. A schematic comparison between GBH and sGBH is given in Figure 1, and the details on sGBH are provided in Section 2.1. Unlike GBH, sGBH first identifies groups of hypotheses of interest and then weights p-values in each such group only. It reduces to GBH when all groups are of interest, and whenever this is the case, results that hold for GBH hold for sGBH also. Compared to TLTA, sGBH does not depend on the groupwise mixture model, and its weights are either induced by estimators of the groupwise proportions or those introduced by Nandi and Sarkar (2018). In essence, sGBH integrates the appealing features of both GBH and TLTA, and is able to account for each of the two features mentioned earlier.

When adapting sGBH to data through estimating the oracle weights, we consider two different ways of doing it and thus produce two adaptive versions of sGBH. In version one, the weights are induced by the estimated groupwise proportions of true nulls, whereas in version two, the weights are borrowed from Nandi and Sarkar (2018). Unlike Hu et al. (2010), who assumed the convergence of various empirical processes related to p-values and the asymptotic conservativeness of each estimator of the groupwise proportion of true nulls, we provide simple conditions on the conservativeness of the first version of the adaptive sGBH. The key to achieve this is a reciprocally conservative or consistent estimator of the proportion of true nulls. As such, our method presents a general strategy to non-asymptotically bound the FDR and show the conservativeness of an adaptively weighted and grouped testing procedure whose weights are induced by the estimated groupwise proportions of true nulls. On the other hand, once groups with false nulls are selected and groups with all true nulls are correctly identified but not selected, the second version of the adaptive sGBH reduces to the one-way GBH procedure of Nandi and Sarkar (2018) and is automatically conservative non-asymptotically under independence.

In addition, we propose a variant of sGBH and justify its conservativeness under similar conditions to those for the first version of the adaptive sGBH. We show that under a “sparse configuration” where each interesting group of hypotheses contains some false nulls and the rest all true nulls, neither the oracle sGBH nor the variant can reject more false nulls than the oracle GBH, even though the oracle GBH cannot be implemented and its power not obtainable in non-asymptotic settings. Specifically, our simulations show that, under nontrivial sparse configurations, both versions of the adaptive sGBH are conservative and usually more powerful than their corresponding versions of the adaptive GBH. Further, we argue that outside nontrivial sparse configurations, the variant and its adaptive version respectively can be more powerful than the oracle GBH and its adaptive version, even though we were not able to theoretically identify conditions that guarantee so. These findings are perhaps the first on better adaptation to group structures of hypotheses for FDR control in multiple testing in the frequentist paradigm.

1.2 Organization of article

The rest of the article is organized as follows. Section 2 introduces sGBH and investigates its conservativeness, and Section 3 provides two versions of the adaptive sGBH. A simulation study on the adaptive sGBH is given in Section 4, and an application in Section 5. The article ends with a discussion in Section 6. Proofs related to sGBH are given in the appendix, and a variant of sGBH and some simulations results are provided in the supplementary material.

2 Grouped hypotheses testing and sGBH

Consider simultaneously testing $m$ null hypotheses $\left\{H_{i}\right\}_{i=1}^{m}$ with their corresponding p-values $\left\{p_{i}\right\}_{i=1}^{m}$. Unless otherwise noted, each $p_{i}$ is assumed to be “super-uniform” under its associated null hypothesis, i.e., its null distribution $F_{i}$ satisfies $F_{i}\left(t\right)\leq t$ for $t\in\left[0,1\right]$. We assume throughout this article that $\min\left\{p_{i}:i\geq 1\right\}>0$ almost surely, in order to avoid the undetermined operation $0\times\infty$ when a p-value is [math] and a weight is $\infty$.

Let $\alpha\in\left(0,1\right)$ be a nominal FDR level. Recall the BH procedure of Benjamini and Hochberg (1995) (“BH” for short) as follows: let $\left\{p_{\left(i\right)}\right\}_{i=1}^{m}$ be the order statistics (non-decreasing in $i$) of $\left\{p_{i}\right\}_{i=1}^{m}$, and $H_{\left(i\right)}$ the null hypothesis associated with $p_{\left(i\right)}$ for each $i$; set $\theta=\max\left\{i:p_{\left(i\right)}\leq\frac{i}{m}\alpha\right\}$, and reject $H_{\left(j\right)}$ for $1\leq j\leq\theta$ if $\left\{i:p_{\left(i\right)}\leq\frac{i}{m}\alpha\right\}\neq\varnothing$ but reject none otherwise.

The BH is an ungrouped testing procedure. In contrast, the settings for grouped hypotheses testing are described as follows. For each natural number $s$, let $\mathbb{N}_{s}=\left\{1,\ldots,s\right\}$. Let $I_{0}$ be the index set of true nulls with cardinality $m_{0}$, $\pi_{0}=m_{0}m^{-1}$ be the proportion of true nulls, and $\pi=1-\pi_{0}$ be the proportion of false nulls. Let the $l$ non-empty sets $\left\{{G_{j}}\right\}_{j=1}^{l}$ be a partition of $\mathbb{N}_{m}$, and accordingly let $\left\{H_{i}\right\}_{i=1}^{m}$ be partitioned into $\mathcal{H}_{j}=\left\{H_{j_{k}}:k\in G_{j}\right\}$ for $j\in\mathbb{N}_{l}$. For each $j$, let $n_{j}$ be the cardinality of $G_{j}$, $\pi_{j0}$ be the proportion of true nulls for $\mathcal{H}_{j}$, and $\pi_{j1}=1-\pi_{j0}$. We refer to the partition and its associated proportions of true nulls and cardinalities as “a hypotheses configuration”. Note that

[TABLE]

2.1 The oracle sGBH and its properties

Let $S$ be a subset of $\mathbb{N}_{l}$, and call each $\mathcal{H}_{j}$, $j\in$ $S$ an “interesting group” and $\mathcal{H}_{j}$, $j\notin$ $S$ an “uninteresting group”. The oracle sGBH is described as follows: (1) set $p_{j_{k}}=\infty$ for each $j\notin S$ and $k\in G_{j}$, i.e., accept the hypothesis set $\mathcal{H}_{S^{\prime}}=\left\{H_{j_{k}}:j\notin S,k\in G_{j}\right\}$; (2) obtain the proportion of true nulls among all interesting groups as

[TABLE]

define the weight

[TABLE]

and weight the p-value $p_{j_{k}}$ into $\tilde{p}_{j_{k}}=p_{j_{k}}v_{j}$ for each $k\in G_{j}$ and $j\in S$; (3) when $\tilde{\pi}_{0}=1$, no rejections are made; otherwise, apply BH to the weighted p-values in the interesting groups, i.e., to the p-value set $\tilde{\mathbf{p}}_{S}=\left\{\tilde{p}_{j_{k}}:j\in S,k\in G_{j}\right\}$ and their corresponding nulls.

We have three remarks regarding the oracle sGBH: firstly, practically speaking $S$ contains groups of hypotheses each of which contains some false nulls; secondly, $v_{j}=\infty$ is set when $\tilde{\pi}_{0}=1$ and/or $\pi_{j0}=1$; thirdly, the oracle GBH of Hu et al. (2010) always sets $S=\mathbb{N}_{l}$ and weighs all $m$ p-values, and is hence subsumed by the oracle sGBH.

Since in practice a group of hypotheses may or may not contain any false nulls, we introduce

Definition 1

A “sparse configuration” for $\left\{H_{i}\right\}_{i=1}^{m}$ is such that $\pi_{j0}=1$ for $\mathcal{H}_{j}$ with $j\notin S$ but $\pi_{j0}<1$ for $\mathcal{H}_{j}$ with $j\in S$ for a subset $S$ of $\mathbb{N}_{l}$. When $S\neq\varnothing$, a sparse configuration is called “nontrivial”; otherwise, it is called “trivial”.

A nontrivial sparse configuration excludes the trivial case where all the $m$ null hypotheses are true, and in most statistical applications a hypotheses configuration is sparse.

Set $\mathbf{p}=\left(p_{1},\ldots,p_{m}\right)$. Recall from Benjamini and Yekutieli (2001) the property “positive regression dependency on each one from $I_{0}$ (PRDS)”, i.e., for any measurable non-decreasing set $D\subseteq\left[0,1\right]^{m}$, the function $t\mapsto\Pr\left(\left.\mathbf{p}\in D\right|p_{i}=t\right)$ is nondecreasing for each $i\in I_{0}$. We are ready to assert the conservativeness of the oracle sGBH under PRDS via the following theorem.

Theorem 1

Consider a sparse configuration for $\left\{H_{i}\right\}_{i=1}^{m}$. If $S=\mathbb{N}_{l}$ or $\varnothing$, then the oracle sGBH coincides with the oracle GBH. However, when $S\neq\varnothing$, the oracle sGBH and the oracle GBH reject the same set of hypotheses at the same nominal FDR level. Further, when $\left\{p_{i}\right\}_{i=1}^{m}$ have the property of PRDS, the oracle sGBH is conservative.

Even though Theorem 1 asserts that the oracle sGBH and the oracle GBH coincide under nontrivial sparse configurations, the former procedure not only reduces the complexity of the latter by applying the BH procedure only to p-values from the interesting groups but also helps gauge, as we will provide some numerical evidence, the reduced power of GBH when the set of interesting groups is not always correctly estimated under a nontrivial sparse configuration. In Section 2.2 we will introduce a quasi-adaptive form of sGBH, as a precursor to the (fully) data-adaptive sGBH to be developed in Section 3.

