September 27, 2022
Multiple testing adjustments, such as the [1] step-up procedure for controlling the false discovery rate (FDR), are typically applied to families of tests that control significance level in the classical sense: for each individual test, the probability of false rejection is no greater than the nominal level. In this paper, we consider tests that satisfy only a weaker notion of significance level control, in which the probability of false rejection need only be controlled on average over the hypotheses. We find that the [1] step-up procedure still controls FDR in the asymptotic regime with many weakly dependent \(p\)-values and an increasing number of rejections, and that certain adjustments for dependent \(p\)-values such as the [2] procedure continue to yield FDR control in finite samples. Our results open the door to FDR controlling procedures in nonparametric and high dimensional settings where weakening the notion of inference may allow for power improvements.
Consider testing \(m\) hypotheses \(H_1,\ldots,H_m\). Let \(\mathcal{H}_0\subseteq\{1,\ldots,m\}\) denote the set of true null hypotheses. Given \(p\)-values \(p_1,\ldots,p_m\) for each of the hypotheses, we wish to form a multiple testing procedure which decides on a subset of hypotheses to reject. A common starting point for multiple testing procedures proposed in the literature is to assume that the \(p\)-values are formed from tests that control significance level in the classical sense, which implies \[\begin{align} \label{eq:level95alpha} \text{for all }t\in [0,1]\text{ and }i\in\mathcal{H}_0,\, P(p_i\le t) \le t. \end{align}\tag{1}\] One then adjusts the critical value so that some notion of multiple testing error, such as the false discovery rate (FDR), is controlled (see formal definitions below).
In this paper, we explore the possibility of forming FDR controlling multiple testing procedures from tests that satisfy a weaker average significance level control criterion: \[\begin{align} \label{eq:average95level95alpha} \text{for all }t\in [0,1], \,\frac{1}{m}\sum_{i\in \mathcal{H}_0} P(p_i\le t) \le t. \end{align}\tag{2}\] Such tests can be formed from confidence intervals (CIs) that weaken the classical definition of a CI by requiring coverage only on average over the reported CIs. Letting \(CI_1(t),\ldots,CI_m(t)\) be CIs for parameters \(\theta_1,\ldots,\theta_m\) with nominal \(100\cdot (1-t)\%\) coverage, the average coverage criterion requires \[\begin{align} \label{eq:average95coverage} \frac{1}{m}\sum_{i=1}^m P(\theta_i\notin CI_i(t))\le t. \end{align}\tag{3}\] Given null hypotheses \(H_i: \theta_i=\theta_{0,i}\), \(p\)-values formed from these CIs will, by definition, satisfy \(p_i\le t\) iff. \(\theta_{0,i}\notin CI_i(t)\). If the CIs satisfy (3 ) for each \(t\in[0,1]\), the resulting \(p\)-values will satisfy (2 ) since \(\frac{1}{m}\sum_{i\in \mathcal{H}_0} P(\theta_{0,i}\notin CI_i(t)) = \frac{1}{m}\sum_{i\in \mathcal{H}_0} P(\theta_i\notin CI_i(t)) \le \frac{1}{m}\sum_{i=1}^m P(\theta_i\notin CI_i(t))\le t\).
CIs satisfying the average coverage criterion (3 ) and related criteria have been developed in a number of settings ([3]; [4]; [5, Ch. 5.8]; [6]; [7]). They are particularly appealing in high dimensional or nonparametric settings involving regularized estimation, where impossibility results [8] severely restrict the scope for constructing classical tests and CIs. Additional settings where the average significance level condition (2 ) can be shown to hold have been considered in recent work by [9], [10] and [11].
We ask whether \(p\)-values satisfying the weaker condition (2 ) can be used as an input to multiple testing procedures used in the literature. We focus on multiple testing procedures designed to control the false discovery rate (FDR) of [1]. We find that average significance level control is indeed sufficient for certain multiple testing procedures to guarantee FDR control. In particular, average significance level control is sufficient to guarantee FDR control of the [1] procedure in the asymptotic regime of weakly dependent \(p\)-values and many hypotheses (\(m\to\infty\)) and of the [2] procedure with fixed \(m\) and arbitrary dependence among \(p\)-values. On the other hand, in contrast to the classical setting, we show by example that the [1] procedure does not in general have FDR control with fixed \(m\) and independent \(p\)-values, and that approaches that estimate the proportion of null hypotheses, such as the procedure of [12], can fail to control FDR even as \(m\to\infty\).