2.2 The quasi-adaptive sGBH

Let the “quasi-adaptive sGBH (qGBH)” be such that $S$ in the oracle sGBH is replaced by its estimate $\hat{S}$ but each $\pi_{j0}$, $j\in\mathbb{N}_{l}$ is retained. Under a nontrivial sparse configuration, qGBH interpolates the oracle GBH, the oracle sGBH and their adaptive versions. In particular, the power difference between the oracle GBH and qGBH reflects the impact of estimating $S$ on the oracle GBH.

We have an interesting result on the conservativeness of qGBH:

Proposition 1

Consider a nontrivial sparse configuration where $S$ has cardinality $1$. Then the qGBH is conservative when p-values satisfy PRDS regardless of how $\hat{S}$ is constructed.

Proposition 1 also implies that when there is only one interesting group, the adaptive sGBH (to be introduced in Section 3) is asymptotically conservative when it employs consistent estimators of each groupwise proportion of true nulls. In contrast to the claim of Proposition 1, when the cardinality of $S$ is greater than $1$ under a nontrivial sparse configuration, the involved probability estimates are rather complicated and somewhat intractable, and it is much harder to derive a concise, relatively tight upper bound on the FDR of qGBH.

A simulation study involving Gaussian distributional setting has been carried out to compare the qGBH with the oracle GBH under the same simulation design provided in Section 4.1, except that the minimal nonzero Normal means has magnitude $\mu_{\ast}=0.1$. For this simulation, $\hat{S}$ is constructed as follows. For each group $\mathcal{H}_{j}$, $j\in\mathbb{N}_{l}$, Simes test (Simes, 1986) to test the global null ${H}_{j}^{\dagger}$ that $\mathcal{H}_{j}$ contains no false nulls is applied at Type I error level $\xi=0.01$, and $\hat{S}$ contains each index $j^{\prime}$ for which ${H}^{\dagger}_{j^{\prime}}$ is rejected. We choose a weak signal setting with $\mu_{\ast}=0.1$ and a relatively more stringent Type I error level $\xi=0.01$ to make it less easy for the qGBH to estimate $S$ correctly, and thus to illustrate the reduced power of GBH under a nontrivial sparse configuration when $S$ has to be estimated.

Figure 2 presents the powers and FDRs of the qGBH and the oracle GBH for $4$ scenarios where $S$ has not always been correctly estimated. For $m=4000$ and $\tilde{\pi}_{0}=0.8$ or $0.9$, $S$ has only been correctly estimated at most $91\%$ of the times, and the power loss of the oracle sGBH can be as large as $5\%$. It can be perceived that the less frequently $S$ is correctly estimated and the less accurately each $\pi_{j0}$ is estimated, the more power loss the adaptive GBH will incur compared to its oracle version. We will study data-adaptive sGBH in Section 3 and empirically show in Section 4 that it is more powerful than the adaptive GBH for nontrivial sparse configurations.

3 Adaptive versions of sGBH

Since in practice there is often prior information on the number of groups for the hypotheses and their sizes, we assume that the partition $\mathcal{H}_{j}=\left\{H_{j_{k}}:k\in G_{j}\right\}$ for $j\in\mathbb{N}_{l}$ is known but that $S$ is unknown. The adaptive sGBH is obtained by replacing $S$ by its estimate $\hat{S}$, setting each p-value in each group $G_{j},j\notin\hat{S}$, to be infinite, replacing each $v_{j}$, $j\in\hat{S}$, by its estimate $\hat{v}_{j}$, and applying BH to the set of weighted p-values $\left\{p_{j_{k}}\hat{v}_{j}:j\in\hat{S},k\in G_{j}\right\}$. The method used to estimate $S$ and the $v_{j}$’s determines the version of adaptive sGBH.

There are mainly two ways to obtain $\hat{v}_{j},j\in S$. Hu et al. (2010) obtained each $\hat{v}_{j}$ by respectively replacing $\pi_{j0}$ and $\pi_{0}$ in the definition of $v_{j}$ by their estimates $\hat{\pi}_{j0}$ and $\hat{\pi}_{0}=m^{-1}\sum\nolimits_{j=1}^{l}\hat{\pi}_{j0}n_{j}$. We refer to these $\hat{v}_{j}$’s as “plug-in” weights and the resulting adaptive sGBH (or adaptive GBH) as the “plug-in” adaptive sGBH (or adaptive GBH). When there are at least two groups, there does not seem to exist any theoretical justification that the plug-in adaptive GBH is conservative non-asymptotically. To overcome this issue, Nandi and Sarkar (2018) estimated each $v_{j}$ by

[TABLE]

where $\lambda\in(0,1)$ is a tuning parameter, $R_{j}\left(\lambda\right)=\sum_{i\in G_{j}}\mathbf{1}_{\left\{p_{i}\leq\lambda\right\}}$ and $R\left(\lambda\right)=\sum_{i=1}^{m}\mathbf{1}_{\left\{p_{i}\leq\lambda\right\}}$. We refer to these weights as “generic weights” and the resulting adaptive sGBH (or adaptive GBH) as the “generic adaptive GBH” (or generic adaptive GBH). In fact, Theorem 2 of Nandi and Sarkar (2018) justifies the non-asymptotic conservativeness of the generic adaptive GBH when p-values are independent and null p-values are uniformly distributed.

We point out that for the generic adaptive sGBH there is no need to estimate the $\tilde{\pi}_{0}$ in (2), and for the plug-in adaptive sGBH (and plug-in adaptive GBH) we will use the same estimator for each $\pi_{j0}$ unless otherwise noted. Note that the plug-in adaptive sGBH makes no rejections when $\hat{\pi}_{0,\hat{S}}=1$, where

[TABLE]

estimates $\tilde{\pi}_{0}$.

To investigate the conservativeness of the plug-in adaptive sGBH, we start with proportion estimators and introduce the following definition.

Definition 2

Let there be $m^{\prime}$ null hypotheses $\left\{H_{i}^{\prime}\right\}_{i=1}^{m^{\prime}}$, for which $I_{0}^{\prime}$ is the index set of true nulls and $p_{i}^{\prime}$ is the p-value associated with $H_{i}^{\prime}$. Let $\pi_{0}^{\prime}$ be the proportion of true nulls for $\left\{H_{i}^{\prime}\right\}_{i=1}^{m^{\prime}}$. An estimator $\hat{\pi}_{0}^{{\dagger}}$ of $\pi_{0}^{\prime}$ based on the p-values $\mathbf{p}^{\prime}=\left(p_{1}^{\prime},\ldots,p_{m}^{\prime}\right)$ is called “non-increasing in $\mathbf{p}^{\prime}$” if $\hat{\pi}_{0}^{{\dagger}}$ is non-increasing in each $p_{i}^{\prime}$. Let $\hat{\pi}_{0,k}^{{\dagger}}$ be the estimator obtained by applying $\hat{\pi}_{0}^{{\dagger}}$ to $\mathbf{p}_{0,k}^{\prime}=\left(p_{1}^{\prime},\ldots,p_{k-1}^{\prime},0,p_{k+1}^{\prime},\ldots,p_{m}^{\prime}\right)$ for each $k\in I_{0}^{\prime}$. $\hat{\pi}_{0}^{{\dagger}}$ is called “reciprocally conservative” (or has the property of “reciprocal conservativeness”) if

[TABLE]

Inequality (5) together with Jensen’s inequality implies $\mathbb{E}\left(\hat{\pi}_{0,k}^{{\dagger}}\right)\geq\pi_{0}$. If $\hat{\pi}_{0}^{{\dagger}}$ is non-increasing, then $\hat{\pi}_{0,k}^{{\dagger}}\leq\hat{\pi}_{0}^{{\dagger}}$ almost surely for any $k\in I_{0}^{\prime}$, and for such $\hat{\pi}_{0}^{{\dagger}}$ reciprocal conservativeness implies conservativeness. When p-values are independent but uniformly distributed under the true nulls, reciprocal conservativeness has been observed and used by Benjamini et al. (2006), Sarkar (2008), Blanchard and Roquain (2009) and Chen et al. (2018) to show the conservativeness of adaptive FDR procedures, and examples of non-increasing and reciprocally conservative estimators, including Storey’s estimator of Storey et al. (2004), are given by Corollary 13 of Blanchard and Roquain (2009). Specifically, Storey’s estimator is defined as

[TABLE]

for a tuning parameter $\lambda\in(0,1)$. However, these reciprocally conservative estimators are usually inconsistent. A consistent estimator will be discussed in Section 3.2.

We will provide simple conditions on the non-asymptotic conservativeness of the plug-in adaptive sGBH. Let $m_{S}$ be the cardinality of $\tilde{\mathbf{p}}_{S}$ (and hence of $\mathbf{p}_{S}=\left\{p_{j_{k}}:j\in S,k\in G_{j}\right\}$), and $\hat{\alpha}_{m}$ the FDR of the plug-in adaptive sGBH. We have

Theorem 2

Consider a nontrivial sparse configuration where $\tilde{\pi}_{0}\in[0,1)$ uniformly in $m$, and assume that $\left\{p_{i}\right\}_{i=1}^{m}$ are mutually independent. If each $\hat{\pi}_{j0},j\in\hat{S}$ is non-increasing and reciprocally conservative and $\Pr\left(\hat{\pi}_{0,\hat{S}}<1\right)>0$, then there exists a constant $\check{\pi}_{0}\in[0,1)$ such that $\Pr\left(\hat{\pi}_{0,\hat{S}}\leq\check{\pi}_{0}\right)>0$ and

[TABLE]

If further $\hat{S}$ and $\hat{\pi}_{0,\hat{S}}$ consistently estimate $S$ and $\tilde{\pi}_{0}$ respectively, then $\limsup_{m\rightarrow\infty}\hat{\alpha}_{m}\leq\alpha$. On the other hand, if $\left\{p_{i}\right\}_{i=1}^{m}$ have the property of PRDS, $\hat{S}$ is consistent for $S$ and $\hat{\pi}_{j0}$ is consistent for $\pi_{j0}$ uniformly in $j\in\hat{S}$ (without necessarily being non-increasing or reciprocally conservative), then $\limsup_{m\rightarrow\infty}\hat{\alpha}_{m}\leq\alpha$.

In Theorem 2, the condition “$\tilde{\pi}_{0}\in[0,1)$ uniformly in $m$” excludes the case “$\tilde{\pi}_{0}=1$ for some $m$”, for which the plug-in adaptive sGBH makes only false rejections (if any), and the assumption $\Pr\left(\hat{\pi}_{0,\hat{S}}<1\right)>0$ excludes the case where the plug-in adaptive sGBH makes no rejections and is conservative. Theorem 2 does not require the number of groups $l$ to be constant in $m$, allows the use of any non-increasing and reciprocally conservative estimator of the proportion of true nulls for each interesting group, and accounts for effects of selecting groups of hypotheses of interest. It generalizes Theorem 3 in Chen et al. (2017). In particular, when all groups are of interest, $\hat{S}=\left\{1,\ldots,l\right\}$ can be set, and the upper bound $\eta_{m}$ on the right hand side of (7) reduces to that provided by Theorem 3 of Chen et al. (2017).

The inequality (7) gives an integrated view on how grouping, groupwise proportions, their estimates and selecting groups for weighting affect the FDR of the plug-in adaptive sGBH. Specifically, $\eta_{m}$ bounds $\hat{\alpha}_{m}$ from above, and $\eta_{m}\leq\alpha$ implies the conservativeness of the procedure. So, for each $\alpha\in\left(0,1\right)$, the solution to $\eta_{m}\leq\alpha$ in terms of the accuracy of $\hat{S}$ and $\hat{\pi}_{0,\hat{S}}$, groupwise proportions $\left\{\pi_{j0}\right\}_{j\in S}$ and the constant $\check{\pi}_{0}$ corresponds to a setting where the plug-in adaptive sGBH is conservative non-asymptotically (as in our simulation studies in Section 4).

3.1 Estimating the subset of interesting groups

We deal with estimating $S$ under a nontrivial sparse configuration. This is handled by the two-sided Kolmogorov-Smirnov (KS) test of Kolmogorov (1933) and Smirnov (1939). Specifically, for each $j\in\mathbb{N}_{l}$, apply the KS test to all p-values in group $G_{j}$ to test their uniformity, i.e., to test if they all follow the uniform distribution, at some Type I error level $\beta>0$, and let $\hat{S}$ contain all $j$ such that the uniformity of p-values in group $G_{j}$ is rejected. For each $j\in\mathbb{N}_{l}$, let $S_{j0}$ be the index set of true nulls among $\mathcal{H}_{j}$. We have

Lemma 1

Consider a nontrivial sparse configuration. Assume $\left\{p_{i}\right\}_{i=1}^{m}$ are independent such that each $p_{i}$ is continuous and each $p_{i},i\in I_{0}$ is uniformly distributed. If $\lim_{m\rightarrow\infty}\inf_{1\leq j\leq l}n_{j}=\infty$ and

[TABLE]

then $\hat{S}$ obtained by the KS test at any Type I error level $\beta>0$ satisfies $\lim_{m\rightarrow\infty}\Pr(\hat{S}=S)=1$.

In Lemma 1, the condition (8) requires that the empirical process associated with p-values corresponding to the false nulls in each interesting group is well separately from the uniform distribution on $\left[0,1\right]$. A condition that helps validate (8) is

[TABLE]

which requires that an interesting group always contain a positive proportion of false nulls. Other tests, such as the higher criticism (HC) of Donoho and Jin (2004) and the Simes test of Simes (1986), on if an $\mathcal{H}_{j}$ contains all true nulls can be used to estimate $S$. However, under a nontrivial sparse configuration, we prefer to theoretically work under the conditions of Lemma 1 and use the KS test to ensure $\lim_{m\rightarrow\infty}\Pr(\hat{S}=S)=1$, so that we may avoid dealing with the sparse case $1-\pi_{j0}\rightarrow 0$ with $j\in S$ better suited for HC and checking if the asymptotic power of Simes test tends to $1$.

Even though the validity of the KS test was proved for a sequence of i.i.d. observations, it essentially requires the supremum of a standardized empirical process to converge to the supremum of a Brownian bridge. Thus, the KS test may still perform well for strongly mixing random variables such as those being autoregressive with order $1$. This has been observed for the Normal means problem (to be defined in Section 3.2) with an autoregressive covariance matrix in the simulation study in Section 4. However, the KS test may be unreliable under moderately strong dependence, and due to its excellent power under independence it may be too stringent on testing uniformity when applied in practice.

3.2 Estimating the null proportions

We discuss consistent proportion estimation, which is related to the asymptotic conservativeness of the plug-in adaptive sGBH. There are only a few consistent proportion estimators which are provided by Swanepoel (1999), Meinshausen and Rice (2006), Jin (2008) and Chen (2019). These estimators complement each other in terms of their scopes of application. In particular, “Jin’s estimator” of Jin (2008) has excellent performance for estimating proportions related to Normal means, and its consistency has been extended by Chen (2018) to hold under a more general type of dependence structure called “principal covariance structure (PCS)”. In contrast, the estimators of Meinshausen and Rice (2006) and Swanepoel (1999) require more stringent conditions than Jin’s estimator in order to be consistent.

We introduce the extended estimator of Chen (2018) (referred to also as “Jin’s estimator” for conciseness of notation). Let $\mathsf{Normal}\left(\mathbf{a},\mathbf{A}\right)$ denote the Normal distribution (or Normal random vector) with mean vector $\mathbf{a}$ and covariance matrix $\mathbf{A}$. Let $\mathbf{y}=\left(y_{1},\ldots,y_{n}\right)$ with $y_{i}\sim\mathsf{Normal}\left(u_{i},s_{ii}\right)$ have covariance matrix $\mathbf{S}=\left(s_{ij}\right)$ and mean vector $\boldsymbol{u}=\left(u_{1},...,u_{n}\right)$. Note that $\mathbf{y}$ itself does not have to be a Normal random vector. Consider the “Normal means problem”, i.e., simultaneously testing the null $H_{i0}:u_{i}=0$ versus the alternative $H_{i1}:u_{i}\neq 0$ for $1\leq i\leq n$. In this setting, $I_{0}^{\ast}=\left\{i:1\leq i\leq n,u_{i}=0\right\}$, the proportion of zero Normal means $\pi_{0}^{\ast}$ is the ratio of the number of zero $u_{i}$’s to $n$, and the proportion of nonzero Normal means is $\pi^{\ast}=1-\pi_{0}^{\ast}$. The extended estimator $\hat{\pi}^{\ast}$ estimates $\pi^{\ast}$ and is defined as follows. Let

[TABLE]

for $\mu\in\mathbb{R}$ and $\sigma>0$, and $\omega$ be an even, real-valued function defined on $\left(-1,1\right)$ that is non-negative and bounded by some finite constant $K>0$ and Lebesgue integrates to $1$. Define

[TABLE]

and

[TABLE]

Then $\varphi_{n}$ usually under-estimates $\pi^{\ast}$.

Let $\left\|\mathbf{S}\right\|_{1}=\sum_{i,j=1}^{n}\left|\mathbf{S}\left(i,j\right)\right|$ and the big $O$ notation be “Landau’s big O”. When each pair of distinct entries of $\mathbf{y}$ is bivariate Normal and $\left\{y_{i}\right\}_{i=1}^{n}$ have a PCS such that

[TABLE]

we can set

[TABLE]

Then, $\hat{\pi}_{0}^{\ast}=1-\hat{\pi}^{\ast}$ estimates $\pi_{0}^{\ast}$. From Theorem 1 and the proof of Theorem 2 of Chen (2018), we can see that, $\hat{\pi}_{0}^{\ast}$ consistently estimates $\pi_{0}^{\ast}$ under PCS when $\pi_{0}^{\ast}\in(0,1]$, $\sup_{n\geq 1}\max_{1\leq i\leq n}s_{ii}\leq 1$ and

[TABLE]

We refer interested readers to Jin (2008) and Chen (2018) for the excellent empirical performances of $\hat{\pi}^{\ast}$ and $\hat{\pi}_{0}^{\ast}$ under PCS and various sparse settings. Note that, when $\mathbf{y}\sim\mathsf{Normal}\left(\boldsymbol{u},\mathbf{S}\right)$ has a PCS with $\max_{1\leq i\leq n}s_{ii}\leq 1$, $\mathbf{S}$ being a correlation matrix represents the most difficult case of estimating $\pi^{\ast}$ among different types of $\mathbf{S}$; see the discussion right after Theorem 2 of Chen (2018). The simulation study in Section 4 for the Normal means problem sets $\mathbf{S}$ to be a correlation matrix.