Much of the literature on FDR controlling multiple testing procedures takes a family of \(p\)-values satisfying the classical significance level control condition (1 ) as a starting point. An important exception is the literature on knockoff based FDR controlling procedures [13], which instead rely on the construction of auxiliary random variables, called knockoffs. Constructing knockoffs typically requires modeling assumptions such as the “model-\(X\)” framework, in which the joint distribution of regression covariates is known or estimated with sufficient accuracy [14], or restricting the procedure to low dimensional settings; see also [15] for an application of this approach under the assumption of a symmetric null distribution.
More recently, [16] have shown that \(e\)-values (random variables \(e_i\) satisfying \(E[e_i]\le 1\) for \(i\in\mathcal{H}_0\)) can be used as an input to FDR controlling procedures, thereby providing another approach to controlling FDR without the use of classical significance level controlling tests. A notion of average error control for \(e\)-values similar to the one used for \(p\)-values in the present paper has arisen independently in this literature; see [17], [18] and [10]. Interestingly, [17] use this idea to draw a connection between \(e\)-values and the knockoff literature cited above.
While we are not aware of previous results applying the average significance control criterion (2 ) to FDR control, the idea of requiring coverage or size control only on average is suggested by empirical Bayes interpretations of the FDR [12] and anticipated in some discussions in this literature [19]. Subsequent to the first draft of this paper, further results and applications involving FDR control using the average significance level control criterion (2 ) have been developed by [9], [10] and [11].
The rest of this paper is organized as follows. Section 2 introduces the setup and provides an overview of results. Section 3 presents finite sample results and their proofs. Section 4 presents results that are asymptotic in the number \(m\) of hypotheses being tested, while the proofs of the asymptotic results are contained in the Supplementary Materials.
A multiple testing procedure is a function that maps the \(p\)-values \(p_1,\ldots,p_m\) to a subset \(\mathcal{R}=\mathcal{R}(p_1,\ldots,p_m)\subseteq\{1,\ldots,m\}\) of rejected null hypotheses. The false discovery proportion (FDP) of a procedure \(\mathcal{R}\) is: \[\begin{align} \label{eq:fdp} \mathop{\mathrm{FDP}}(\mathcal{R},\mathcal{H}_0)=\frac{\#\left( \mathcal{R}\cap\mathcal{H}_0 \right)}{\#\mathcal{R}\vee 1} \end{align}\tag{4}\] where \(\#\mathcal{A}\) is the cardinality of \(\mathcal{A}\) and \(a\vee b\) denotes the maximum of \(a\) and \(b\). The false discovery rate (FDR) of this procedure is the expectation of the FDP: \[\begin{align} \label{eq:fdr} \mathop{\mathrm{FDR}}(\mathcal{R},\mathcal{H}_0,P) =E_P\mathop{\mathrm{FDP}}(\mathcal{R},\mathcal{H}_0) =E_P\left[ \frac{\#\left( \mathcal{R}\cap\mathcal{H}_0 \right)}{\#\mathcal{R}\vee 1} \right] \end{align}\tag{5}\] where \(E_P\) denotes expectation under the distribution \(P\) of the \(p\)-values. We say that \(\mathcal{R}\) controls the false discovery rate at level \(q\) if \(\mathop{\mathrm{FDR}}(\mathcal{R},\mathcal{H}_0,P)\le q\).
While some of our results are more general, our main focus is on the [1] step-up procedure, and generalizations such as those considered by [2], [12] and [20]. To describe these procedures, let \[\begin{align} \label{eq:fixed95t95R} \mathcal{R}^{\operatorname{fixed}}_t(p_1,\ldots, p_n) = \left\{ i: p_i\le t \right\}. \end{align}\tag{6}\] denote the fixed rejection region procedure with cutoff \(t\). That is, we reject all hypotheses with \(p\)-value less than \(t\). Let \[\begin{align} \label{eq:VtStRt} &V(t)=\sum_{i\in\mathcal{H}_0} I(p_i\le t) =\#\left( \mathcal{R}^{\operatorname{fixed}}_t \cap \mathcal{H}_0 \right), \quad S(t)=\sum_{i\notin\mathcal{H}_0} I(p_i\le t) =\#\left( \mathcal{R}^{\operatorname{fixed}}_t \backslash \mathcal{H}_0 \right) \nonumber \\ &\text{and }R(t)=V(t)+S(t)=\# \mathcal{R}^{\operatorname{fixed}}_t. \end{align}\tag{7}\] The FDP of \(\mathcal{R}^{\operatorname{fixed}}_t\) is given by \(V(t)/[R(t)\vee 1]\). The BH procedure can be motivated by noting that, while \(V(t)\) cannot be observed, one can form a conservative estimate by replacing it with \(m\cdot t\). This gives an estimate of the fixed rejection region FDR: \[\begin{align} \label{eq:fdr95hat} \widehat{\mathop{\mathrm{FDR}}}(t) = \frac{m\cdot t}{\# \mathcal{R}^{\operatorname{fixed}}_t\vee 1} = \frac{m\cdot t}{R(t)\vee 1}. \end{align}\tag{8}\] The BH procedure at nominal FDR level \(q\) uses a cutoff \(\hat{t}_{\operatorname{BH},q}\) based on this estimate: \[\begin{align} \label{eq:bhq} \mathcal{R}_{\operatorname{BH},q}(p_1,\ldots,p_m) = \{i: p_i\le \hat{t}_{\operatorname{BH},q}\} \quad\text{where}\quad \hat{t}_{\operatorname{BH},q} = \max\{t : \widehat{\mathop{\mathrm{FDR}}}(t)\le q\}. \end{align}\tag{9}\] A more general class of step-up procedures can be formed by using an estimate of the form \(\pi m t\) for \(V(t)\) and modifying the denominator using a nondecreasing function \(\beta\), called a shape function: \[\begin{align} \label{eq:step-up} \mathcal{R}_{\pi,\beta(\cdot),q}(p_1,\ldots,p_m) = \{i: p_i\le \hat{t}_{\pi,\beta(\cdot),q}\} \text{ where } \hat{t}_{\pi,\beta(\cdot),q} = \max\left\{t : \frac{\pi m t}{\beta(R(t))}\le q\right\}. \end{align}\tag{10}\] Such procedures have been considered by, among others, [2], [12] and [20].
When the \(p\)-values satisfy the classical significance level control condition (1 ), these procedures are known to have the following properties.
The BH procedure controls FDR when \(p\)-values are independent [1].
The estimate \(\widehat{\mathop{\mathrm{FDR}}}(t)\) is upwardly biased for the FDR of the fixed rejection region procedure \(\mathcal{R}^{\operatorname{fixed}}_t\) when \(p\)-values are independent [21], [22].
The procedure \(\mathcal{R}_{1,\beta(\cdot),q}\) (with \(\pi=1\)) controls FDR under arbitrary dependence for the shape function \(\beta(k)=k\left( \sum_{i=1}^m i^{-1} \right)^{-1}\) [2] and, more generally, when \(\beta(k)=\int_0^k xd\nu(x)\) for an arbitrary probability distribution \(\nu\) on \((0,\infty)\) [20].
The BH procedure controls FDR asymptotically (as \(m\to\infty\)) when the \(p\)-values satisfy a weak dependence condition [21], [23].
The procedure \(\mathcal{R}_{\hat{\pi}, \beta(\cdot), q}\), where \(\beta(t)=t\) and \(\hat{\pi}=(\sum_{i=1}^mI(p_i>\lambda)+1)/((1-\lambda) m)\) is an estimate of \(\#\mathcal{H}_0/m\), controls FDR (a) under fixed \(m\) with independent \(p\) values using a slight modification of the procedure [21]2 and (b) asymptotically as \(m\to\infty\) when the \(p\)-values satisfy a weak dependence condition [21], [23].
Our results can be summarized as showing that, when the \(p\)-values only satisfy the weaker average significance level control condition (2 ), properties (ii), (iii) and (iv) continue to hold, but that properties (i) and (v)(a) and (v)(b) in general do not. Section 3.1 shows property (iii) and provides a counterexample to property (i). Section 3.2 shows property (ii). Section 4 shows property (iv). A counterexample for property (v) is given in the Supplementary Materials.
This section considers finite sample control of FDR for step-up procedures (Section 3.1) and point estimation of FDR of the fixed rejection region procedure \(\mathcal{R}^{\operatorname{fixed}}_t\) (Section 3.2).
Our result on FDR control for step-up procedures is a corollary of a more general result that uses an invariance assumption on an oracle version of a multiple testing procedure. The basic idea is that, if the \(p\)-values satisfy the average significance level control condition (2 ), then one can form another multiple testing problem in which the classical condition (1 ) holds by randomly permuting the \(p\)-values of the true null hypotheses and multiplying them by \(m/\#\mathcal{H}_0\). One can then apply results from the literature to this new setting.