We remark on the relationship between PRDS and PCS. Consider a variant of the Normal means problem where $H_{i0}:u_{i}=0$ and $H_{i1}:u_{i}>0$ for each $1\leq i\leq n$. If $\mathbf{y}\sim\mathsf{Normal}\left(\boldsymbol{u},\mathbf{S}\right)$ has a PCS and $\mathbf{S}$ has nonnegative entries, then the distribution of $\mathbf{y}$ is PRDS on $I_{0}^{\ast}$, and so are the one-sided p-values $1-\Phi\left(y_{i}\right)$ for $1\leq i\leq n$, where $\Phi$ is the CDF for $\mathsf{Normal}\left(0,1\right)$; see Benjamini and Yekutieli (2001) for a justification on this. In other words, under these settings PRDS and PCS are compatible with each other to enable the theory presented by Theorem 2 on asymptotic FDR control of the plug-in adaptive sGBH.

4 Simulation study

We will employ Storey’s estimator or Jin’s estimator to compare the performances of the plug-in adaptive sGBH and plug-in adaptive sGBH, and also compare the performances of the generic adaptive GBH and generic adaptive sGBH. Let $\mathbf{z}\sim\mathsf{Normal}\left(\boldsymbol{\mu},\mathbf{\Sigma}\right)$, where $\mathbf{z}=\left(z_{1},\ldots,z_{m}\right)$, $\mathbf{\Sigma}=\left(\sigma_{ij}\right)$ and $\boldsymbol{\mu}=\left(\mu_{1},...,\mu_{n}\right)$. We consider the Normal means problem under PCS where $\mathbf{\Sigma}$ is a correlation matrix and for each $H_{i0}:\mu_{i}=0$, the two-sided p-value $p_{i}=2\Phi\left(-\left|z_{i}\right|\right)$ and the one-sided p-value ${p}_{i}=1-\Phi\left(z_{i}\right)$. Since sGBH coincides with GBH when $S=\varnothing$ or $S=\mathbb{N}_{l}$, a sparse configuration with $S\neq\varnothing$ and $S\neq\mathbb{N}_{l}$ will be used for the hypotheses.

4.1 Simulation design

For $\mathbf{z}\sim\mathsf{Normal}\left(\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$ with $\boldsymbol{\Sigma}$ being a correlation matrix, we consider $6$ values for $m$ as $4\times 10^{3}$, $10^{4}$, $2\times 10^{4}$, $4\times 10^{4}$, $8\times 10^{4}$ and $10^{5}$, and $2$ types of dependence encoded by $\boldsymbol{\Sigma}$ that satisfy PCS defined by (10). Specifically, $\boldsymbol{\Sigma}=\mathsf{diag}\left(\boldsymbol{\Sigma}_{1},\boldsymbol{\Sigma}_{2},\boldsymbol{\Sigma}_{3},\boldsymbol{\Sigma}_{4}\right)$ is block diagonal with $4$ blocks of equal sizes and $\boldsymbol{\Sigma}_{k}=\left(\sigma_{ij,k}\right)$ for $1\leq k\leq 4$, set as follows:

• •

“Independent”: $\sigma_{ij}=0$ when $i\neq j$, i.e., $\boldsymbol{\Sigma}$ is the identity matrix.

• •

“Autoregressive”: $\sigma_{ij,k}=\rho_{k}^{\left|i-j\right|}1_{\left\{i\neq j\right\}}$ with $\rho_{k}=0.1k$ for $k=1,\ldots,4$. Each $\boldsymbol{\Sigma}_{k}$ is the autocorrelation matrix of an autoregressive model of order $1$, such that

[TABLE]

The Autoregressive dependence given above is strongly mixing and interpolates block dependence and short-range dependence. Note that $\boldsymbol{\Sigma}$ itself encodes block dependence. Since each type of $\boldsymbol{\Sigma}$ has nonnegative entries, $\mathbf{z}$ satisfies PRDS on $I_{0}$. By the discussion at the end of Section 3.2, we see that the one-sided p-values satisfy PRDS on $I_{0}$ whereas the two-sided ones may not.

For sGBH, the configuration for $\left\{H_{i}\right\}_{i=1}^{m}$ is as follows. There are $4$ groups of hypotheses

[TABLE]

for $k=1,\ldots,4$, such that $S=\left\{1\right\}$ and $\tilde{\pi}_{0}=\pi_{10}=0.7,0.8$ or $0.9$. Namely, each $\mathcal{H}_{k}$ matches the corresponding $\boldsymbol{\Sigma}_{k}$ and only $\mathcal{H}_{1}$ contains false nulls. This particular partition for the hypotheses is not tailored for the adaptive sGBH to be more powerful than the adaptive GBH but is meant for easy computer simulation. However, it will reveal a general phenomenon about GBH in non-asymptotic settings as we will explain in Section 4.2.

The nonzero $\mu_{i}$’s are generated independently such that their absolute values $|\mu_{i}|$ are from the uniform distribution on the compact interval $\left[0.6,3.6\right]$ but each $\mu_{i}$ has probability $0.5$ to be negative or positive. Let $\mu_{\ast}=\min\left\{\left|\mu_{i}\right|:\mu_{i}\neq 0\right\}$, and recall the sufficient condition (12) needed to ensure the consistency of Jin’s estimator $\hat{\pi}_{0}^{\ast}$ when it is applied to estimate the $\pi_{j0}$’s. Here $\mu_{\ast}=0.6$, and (12) is satisfied. Further, $\hat{\pi}_{0}^{\ast}$ with $\gamma=0.5$ is used to estimate $\pi_{j0}$ for each $j\in\mathbb{N}_{l}$. Note that choosing $\gamma=2^{-1}\delta$, where $\delta$ is the “PCS index” appearing in (10) and $\delta=1$ here for each $\boldsymbol{\Sigma}_{k}$, leads to relatively fast convergence of $\hat{\pi}_{0}^{\ast}$ to achieve consistency but may cause $\hat{\pi}_{0}^{\ast}$ not to have relatively small variance.

There are $4$ nominal FDR levels, i.e., $\alpha=0.05,0.1,0.15$ or $0.2$. The simulation is implemented by independently repeating $200$ times each experiment determined by the quintuple $\left(\alpha,m,\tilde{\pi}_{0},\boldsymbol{\Sigma},p\right)$ for a total of $288=144\times 2$ scenarios, where $p$ denotes a one-sided or two-sided p-value. Storey’s estimator is implemented by the pi0est function with parameter ‘smoother’ from the q-value package. For the generic adaptive sGBH and generic adaptive GBH, the weights in (4) are obtained with $\lambda=0.25$, $0.5$ or $0.75$. When the KS test is used to estimate $S$, it is implemented at Type I error $\beta=0.025$.

4.2 Simulation results

We will visualize major results based on two-sided p-values in the main text but gather in the supplementary material those based on one-sided p-values. To measure the power of an FDR procedure, we use the expectation of the true discovery proportion (TDP), defined as the ratio of the number of rejected false nulls to the total number of false nulls. Note that FDR is the expectation of the false discovery proportion (FDP), defined as the ratio of the number of rejected true nulls to the total number of rejections. We also report but do not focus on the standard deviations of the FDP and TDP since smaller standard deviations for these quantities mean that the corresponding procedure is more stable in FDR and power.

Figure 3 presents the FDRs and powers of the plug-in adaptive sGBH and adaptive GBH that employ Jin’s estimator, Figure 4 those of the generic adaptive sGBH and adaptive GBH with tuning parameter $\lambda=0.5$, and Figure 5 those of the generic adaptive sGBH when the tuning parameter $\lambda$ (appearing in (4)) ranges in $\{0.25,0.5,0.75\}$, all based on two-sided p-values. For notational simplicity, the “adaptive GBH (or adaptive sGBH)” refers to the two versions, the plug-in and generic, adaptive GBH (or adaptive sGBH). The following can be observed: (i) the adaptive sGBH is conservative and always has smaller FDR than the adaptive GBH for all scenarios; (ii) the adaptive sGBH is more powerful than the adaptive GBH for all scenarios related to two-side p-values, and it is so for one-sided p-values except when $\tilde{\pi}_{0}=0.9$ and $m=4000$; (iii) as the number of hypotheses increases, the improvement in power of the plug-in adaptive sGBH upon the plug-in adaptive GBH tends to decrease; (iv) the FDR and power of the generic adaptive sGBH change little even if we change $\lambda$; (v) for two-sided p-values, the plug-in adaptive sGBH with Jin’s estimator has similar power to that of the plug-in adaptive sGBH with Storey’s estimator but is more powerful than the generic adaptive sGBH. Similar power characteristics can be observed for the plug-in adaptive GBH for two-sided p-values; (vi) the plug-in adaptive GBH with Storey’s estimator can be anti-conservative under independence, as indicated by the scenario $\tilde{\pi}_{0}=0.9$ and $m=4000$, and it can have very low power for one-sided p-values (since for a point null a one-sided p-value corresponds to a misspecified test and each of the means, i.e., $\mu_{i}$’s, has equal probability to be positive or negative in our simulation). The explanations for these observations are provided below.