To state our result, we explicitly introduce notation \(\mathcal{R}(p_1,\ldots,p_m;\mathcal{H}_0)\) for oracle procedures that depend on the set of true null hypotheses \(\mathcal{H}_0\) (typically through the cardinality \(\#\mathcal{H}_0\) of this set). We use a permutation invariance condition \[\begin{align} \label{eq:permutation95invariance} i\in \mathcal{R}(p_1,\ldots,p_m) \quad\text{iff.}\quad \sigma(i)\in \mathcal{R}(p_{\sigma(1)},\ldots,p_{\sigma(m)}) \end{align}\tag{11}\] for any permutation \(\sigma\) of the indices \(1,\ldots,m\) of the tests. This includes the class of step-up procedures (10 ), so long as \(\pi\) is either a fixed number or a permutation invariant function of the \(p\)-values.
Theorem 1. Let \(\mathcal{R}\) be a multiple testing procedure that satisfies the permutation invariance condition (11 ), and suppose that the oracle procedure \(\widetilde{\mathcal{R}}(p_1,\ldots,p_m;\mathcal{H}_0) =\mathcal{R}(p_1(m_0/m),\ldots,p_m(m_0/m))\) (where \(m_0=\#\mathcal{H}_0\)) controls FDR at level \(q\) for any \((P,\mathcal{H}_0)\) satisfying the classical significance level control condition (1 ), regardless of the dependence structure of \(p_1,\ldots,p_m\) under \(P\). Then \(\mathcal{R}\) controls FDR at level \(q\) for any \((P,\mathcal{H}_0)\) such that the average significance level control condition (2 ) holds.
Proof. Given \((P,\mathcal{H}_0)\) such that (2 ) holds and \(p_1,\ldots,p_n\) drawn from \(P\), define \(\tilde{p}_i\) as follows. Let \(\sigma\) be a permutation of \(\mathcal{H}_0\), taken at random from the set of all permutations of \(\mathcal{H}_0\) with equal probability, independently of \(p_1,\ldots,p_m\). Extend \(\sigma\) to a permutation on \(\{1,\ldots,m\}\) by taking \(\sigma(i)=i\) for \(i\notin \mathcal{H}_0\). Let \(\tilde{p}_i=(m/m_0)p_{\sigma(i)}\), where \(m_0=\#\mathcal{H}_0\). Then, for \(i\in\mathcal{H}_0\) and \(t\in[0,1]\), \[\begin{align} P(\tilde{p}_i\le t) = \sum_{j\in\mathcal{H}_0} P(\sigma(i)=j)P(p_j(m/m_0)\le t|\sigma(i)=j) =\frac{1}{m_0}\sum_{j\in\mathcal{H}_0} P(p_j(m/m_0)\le t) \end{align}\] where we use independence of \(\sigma\) and \(p_j\) and the fact that \(P(\sigma(i)=j)=1/m_0\). Since \(p_1,\ldots,p_m\) satisfy (2 ) under \((P,\mathcal{H}_0)\), this is bounded by \((m/m_0)\cdot t m_0/m=t\). Thus, letting \(\tilde{P}\) denote the distribution of \(\tilde{p}_1,\ldots,\tilde{p}_n\) under \(P\), \((\tilde{P},\mathcal{H}_0)\) satisfies the classical significance level control condition (1 ). It follows by the assumptions of the theorem that the oracle procedure \(\widetilde{\mathcal{R}}(\tilde{p}_1,\ldots,\tilde{p}_m;\mathcal{H}_0)=\mathcal{R}(\tilde{p}_1(m_0/m),\ldots,\tilde{p}_m(m_0/m))=\mathcal{R}(p_{\sigma(1)},\ldots,p_{\sigma(m)})\) controls FDR at level \(q\) under \(\mathcal{H}_0\) when \(p_1,\ldots,p_m\) are drawn according to \(P\). But by permutation invariance of \(\mathcal{R}\) and the fact that \(\sigma\) maps \(\mathcal{H}_0\) to itself, we have \(\#\left( \mathcal{R}(p_{\sigma(1)},\ldots,p_{\sigma(m)}) \cap \mathcal{H}_0 \right) = \#\left( \mathcal{R}(p_{1},\ldots,p_{m}) \cap \mathcal{H}_0 \right)\). Also, \(\# \mathcal{R}(p_{\sigma(1)},\ldots,p_{\sigma(m)})=\# \mathcal{R}(p_{1},\ldots,p_{m})\) by permutation invariance. Thus, the FDR of \(\mathcal{R}(p_{m},\ldots,p_{m})\) is the same as the FDR of \(\mathcal{R}(p_{\sigma(1)},\ldots,p_{\sigma(m)})\), and is therefore bounded by \(q\). ◻
As a special case, applying Proposition 2.7 and Lemma 3.2(iii) in [20] gives the following.
Corollary 1. The class of dependence controlling step-up procedures of [20], given by (10 ) with \(\pi=1\) and \(\beta(r)=\int_{0}^{r}xd\nu(x)\) for some probability measure \(\nu\), controls FDR at level \(q\) for any \((P,\mathcal{H}_0)\) such that the average significance level control condition (2 ) holds. In particular, the step-up procedure of [2], which is given by (10 ) with \(\pi=1\) and \(\beta(r)=r/\left( \sum_{i=1}^m 1/i \right)\), controls FDR at level \(q\) for any \((P,\mathcal{H}_0)\) such that the average significance level control condition (2 ) holds.
Key requirements here are that the original procedure (a) controls FDR under arbitrary dependence and (b) can incorporate the \(m/m_0\) adjustment through an oracle result. In particular, (b) rules out procedures of the form \(\mathcal{R}_{\hat{\pi},\beta(\cdot),q}\) with \(\hat{\pi}\) an estimate of \(m_0/m\), as in [12]. Clearly, ruling out estimates of \(m_0/m\) is necessary, since such estimates attempt to use a bound \(m_0\cdot t\) on \(V(t)\), whereas average coverage only gives a bound of \(m\cdot t\) on the expectation of \(V(t)\) (see the Supplementary Materials for a counterexample). Regarding (a), note that even if the \(p\)-values satisfy some dependence structure that would guarantee FDR control under classical significance level control (e.g. independence or the positive regression dependency on a subset condition used by [2]), FDR control is not guaranteed. The following counterexample shows that (a) is necessary in general. In particular, the BH procedure need not control FDR under the average significance level control condition (2 ) and independent \(p\)-values.
Suppose \(m\ge 2\) and \(q<2/3\). Let \(P(p_1\le t)=t\cdot m\) for \(0\le t\le (3/2)\cdot (q/m)\) and let \(P(p_1\in ((3/2)\cdot (q/m),2q/m])=0\). Let \(P(p_2\in [a,b])=(b-a)\cdot m\) for any \((3/2)\cdot (q/m)\le a\le b\le 2q/m\) and let \(P(p_2\in [0,(3/2)\cdot (q/m)))=0\). We can then distribute the remaining probability mass of \(p_1,p_2\) and \(p_3,\ldots,p_m\) over the set \((2q/m,1]\) so that \(\frac{1}{m}\sum_{i=1}^m P(p_i\le t)\le t\) for all \(t\in [0,1]\) (for example, we can set \(p_3,\ldots,p_m\) to be equal to \(1\) with probability one, and we can set the remaining probability mass for \(p_1\) and \(p_2\) to point masses at \(1\)). Thus, the average significance level condition (2 ) holds, with \(\mathcal{H}_0=\{1,\ldots,m\}\). Now consider the FDR of the Benjamini-Hochberg procedure, which rejects all hypotheses \(i\) such that \(p_i\le q\hat{r}/m\) where \(\hat{r}\) is the number of rejected hypotheses. The FDR is equal to the probability of at least one rejection in this case (since \(\mathcal{H}_0=\{1,\ldots,m\}\)). Note that the event \(p_1\le q/m\) implies that hypothesis \(1\) is rejected, and this has probability \(q\). But the event \(q/m< p_1\le (3/2)\cdot (q/m)\) and \(p_2\le 2q/m\) has probability \((q/2)\cdot (q/2)\), and it is disjoint with the event \(p_1\le q/m\). This gives a lower bound of \(q+[(q/m)\cdot(1/2)]^2>q\) for the FDR. Thus, the FDR is not controlled at level \(q\).