Recall $m_{S}$ as the cardinality of $\mathbf{p}_{S}=\left\{p_{j_{k}}:j\in S,k\in G_{j}\right\}$. Under a nontrivial sparse configuration, we have $1>\pi_{0}\geq\tilde{\pi}_{0}$, $m\geq m_{S}$ and

[TABLE]

First, consider the two oracle procedures. Then

[TABLE]

for $j\in S$ and $k\in G_{j}$ respectively for the oracle GBH and the oracle sGBH. So

[TABLE]

Since neither of the oracle procedures rejects any hypothesis $H_{j}$ with $j\notin S$ and $k\in G_{j}$, the identities (14) and (16) force them to reject the same set of hypotheses. This is what Theorem 1 asserts.

Now consider the adaptive versions of the two oracle procedures when the adaptive sGBH correctly estimates $S$. Set

[TABLE]

which estimates $\tilde{\pi}_{0}$. Then, for $j\in S$ and $k\in G_{j}$, ${p_{j_{k}}^{\ast}}$ and ${\tilde{p}_{j_{k}}}$ respectively become

[TABLE]

Note that $\pi_{j0}=1$ for $j\notin S$. If $\hat{\pi}_{j^{\prime}0}\neq 1$ when $\pi_{j^{\prime}0}=1$ for some $j^{\prime}\notin S$, then the plug-in adaptive GBH will likely reject a hypothesis, say, $H_{j_{k}^{\prime}}$, from the group $G_{j^{\prime}}$, potentially leading to increased FDR and anti-conservativeness. However, when the adaptive sGBH correctly estimates $S$, it will never reject this $H_{j_{k}^{\prime}}$ and will never reject any $H_{j}$ with $j\notin S$ and $k\in G_{j}$, likely leading to conservativeness and potentially smaller FDR than the adaptive GBH. Further, due to the potential inconsistency of proportion estimation in non-asymptotic settings, the order between ${p_{j_{k}}^{\ast}}$ and ${\tilde{p}_{j_{k}}}$ for $j\in S$ and $k\in G_{j}$ given by (16) for the oracles no longer necessarily holds for ${\hat{p}_{j_{k}}^{\ast}}$ and ${\hat{\tilde{p}}_{j_{k}}}$, and (14) becomes $\left(1-\hat{\pi}_{0,S}\right)m_{S}\left(\left(1-\hat{\pi}_{0}\right)m\right)^{-1}$, not necessarily being $1$. This allows the plug-in adaptive sGBH to be uniformly more powerful than the plug-in adaptive GBH in non-asymptotic settings. However, as $m$ increases and becomes sufficiently large, the effect of asymptotic theory comes into play, the estimates of $S$ and each $\pi_{j0}$ become more accurate, and eventually both plug-in adaptive procedures converge to their oracle versions, having identical performance.

We have observed that the subset $S$ of interesting groups has been correctly estimated in each repetition of each simulation scenario except when $\tilde{\pi}_{0}=0.9$ and $m=4000$ and one-sided p-values were used, and for a few repetitions of some experiments one of the uninteresting groups has been identified as an interesting group. In other words, it is easier to consistently estimate $S$ than to consistently estimate $\pi_{j0}$ for each $j\in\mathbb{N}_{l}$. So, under a nontrivial sparse configuration, the adaptive sGBH tends to perform better than the adaptive GBH. The estimated groupwise proportions for the uninteresting groups and the estimate of $\tilde{\pi}_{0}$ are provided in the supplementary material. Regardless of if Jin’s estimator or Storey’s estimate is used, for each uninteresting group the frequency of its estimated null proportion being $1$ is considerably less than $1$, and the estimated $\pi_{0}$ and $\tilde{\pi}_{0}$ are often smaller than $1$. Therefore, the plug-in weights for these uninteresting groups are often finite, and the plug-in adaptive GBH tends to make more false discoveries and be less powerful than the plug-in adaptive sGBH.

Further, we explain why for one-sided p-values, the plug-in adaptive sGBH is less powerful than the plug-in adaptive GBH when $\tilde{\pi}_{0}=0.9$ and $m=4000$. When $\mu_{i}\neq 0$, its associated one-sided p-value $\check{p}_{i}=1-\Phi\left(z_{i}\right)$ tends to be larger when $\mu_{i}<0$ than it is when $\mu_{i}\geq 0$. So, many $\check{p}_{i}$’s will be relatively large when their corresponding $\mu_{i}$’s are negative and small, there is a power loss when conducting multiple testing based on $\check{p}_{i}$’s, the KS test will not have enough power based on $\check{p}_{i}$’s when $m$ is small, and the proportion estimators will have inflated biases. In fact, in this scenario, the subset $S$ of interesting groups is only correctly estimated for at most $39\%$ of the times, $\hat{\pi}_{0,\hat{S}}$ over-estimates $\tilde{\pi}_{0}$ and is often close to $1$ when $S$ is correctly estimated, $\hat{\pi}_{0}$ is often larger than $\hat{\pi}_{0,\hat{S}}$, and neither $\hat{\pi}_{0,\hat{S}}$ nor $\hat{\pi}_{0}$ is identically $1$. This leads the adaptive plug-in sGBH to make less rejections and hence be less powerful than the plug-in adaptive GBH since the latter is applied to all p-values whereas the former only to those in the estimated interesting groups.

Finally, we explain (a) why the two versions of the plug-in adaptive sGBH (or GBH) based on Jin’s estimator and Storey’s estimator have similar powers and (b) why the plug-in adaptive sGBH (or GBH) is more powerful than the generic adaptive sGBH (or GBH). Even though Jin’s estimator provides more accurately estimates of $\tilde{\pi}_{0}$ than Storey’s estimator, it estimates the groupwise proportions less accurately than Storey’s estimator. This explains (a). On the other hand, the plug-in weights are more aggressive estimates than the generic weights in terms of estimating the oracle weights $\{v_{j}\}_{j=1}^{l}$. So, the plug-in adaptive sGBH (or GBH) is often more powerful than the generic adaptive sGBH (or GBH), even though at the risk of being anti-conservative non-asymptotically. This explains (b).

We also examined the generic adaptive sGBH under the same settings given in Section 4.1 but used Simes test to identify the set $S$ of interesting groups at Type I error level $\xi=0.05,0.1$ or $0.2$. For each repetition of each of the $288$ experiments and each of the three $\xi$ values, $S$ was correctly identified. However, similar to the KS test, for a few repetitions of some experiments one of the uninteresting groups has been identified as an interesting group. Figure 6 presents the comparison between the generic adaptive sGBH and GBH, and Figure 7 the performance of the generic adaptive sGBH as $\xi$ changes, both for two-sided p-values. The generic adaptive sGBH is conservative and always more powerful than the generic adaptive GBH for all $3$ values of $\xi$, and the two procedures have competitive FDRs. The former procedure has more power improvement over and has smaller FDR than the latter when $\xi$ is smaller. This is reasonable since the smaller $\xi$ is, the less likely an uninteresting group that has no false nulls will be identified as an interesting group. We did not examine the generic adaptive sGBH that employs Simes test to select groups of hypotheses with a fixed Type I error level $\xi$ but as the tuning parameter $\lambda$ changes since we suspect that the FDR and power performances of the procedure under this setting should be similar as $\lambda$ changes, in view of the performances of the generic adaptive sGBH that employs the KS test at a fixed Type I error level but as $\lambda$ changes.

5 An application of sGBH

We apply the adaptive sGBH to the prostate cancer data set of Singh et al. (2002), with comparison to the adaptive GBH. The data set contains expressions of approximately $12600$ genes from $52$ patients with prostate tumors and $50$ normal specimens, and the target is to identify genes that are differentially expressed between the two biological conditions. Under each biological condition, each gene expression is modelled by a Normal random variable. The detailed analysis is given below.

For gene $i$, p-value $p_{i}$ from a two-sided two-sample t-test and the z-score $z_{i}=\Phi^{-1}(p_{i})$ are obtained. A total of $4374$ hypotheses are selected and partitioned into $3$ groups as follows: $\mathcal{H}_{1}$ contains hypotheses whose associated p-values are bigger than $0.7$ and has $1374$ elements; $\mathcal{H}_{2}$ contains $1500$ hypotheses that are randomly sampled from hypotheses whose associated p-values are between $0.15$ and $0.7$; $\mathcal{H}_{3}$ contains $1500$ hypotheses that are randomly sampled from hypotheses whose associated p-values are less than $0.15$. For such a configuration, $\mathcal{H}_{1}$ likely will contain many more true nulls than false nulls, and $\mathcal{H}_{3}$ many false nulls than true nulls. The PCS index $\delta$ defined by (10) for the z-scores are respectively $0.1826$, $0.0880$ and $0.1546$, revealing dependencies much stronger than covered by our simulation study. In fact, the KS test at Type I error level $0.1$ asserts that each group contains some false nulls. So, instead of the KS test, we use Simes test to select groups of interesting hypotheses as done in Section 2.2 but at Type I error level $0.1$, which claims $\mathcal{H}_{2}$ as interesting and $\hat{S}=\left\{2\right\}$. The estimated groupwise proportions are $\hat{\pi}_{10}=0.9317$, $\hat{\pi}_{20}=0.8171$ and $\hat{\pi}_{30}=1$, where $\gamma=(\delta-0.0001)/2$ is used for Jin’s estimator. This gives $\hat{\pi}_{0}=0.9158$ and $\hat{\pi}_{0,\hat{S}}=\hat{\pi}_{20}=0.8171$. At nominal FDR level $0.05$, the plug-in adaptive sGBH claims that $759$ are differentially expressed as compared to $487$ claimed by the plug-in adaptive GBH, showing a considerable improvement.

We are well aware that different configurations may affect the performances of the two plug-in adaptive procedures. So, we have tried other schemes of selecting from the $12600$ hypotheses and then partitioning the selected hypotheses with configurations different than the one given above. However, for all schemes we have tried the plug-in adaptive sGBH had no less rejections than the plug-in adaptive GBH, and for some the former had more rejections than the latter, all at the same FDR level. This, together with Proposition 1, the discussion right after it, and the simulation results in Section 4, suggests that with the same partition of hypotheses and the same proportion estimators, the plug-in adaptive sGBH usually does not have less rejections than the plug-in adaptive GBH. Maintaining $\hat{S}=\left\{2\right\}$ (when Simes test is used) or $\hat{S}=\left\{1,2,3\right\}$ (when the KS test is used) and the same nominal FDR level $0.05$, we applied the generic adaptive sGBH and generic adaptive GBH, and they identified the same number of differentially expressed genes. We caution that the FDRs of the adaptive GBH and sGBH may exceed the specified normal level $0.05$ due to a potential violation of the assumptions that ensure their non-asymptotic conservativeness.

6 Discussion

To better adapt to a group structure among hypotheses, we have proposed a grouped, selectively weighted FDR procedure, sGBH, that is a refinement and extension of the GBH and wFDR procedures of Hu et al. (2010), Chen et al. (2017) and Nandi and Sarkar (2018) and that accommodates scenarios where only a few groups are likely to be interesting. For the plug-in adaptive sGBH, we have provided simple conditions to ensure and some empirical evidence on its conservativeness, together with an FDR upper bound that quantifies the effect of estimating the interesting groups. Further, we have provided numerical evidence on conservativeness and improved power of the generic adaptive sGBH that employs Simes test to select interesting groups. These two versions of the adaptive sGBH have been numerically shown to be robust to the Type I error level of the test that is used to select interesting groups and the tuning parameter that is used to construct the weights, and to be robust to strongly-mixing dependence. As with any grouped FDR procedure, how hypotheses are partitioned affects inferential results, and we argue that this should usually be done carefully using a practitioner’s domain knowledge.

There are four issues left for future investigation. Firstly, it is worth developing an adaptive sGBH for multiple testing based on discrete p-values that may utilize the consistent proportion estimators proposed by Chen (2019) for discrete statistics. Secondly, we have not numerically examined the relative performances of the variant of sGBH and GBH since we were not able to identify configurations under which the former is more powerful than the latter. However, we believe they do exist. Thirdly, it is quite challenging to design a scheme to correctly, non-asymptotically identify interesting and uninteresting groups under a nontrivial sparse configuration and show the non-asymptotic conservativeness of the resulting adaptive sGBH. Fourthly, an unsettled issue is how to design data-adaptive weights that ensure the non-asymptotic conservativeness and improved power (with respect to BH) of a weighted FDR procedure under dependence (or even under PRDS). For a proposal for this under block dependence, we refer the readers to Guo and Sarkar (2016).

Appendix A Proofs related to sGBH

In the proofs here and in the supplementary material, the indicator function $\mathbf{1}_{A}$ of a set $A$ will be written as $\mathbf{1}A$ if $A$ is described by a proposition, and $\left|A\right|$ is the cardinality of $A$.

A.1 Proof of Theorem 1

The first claim is obvious. Now we show the third claim. When $S=$ $\varnothing$, the oracle sGBH reduces to the oracle GBH with $\tilde{\pi}_{0}=\pi_{0}=1$, makes no rejections and is thus conservative. So, it is left to consider the case $S\neq$ $\varnothing$, which implies $1>\pi_{0}>\tilde{\pi}_{0}$. In this case, the weights for p-values in the set $\mathbf{p}_{S}=\left\{p_{j_{k}}:j\in S,k\in G_{j}\right\}$ are all finite, whereas $p_{j_{k}}$ is set to be $\infty$ for each $j\notin S$ and $k\in G_{j}$. Therefore, the oracle sGBH makes no false discoveries from the set $\mathcal{H}_{S^{\prime}}=\left\{H_{j_{k}}:j\notin S,k\in G_{j}\right\}$, and we only need to study the oracle GBH applied to $\mathbf{p}_{S}$.

Even though Theorem 1 of Hu et al. (2010) showed the conservativeness of the oracle GBH when it is applied to $\mathbf{p}_{S}$, missing is the justification that the PRDS property of $\mathbf{p}_{S}$ is preserved when each $p_{i^{\prime}}\in\mathbf{p}_{S}$ is weighted by a finite, nonnegative deterministic number. Here we provide it. Recall $m_{S}$ as the cardinality of $\mathbf{p}_{S}$, denote also by $\mathbf{p}_{S}$ the vector formed by enumerating elements of $\mathbf{p}_{S}$, and let $R$ be the number of rejections made by the oracle GBH when it is applied to the set $\tilde{\mathbf{p}}_{S}$ of weighted p-values associated with the interesting groups. Clearly, $R$ is non-increasing in each p-value in $\mathbf{p}_{S}$. In particular, for each $r\in\left\{0,\ldots m_{S}\right\}$,

[TABLE]

is a non-decreasing set. Let $I_{0,S}$ be the index set of true nulls among $\mathcal{H}_{S}=\left\{H_{j_{k}}:j\in S,k\in G_{j}\right\}$. Then, the PRDS property of $\mathbf{p}_{S}$ and the arguments in the proof of Proposition 3.6 of Blanchard and Roquain (2008) imply that the function

[TABLE]

is nondecreasing for each $i^{\prime}\in I_{0,S}$ and $r\in\left\{0,\ldots m_{S}\right\}$. Thus, Theorem 1 of Hu et al. (2010) yields the claim that the oracle sGBH is conservative under PRDS. We remark that the conservativeness of the oracle sGBH follows by slightly adapting the arguments of Theorem 1 of Nandi and Sarkar (2018) to super-uniform null p-values.

Finally, we show the second claim. Note that $m_{S}$ is also the cardinality of $\mathcal{H}_{S}$. When $S\neq\varnothing$, we have

[TABLE]

By the arguments presented above, to see which among the oracle GBH and the oracle sGBH rejects more, we only need to check them based on $\tilde{\mathbf{p}}_{S}$. Each $p_{j_{k}}\in\mathbf{p}_{S}$ has been weighted into

[TABLE]

respectively by the oracle GBH and oracle sGBH. Let $G_{S}=\left\{j_{k}:{j}\in S\text{,}\ k\in G_{j}\right\}$. Then (19), (20) and no p-value taking value [math] together imply ${p_{j_{k}}^{\ast}\leq\tilde{p}_{j_{k}}}$ and

[TABLE]

Let the $i^{\prime}$-th order statistic among $\left\{\frac{{p}_{j_{k}}\pi_{j0}}{1-\pi_{j0}}:j_{k}\in G_{S}\right\}$ be $\vartheta_{\left(i^{\prime}\right)}$. Then the $i^{\prime}$-th order statistic among $\left\{{p_{j_{k}}^{\ast}:}j_{k}\in G_{S}\right\}$, denoted by ${p_{\left(i^{\prime}\right)}^{\ast}}$, is $\vartheta_{\left(i^{\prime}\right)}\left(1-\pi_{0}\right)$, and the $i^{\prime}$-th order statistic among $\left\{{\tilde{p}_{j_{k}}:}j_{k}\in G_{S}\right\}$, denoted by ${\tilde{p}_{\left(i^{\prime}\right)}}$, is $\vartheta_{\left(i^{\prime}\right)}\left(1-\tilde{\pi}_{0}\right)$, and ${p_{\left(i^{\prime}\right)}^{\ast}\leq\tilde{p}_{\left(i^{\prime}\right)}}$.

Let $R^{\ast}$ be the number of rejections made by the oracle GBH and $\tilde{R}$ that by the oracle sGBH. Then $R^{\ast}\leq m_{S}$ and

[TABLE]

With this, we see from (21), the identity

[TABLE]

and the orderings discussed in the previous paragraph that

[TABLE]

and ${\tilde{p}_{\left(r\right)}}>\kappa\frac{r\alpha}{m}=\frac{r\alpha}{m_{S}}$ for $m_{S}\geq r>R^{\ast}$. On the other hand, $\tilde{R}\leq m_{S}$ and

[TABLE]

So, from (21), (22) and the orderings discussed previously, we see

[TABLE]

and $p_{\left(s\right)}^{\ast}>\frac{1}{\kappa}\frac{s\alpha}{m_{S}}=\frac{s\alpha}{m}$ for all $m_{S}\geq s>\tilde{R}$. Consequently, both procedures reject the same set of hypotheses.