It is worth mentioning here some further results that have been obtained subsequent to the first draft of this paper. First, [10] have developed results for \(e\)-values using a related notion of average error control: they show that if the \(e\)-values \(e_1,\ldots,e_m\) satisfy \((1/m)\sum_{i\in\mathcal{H}_0}E[e_i]\le 1\), then the \(e\)-BH procedure of [16] controls FDR at the nominal level. Using this result and a certain calibration of \(p\)-values to \(e\)-values, they provide an alternative proof of the main conclusion of Corollary 1. Second, [11] have obtained several results characterizing the FDR properties of the BH procedure under (2 ) and various dependence assumptions on the \(p\)-values. In particular, they show that, when \(p\)-values are independent, applying the procedure (10 ) with \(\pi=1\) and \(\beta(r)=r/1.93\) (i.e. the BH procedure at nominal level \(q/1.93\)) controls FDR. This provides a much less conservative procedure compared to those in Corollary 1 in the case of independent \(p\)-values.
We now consider using the BH cutoff as an estimate of the FDR for a fixed rejection region multiple testing procedure. Under independent \(p\)-values, it is known that \(\widehat{\mathop{\mathrm{FDR}}}(t)\) is an upwardly biased estimate of \(\mathop{\mathrm{FDR}}(\mathcal{R}^{\operatorname{fixed}}_t)\) under the classical significance level control condition (1 ) (see [21] and the correction by [22]). Indeed, a slightly weaker assumption of independence between null and non-null \(p\)-values suffices.3 We now show that this property continues to hold under the weaker average significance level control condition (2 ). The result essentially follows from the same arguments as in the case where the \(p\)-values satisfy the classical significance level control condition.
Theorem 2. Suppose that \((P,\mathcal{H}_0)\) satisfies the average significance level control condition (2 ) and that the null \(p\)-values \(\{p_i\}_{i\in\mathcal{H}_0}\) are statistically independent of the non-null \(p\)-values \(\{p_i\}_{i\notin\mathcal{H}_0}\). Then \(E_P\widehat{\mathop{\mathrm{FDR}}}(t)\ge \mathop{\mathrm{FDR}}(\mathcal{R}^{\operatorname{fixed}}_t,\mathcal{H}_0,P)\).
Proof. For \(V(t)\) and \(S(t)\) defined in (7 ), we have \[\begin{align} E_P\widehat{\mathop{\mathrm{FDR}}}(t) - \mathop{\mathrm{FDR}}(\mathcal{R}^{\operatorname{fixed}}_t,\mathcal{H}_0,P) =E_P\frac{m\cdot t-V(t)}{[V(t)+S(t)]\vee 1} \ge E_P\frac{m\cdot t-V(t)}{[m\cdot t+S(t)]\vee 1} \end{align}\] (the last step follows by noting that replacing \(V(t)\) with \(m\cdot t\) in the denominator weakly decreases the denominator when the numerator is negative and weakly increases the denominator when the numerator is positive). The result then follows by noting that \(S(t)\) and \(V(t)\) are independent by the independence assumption on \(p\)-values, and that \(E_PV(t)\le m\cdot t\) by the assumption that the \(p\)-values satisfy the average significance level control condition (2 ). ◻
We now consider asymptotic FDR control, under a sequence \(P=P^{(m)}\) of probability measures and \(\mathcal{H}_0=\mathcal{H}_0^{(m)}\) and \(m\to\infty\). We suppress the dependence on \(m\) whenever it doesn’t cause confusion, but we note that the \(p\)-values form a triangular array, since the distribution (and the set \(\mathcal{H}_0\) of true null hypotheses) can change with \(m\). Recall the definitions of \(V(t)\), \(S(t)\) and \(R(t)\) in (7 ). If the average significance level control condition (2 ) holds, and the \(p\)-values do not exhibit too much statistical dependence, we will have \[\begin{align} \label{eq:asymptotic95average95level} \frac{1}{m}V(t) \le t+o_P(1) \text{ for all }t\in [0,1]. \end{align}\tag{12}\] For some results, we also assume a law of large numbers for the total rejections and rejected true nulls: \[\begin{align} \label{eq:total95rejections95lln} \frac{1}{m}V(t) \stackrel{p}{\to} G(t)\le t \quad\text{and}\quad \frac{1}{m}R(t) \stackrel{p}{\to} F(t) \text{ for all }t\in [0,1]. \end{align}\tag{13}\] These assumptions are analogous to assumptions made for asymptotic FDR control under classical significance level control in the literature e.g. [21] Eq. (7)-(9). The difference here is that the conditions are weaker, since the upper bound in (12 ) is given by \(t\) rather than \(t\pi_0\) where \(\pi_0\) is the limit of \(\#\mathcal{H}_0/m\). As one might expect, this will lead to problems for “adaptive” procedures that attempt to estimate \(\pi_0\) (see the Supplementary Materials for a counterexample). However, as we now show, it is not a problem for the Benjamini-Hochberg procedure, which uses the conservative upper bound of \(1\). We first show conservative consistency of the BH cutoff (8 ) for the FDR (and FDP) of the fixed rejection region procedure.