A.2 Proof of Proposition 1

Let $R^{\natural}$ and $\alpha_{m}^{\natural}$ be the number of rejections and FDR of qGBH, respectively. Let $m_{S}=\sum\nolimits_{j\in S}n_{j}$, $m_{\hat{S}}=\sum\nolimits_{j\in\hat{S}}n_{j}$ and

[TABLE]

Note that each $\pi_{j0}$ is known to qGBH and $\pi_{j0}=1$ for $j\notin S$. If $\hat{S}\cap S=\varnothing$, then $\pi_{0,\hat{S}}=1$, $R^{\natural}=0$ and $\alpha_{m}^{\natural}=0$. On the other hand, if $\hat{S}\cap S\neq\varnothing$ and $\hat{S}\cap\left(\mathbb{N}_{l}\setminus S\right)\neq\varnothing$, then $\pi_{0,\hat{S}}<1$ but no rejections will be made from any group $\mathcal{H}_{j}$ for $j\notin S$. Without loss of generality, let $S=\left\{1\right\}$. Then $R^{\natural}\leq n_{1}$ almost surely. Observing the identity $\left(1-\pi_{10}\right)n_{1}=\left(1-\pi_{0,\hat{S}}\right)m_{\hat{S}}$ when $\hat{S}\cap S\neq\varnothing$, we have

[TABLE]

where $S_{j0}$ is the index set of true nulls in $\mathcal{H}_{j}$. However, the quantity in (23) is upper bounded by $\alpha$ by the oracle property of sGBH under PRDS. Thus, the claim holds.

A.3 Proof of Theorem 2

We will show the claims in two steps: Step 1. “filter out irrelevant cases from the analysis” and Step 2. “obtain upper bounds on $\hat{\alpha}_{m}$”.

Step 1. Recall $\mathbf{p}=\left(p_{1},\ldots,p_{m}\right)$ and $R=R\left(\mathbf{p}\right)$ as the number of rejections made by the procedure. Then, we can assume $R\left(\mathbf{p}\right)\geq 1$. If $\pi_{j0}=0$, then the null hypotheses in group $G_{j}$ are all false and do not contribute to $\hat{\alpha}_{m}$. So, we can assume $\pi_{j0}>0$ for each $j$. Since $\min\left\{p_{i}:i\geq 1\right\}>0$ almost surely, a weighted p-value is [math] only when its associated weight is [math]. Further, it suffices to consider the case $\hat{S}\neq\varnothing$.

Step 2. Before we proceed further, we need to set up some notations. For each $j\in\mathbb{N}_{l}$, let $\mathbf{q}_{j}$ be the vector of p-values whose indices are in group $G_{j}$, and for each $k\in G_{j}$, let $\mathbf{q}_{j,-k}$ be the vector obtained by removing $p_{j_{{}^{k}}}$ from $\mathbf{q}_{j}$, and $\mathbf{q}_{j,0,k}$ the vector obtained by setting $p_{j_{k}}=0$ in $\mathbf{q}_{j}$. Let the plug-in adaptive sGBH employ the same proportion estimator $\hat{\pi}_{0}^{\dagger}$ to estimate $\pi_{j0}$ for each needed $j$. We will denote by $\hat{\pi}_{j0}$ and $\hat{\pi}_{j0,-j_{k}}$ respectively the estimates obtained by applying $\hat{\pi}_{0}^{\dagger}$ to $\mathbf{q}_{j}$ and $\mathbf{q}_{j,0,k}$ for a $j$ and $k\in G_{j}$. Recall $\hat{\pi}_{0}=m^{-1}\sum_{j=1}^{l}\hat{\pi}_{j0}n_{j}$. For any two vectors $\mathbf{\tilde{p}}$ and $\mathbf{\hat{p}}$ whose entries together partition $\left\{p_{1},\ldots,p_{m}\right\}$, we write $R\left(\mathbf{p}\right)$ equivalently as $R\left(\mathbf{p}\right)=R\left(\mathbf{\tilde{p}},\mathbf{\hat{p}}\right)$. Let $V\left(\mathbf{p}\right)$ be the number of false discoveries of the procedure, which is also written as $V$ for notational simplicity. Let $m_{\hat{S}}=\sum_{j\in\hat{S}}n_{j}$ and

[TABLE]

where we recall

[TABLE]

The rest of the proof will be divided into 2 parts: Part I for the first claim and Part II the second.

Part I: Since $\Pr\left(\hat{\pi}_{0,\hat{S}}<1\right)>0$, there must be a constant $\check{\pi}_{0}\in[0,1)$ such that the event $\mathcal{B}_{m}=\left\{\hat{\pi}_{0,\hat{S}}\leq\check{\pi}_{0}\right\}$ has positive probability. Let $\mathcal{B}_{m}^{\prime}$ be the complement of $\mathcal{B}_{m}$. Further, let $\mathcal{A}_{m}=\left\{\hat{S}=S\right\}$ with complement $\mathcal{A}_{m}^{\prime}$. Then

[TABLE]

For each $j\in S$ and $k\in G_{j}$, let

[TABLE]

Since $\hat{\pi}_{0}^{\dagger}$ is a non-increasing estimator, we have almost surely

[TABLE]

for each $j\in\hat{S}$ and $k\in S_{j0}$. Therefore,

[TABLE]

where for $j\in S$, $k\in S_{j0}$ and $1\leq r\leq m_{S}$,

[TABLE]

However, when $\left\{p_{i}\right\}_{i=1}^{m}$ are independent,

[TABLE]

Thus, $\gamma_{m}$ in (26) satisfies

[TABLE]

where the inequality in (27) holds since $\hat{\pi}_{0}^{\dagger}$ is reciprocally conservative, i.e., $\mathbb{E}\left[\left.1\right/\hat{\pi}_{j0,-j_{k}}\right]\leq\left.1\right/\pi_{j0}$. Combining (24), (26) and (27) gives (7), i.e.,

[TABLE]

If in addition both $\hat{S}$ and $\hat{\pi}_{0,\hat{S}}$ are consistent, then $\Pr\left(\hat{\pi}_{0,\hat{S}}=\tilde{\pi}_{0}\right)\rightarrow 1$, $\Pr\left(\hat{S}=S\right)\rightarrow 1$ and the event $\mathcal{B}_{m}=\left\{\hat{\pi}_{0,\hat{S}}\leq\check{\pi}_{0}\right\}$ can be set as $\left\{\hat{\pi}_{0,\hat{S}}\leq\tilde{\pi}_{0}\right\}$. By almost identical arguments used to obtain (7), we have

[TABLE]

Therefore, $\limsup_{m\rightarrow\infty}\hat{\alpha}_{m}\leq\alpha$.

Part II: To show $\limsup_{m\rightarrow\infty}\hat{\alpha}_{m}\leq\alpha$ when $\left\{p_{i}\right\}_{i=1}^{m}$ have the property of PRDS, $\Pr(\hat{S}\neq S)\rightarrow 0$ and $\Pr\left(\bigcap\nolimits_{j\in\hat{S}}\left\{\hat{\pi}_{j0}=\pi_{j0}\right\}\right)\rightarrow 1$. Let $\mathcal{C}_{m}=\bigcap\nolimits_{j\in S}\left\{\hat{\pi}_{j0}=\pi_{j0}\right\}$ and $\mathcal{D}_{m}$ be the complement of $\mathcal{C}_{m}$. Then,

[TABLE]

where the second inequality follows from the conservativeness of the oracle sGBH under PRDS. So, the claim holds.

A.4 Proof of Lemma 1

For each $j\in\left\{1,\ldots,l\right\}$, let $F_{j}$ be the empirical distribution of the p-values whose indices are in group $G_{j}$. Let $m_{j0}=n_{j}\pi_{j0}$ and $m_{j1}=n_{j}\left(1-\pi_{j0}\right)$. First of all, for each $j$,

[TABLE]

where $d_{j,0,m}\left(t\right)={m_{j0}^{-1}}\sum\nolimits_{i\in S_{j0}}\left(\mathbf{1}_{\left\{p_{i}\leq t\right\}}-t\right)$ and

[TABLE]

By the independence between $\left\{p_{i}\right\}_{i=1}^{m}$, we see that $\sup_{t\in\left[0,1\right]}\left|d_{j,0,m}\left(t\right)\right|\rightarrow$ [math] almost surely uniformly in $j\notin S$. By assumption (8) on $d_{j,1,m}\left(t\right)$, we see that

[TABLE]

By Massey (1950), (28) implies that the power of the KS test tends to $1$ as $m\rightarrow\infty$ uniformly in $j\in S$. Therefore, $\Pr(\hat{S}=S)\rightarrow 1$.

Supplementary material for “A grouped, selectively weighted false discovery rate procedure”

Xiongzhi Chen***Corresponding author: Department of Mathematics and Statistics, Washington State University, Pullman, WA 99164, USA; Email: xiongzhi.chen@wsu.edu.  and Sanat K. Sarkar†††Department of Statistical Science and Fox School of Business, Temple University, Philadelphia, PA 19122, USA; Email: sanat@temple.edu.

We provide in Appendix B a variant of sGBH and its properties, and in Appendix C additional simulation results. The powers of the plug-in adaptive sGBH and plug-in adaptive GBH based on Storey’s estimator and one-sided p-values are very close to zero and hence not reported.

Appendix B A variant of sGBH

In this section, we introduce a variant of the oracle sGBH outside the setting of sparse configuration. The notations in this section bear the same meanings as those for sGBH unless otherwise noted or defined. The main message is that, for non-sparse configurations, the variant can be more powerful than the GBH, whereas for sparse configurations it cannot.

Let $S$ be a subset of $\mathbb{N}$l, i.e., $S$ is the set of interesting groups of hypotheses, for which $\pi_{j0}=1$ is not necessarily required for any $j\notin S$. Set the weights as

[TABLE]

where $w_{j}=\infty$ is set when $\tilde{\pi}_{0}=1$ and/or $\pi_{j0}=1$. Weight each p-values $p_{i}$ into $\tilde{p}_{i}=p_{i}w_{j}$ for $i\in G_{j}$ for each $j\in\mathbb{N}_{l}$. Apply the BH procedure to the $m$ weighted p-values $\left\{{\tilde{p}_{i}}\right\}_{i=1}^{m}$. Unless otherwise noted, we will refer to the above procedure as the “variant”. Note that $S$ is preselected and do not have to be estimated, and that only p-values in the preselected groups are effectively weighted.