Theorem 3. Let \(\widehat{\mathop{\mathrm{FDR}}}(t)\) be the BH estimate, given in (8 ), of the FDR of the fixed rejection region procedure \(\mathcal{R}^{\text{fixed}}_t\) given in (6 ) and suppose that (12 ) holds. Let \(\underline t\) be such that there exists \(\eta>0\) with \(\frac{1}{m}\sum_{i=1}^m I(p_i\le \underline t)\ge \eta + o_P(1)\). Then \[\begin{align} \inf_{t\in [\underline t, 1]}\left[ \widehat{\mathop{\mathrm{FDR}}}(t)-\mathop{\mathrm{FDP}}(\mathcal{R}^{\text{fixed}}_t,\mathcal{H}_0) \right] \ge o_P(1). \end{align}\] If, in addition, (13 ) holds for continuous functions \(G\) and \(F\), then, letting \(\mathop{\mathrm{FDR}}_\infty(t)=G(t)/F(t)\), we have \[\begin{align} &\sup_{t\in [\underline t, 1]} \left| \mathop{\mathrm{FDP}}(\mathcal{R}^{\text{fixed}}_t,\mathcal{H}_0) - \mathop{\mathrm{FDR}}_\infty(t) \right|\stackrel{p}{\to } 0, \quad \sup_{t\in [\underline t, 1]} \left| \mathop{\mathrm{FDR}}(\mathcal{R}^{\text{fixed}}_t,\mathcal{H}_0,P) - \mathop{\mathrm{FDR}}_\infty(t) \right|\to 0 \\ &\text{ and } \inf_{t\in [\underline t, 1]}\left[ \widehat{\mathop{\mathrm{FDR}}}(t)-\mathop{\mathrm{FDR}}(\mathcal{R}^{\text{fixed}}_t,\mathcal{H}_0,P) \right] \ge o_P(1). \end{align}\]
The proof of Theorem 3 is given in the Supplementary Materials. Next, we state a result showing asymptotic control of FDR for the BH procedure \(\mathcal{R}_{\operatorname{BH},q}\) defined in (9 ).
Theorem 4. Suppose Assumptions (12 ) and (13 ) hold for continuous functions \(G\) and \(F\) and that there exists \(t^*>0\) such that \(F(t^*)>0\) and \(G(t^*)/F(t^*)<q\). Then \[\begin{align} \mathop{\mathrm{FDP}}(\mathcal{R}_{\operatorname{BH},q},\mathcal{H}_0)\le q + o_P(1) \quad\text{and}\quad \mathop{\mathrm{FDR}}(\mathcal{R}_{\operatorname{BH},q},\mathcal{H}_0,P)\le q + o(1). \end{align}\]
The proof of Theorem 4 is given in the Supplementary Materials. The condition on \(t^*\) used in Theorem 4 imposes a lower bound on the proportion of total rejections relative to null rejections at nominal level \(t^*\). This condition requires that the hypothesis tests have sufficient power on average, and that the proportion \((m-\#\mathcal{H}_0)/m\) of non-null hypotheses is not too small as \(m\to\infty\). Similar conditions have been used in the asymptotic analysis of multiple hypothesis testing procedures under classical significance level control [21].
This supplementary material contains proofs of Theorems 3 and 4 from the main text, as well as a counterexample showing that procedures that use an estimate of the proportion of true null hypotheses may not control FDR when applied to average significance level controlling \(p\)-values.
We begin with a lemma regarding uniform convergence.