We assume that the same hypotheses configuration is used for both the oracle GBH and the variant and that the same proportion estimator is employed by the plug-in adaptive version of the variant (“adaptive variant” for short) and the plug-in adaptive GBH to obtain the plug-in weights.

Theorem 3

If $S=\mathbb{N}$l, then the variant coincides with the oracle GBH. However, under a nontrivial sparse configuration, the variant never rejects more false nulls than the oracle GBH at the same nominal FDR level. When $\left\{p_{i}\right\}_{i=1}^{m}$ have the property of PRDS and $\tilde{\pi}_{0}<1$, the variant is conservative.

Theorem 3 implies that one should not attempt a method as the variant under a nontrivial sparse configuration in order to reject more false nulls than the oracle GBH. On the other hand, one may attempt to identify conditions under which the uniform dominance

[TABLE]

holds and thus the variant rejects no less hypotheses than the oracle GBH, where $p_{j_{k}}^{\ast}=p_{j_{k}}v_{j}$ and $\tilde{p}_{j_{k}}=p_{j_{k}}w_{j}$ for $j\in\mathbb{N}_{l}$ and $k\in G_{j}$. We will present two scenarios where (30) can never hold. Set the proportion of true nulls for the uninteresting groups as

[TABLE]

and let

[TABLE]

Proposition 2

Assume $S\neq\varnothing$ and $S\neq\mathbb{N}_{l}$. If $\max\left\{\pi_{0},\tilde{\pi}_{0}\right\}<1$, then it cannot hold that $\varsigma\geq 1$ and $\tilde{\pi}_{0}\geq\rho_{0}$. On the other hand, if $\tilde{\pi}_{0}=1$ but $\pi_{0}<1$, then $\varsigma\geq 1$ cannot hold.

In Proposition 2, the opposite of either claim under its corresponding settings implies (30). We remark that the pessimistic conclusion from Proposition 2 does not mean that there is no other setting where the variant is more powerful than the oracle GBH. However, we find it very challenging to identify such settings. Recall

[TABLE]

and let $\tilde{\alpha}_{m}$ be the FDR of the adaptive variant. We have

Theorem 4

Assume $\tilde{\pi}_{0}\in[0,1)$ uniformly in $m$ and that $\left\{p_{i}\right\}_{i=1}^{m}$ are mutually independent. If each $\hat{\pi}_{j0},j\in S$ is non-increasing and reciprocally conservative and $\Pr\left(\hat{\pi}_{0,S}<1\right)>0$, then there exits a constant $\check{\pi}_{0}\in[0,1)$ such that $\Pr\left(\hat{\pi}_{0,S}\leq\check{\pi}_{0}\right)>0$ and

[TABLE]

If in addition $\hat{\pi}_{0,S}$ consistently estimates $\tilde{\pi}_{0}$, then $\limsup_{m\rightarrow\infty}\tilde{\alpha}_{m}\leq\alpha$. On the other hand, if $\left\{p_{i}\right\}_{i=1}^{m}$ have the property of PRDS and $\hat{\pi}_{j0}$ is consistent for $\pi_{j0}$ uniformly in $j\in S$ (with necessarily being non-increasing or reciprocally conservative), then $\limsup_{m\rightarrow\infty}\hat{\alpha}_{m}\leq\alpha$.

Theorem 4 bears the same spirit as Theorem 2 and generalizes Theorem 3 of Chen et al. (2017). The (adaptive) variant allows a user to select groups of hypotheses of interest and is very flexible.

B.1 Proof of Theorem 3

The first claim is obvious. Now we show the third claim. Let $R$ be the number of rejections made by the variant. From the proof of Theorem 1, we see that

[TABLE]

is nondecreasing for each $i\in I_{0}$ and $r\in\left\{0,\ldots m\right\}$. Recall $w_{j}=\frac{\pi_{j0}\left(1-\tilde{\pi}_{0}\right)}{1-\pi_{j0}}$ for $j\in S$ and $w_{j}=1$ for $j\notin S$. Since $\tilde{\pi}_{0}<1$, each weight $w_{j},j=1,\ldots,l$ is positive and finite. Let $\hat{\alpha}$ be the FDR of the variant. Then,

[TABLE]

From (34), we obtain, for each fixed $j\in\left\{1,\ldots,l\right\}$ and $k\in G_{j}$,

[TABLE]

Thus, (35) and (36) together imply $\hat{\alpha}\leq\frac{\alpha}{m}\sum_{k\in S_{j0}}\sum_{j=1}^{l}\frac{1}{w_{j}}$. However,

[TABLE]

So, $\hat{\alpha}\leq\alpha$, and the variant is conservative.

Finally, we show the second claim. Under the assumptions, the weights $v_{j}=\infty$ for $j\notin S$, i.e., ${p_{j_{k}}^{\ast}=\infty}$ for $j\notin S$ and $k\in G_{j}$, the weights $w_{j}$ and $v_{j}$ for $j\in S$ are all positive and finite, and both $\pi_{0}$ and $\tilde{\pi}_{0}$ are less than $1$. It is natural to look at the relative orders between

[TABLE]

for ${j}\in S$ and $k\in G_{j}$ and $p_{j_{k^{\prime}}^{\prime}}$ for ${j}^{\prime}\notin S$ and $k^{\prime}\in G_{j^{\prime}}$. Under a nontrivial sparse configuration, $\rho_{0}=1$ and the identity $\pi_{0}-\tilde{\pi}_{0}=m^{-1}\left(1-\tilde{\pi}_{0}\right)\sum\nolimits_{j\notin S}n_{j}$ implies $\pi_{0}\geq\tilde{\pi}_{0}$. This, together with (37), implies ${p_{j_{k}}^{\ast}\leq\tilde{p}_{j_{k}}}$ for all ${j}\in S$ and $k\in G_{j}$. So, the variant can never reject more false nulls than the oracle GBH, and the former may reject more hypotheses than the latter only by rejecting some true nulls.

B.2 Proof of Proposition 2

Let $s_{0}=\sum\nolimits_{j\in S}n_{j}$ and $q_{0}=\sum\nolimits_{j\notin S}n_{j}$. Consider the first claim. Let $a=\left(1-\pi_{0}\right)^{-1}$. Then $\frac{a}{1+a}=\frac{1}{2-\pi_{0}}$. Suppose $\varsigma\geq 1$ and $\tilde{\pi}_{0}\geq\rho_{0}$. Then $\min_{j\notin S}\pi_{j0}\geq\frac{1}{2-\pi_{0}}$ and $\rho_{0}\geq\frac{1}{2-\pi_{0}}$. When $\tilde{\pi}_{0}\geq\rho_{0}$, we must have $\rho_{0}\leq\pi_{0}\leq\tilde{\pi}_{0}$. In other words, (32) implies $\pi_{0}\geq\frac{1}{2-\pi_{0}}$. However, this forces $\pi_{0}=1$, contradicting $\max\left\{\pi_{0},\tilde{\pi}_{0}\right\}<1$. Now consider the second claim. Suppose $\varsigma\geq 1$. Then

[TABLE]

has to hold. This forces $\pi_{0}\in\left[1,2\right]$, contradicting $\pi_{0}<1$.

B.3 Proof of Theorem 4

The arguments to be presented next are very similar to those in the proof of Theorem 2, and the notations here bear the same meanings there unless otherwise defined or noted. It suffices to consider the setting where the number of rejections of the procedure $R=R\left(\mathbf{p}\right)\geq 1$ and $\pi_{j0}>0$ for each $j$. The rest of the proof will be divided into 2 parts: Part I for the first claim and Part II the second.

Part I: Let $\check{w}_{j}=c_{j_{k}}$ for $j\in S$ and $\check{w}_{j}=1$ for $j\notin S$, where $c_{j_{k}}$ is defined by (25). For each $j\in S$ and $k\in S_{j0}$, define

[TABLE]

The same strategy in Part I of the proof of Theorem 2 implies

[TABLE]

and

[TABLE]

Therefore, we have (33), i.e.,

[TABLE]

If in addition $\Pr\left(\hat{\pi}_{0,S}=\tilde{\pi}_{0}\right)\rightarrow 1$, then the same strategy in Part I of the proof of Theorem 2 gives

[TABLE]

which yields $\limsup_{m\rightarrow\infty}\hat{\alpha}_{m}\leq\alpha$.

Part II: To show $\limsup_{m\rightarrow\infty}\hat{\alpha}_{m}\leq\alpha$ when $\left\{p_{i}\right\}_{i=1}^{m}$ have the property of PRDS and $\Pr\left(\bigcap\nolimits_{j\in S}\left\{\hat{\pi}_{j0}=\pi_{j0}\right\}\right)\rightarrow 1$. Let $\mathcal{D}_{m}$ be the complement of the event $\bigcap\nolimits_{j\in S}\left\{\hat{\pi}_{j0}=\pi_{j0}\right\}$. Then

[TABLE]

where the second inequality follows from the conservativeness of the variant under PRDS. So, the claim holds.

Appendix C Additional simulation results