Lemma 1. If (12 ) holds, then \[\begin{align} \label{eq:uniform95t95asymptotic95size} \inf_{t\in [0,1]}\left[ t - V(t)/m \right]\ge o_P(1). \end{align}\tag{14}\] If (13 ) holds for continuous functions \(F\) and \(G\), then \[\begin{align} \label{eq:uniform95t95lln} \sup_{t\in [0,1]}\left| V(t)/m - G(t) \right|\stackrel{p}{\to } 0 \quad\text{and}\quad \sup_{t\in [0,1]}\left| R(t)/m - F(t) \right|\stackrel{p}{\to } 0. \end{align}\tag{15}\]
Proof. The result follows by arguing as in the proof of the Glivenko-Cantelli Theorem. Let \(\overline{t}_K(t)\) be the least element in \(\{0,1/K,\ldots,(K-1)/K,1\}\) that is greater than or equal to \(t\). To show (14 ), note that \(t-V(t)/m\ge t-V(\overline{t}_K(t))/m\ge \overline{t}_K(t)-V(\overline{t}_K(t)/m)-1/K\). Thus, \(\inf_{t\in [0,1]}\left[ t - V(t)/m \right]\ge \min_{t\in \{0,1/K,\ldots,(K-1)/K,1\}}\left[ t - V(t)/m \right]-1/K\). Applying (12 ) and taking \(K\to\infty\) gives (14 ).
Similarly, to show that (15 ) holds so long as (13 ) holds for continuous \(G\) and \(F\), note that \(V(t)/m - G(t) \le V(\overline{t}_K(t))/m - G(t) \le V(\overline{t}_K(t))/m - G(\overline{t}_K(t)) +\omega(1/K)\) where \(\omega(\varepsilon)=\sup_{s,t\in [0,1], |s-t|\le \varepsilon} |G(s)-G(t)|\) satisfies \(\lim_{\varepsilon\to 0}\omega(\varepsilon)=0\) by uniform continuity of \(G\) on the interval \([0,1]\). From this and an analogous lower bound, it follows that \(\sup_{t\in [0,1]} |V(t)/m-G(t)|\le \max_{t\in \{0,1/K,\ldots,(K-1)/K,1\}}|V(t)/m-G(t)| + \omega(1/K)\). Applying (13 ) and taking \(K\to 0\) gives the first part of (15 ). The same arguments applied to \(R(t)/m-F(t)\) give the second part of (15 ). ◻
We now prove Theorem 3. We begin by showing that the first claim in the theorem holds under (12 ). For any \(\varepsilon>0\), the event \(\inf_{t\in [\underline t, 1]}\left[\widehat{\mathop{\mathrm{FDR}}}(t)-\mathop{\mathrm{FDP}}(\mathcal{R}^{\text{fixed}}_t,\mathcal{H}_0)\right]<-\varepsilon\) implies that there exists \(t\in [\underline t, 1]\) such that \(t-V(t)/m<-\varepsilon(R(t)\vee 1)/m\le -\varepsilon (R(\underline t)\vee 1)/m\). This implies \(\inf_{t\in [\underline t, 1]}[t-V(t)/m]\le -\varepsilon\cdot\eta/2\) on the event \(R(\underline t)/m=\frac{1}{m}\sum_{i=1}^mI(p_i\le \underline t)\ge\eta/2\), which holds with probability approaching one by assumption. The first claim of Theorem 3 now follows by noting that the probability of this event converges to zero by (14 ).
Next, we show that the second claim of the theorem holds if (12 ) holds and (13 ) holds for some continuous \(F\) and \(G\). The first part of the second claim follows immediately from (15 ), using uniform continuity of the function \((a,b)\mapsto a/b\) over \(b\in [F(\underline t),1]\) since \(F(\underline t)\ge \eta>0\). The second part of the second claim then follows immediately from the dominated convergence theorem. The third part of the second claim follows from combining the first claim in the theorem with the first and second part of the second claim in the theorem.
We have \[\begin{align} \mathop{\mathrm{FDP}}(\mathcal{R}_{\operatorname{BH},q},\mathcal{H}_0) = \frac{V(\hat{t}_{\operatorname{BH},q})/m}{[R(\hat{t}_{\operatorname{BH},q})\vee 1]/m} \le \widehat{\mathop{\mathrm{FDR}}}(\hat{t}_{\operatorname{BH},q}) + I(\hat{t}_{\operatorname{BH},q}< t^*) + o_P(1) \end{align}\] using the fact that \(\sup_{t\in [t^*,1]}\left[ \frac{V(t)/m}{[R(t)\vee 1]/m} - \widehat{\mathop{\mathrm{FDR}}}(t) \right]\le o_P(1)\) by Theorem 3. Since \(\widehat{\mathop{\mathrm{FDR}}}(\hat{t}_{\operatorname{BH},q})\le q\) by construction, it suffices to show \(P(\hat{t}_{\operatorname{BH},q}\ge t^*)\to 1\). But this follows since \(\widehat{\mathop{\mathrm{FDR}}}(t^*)\le q\) implies \(\hat{t}_{\operatorname{BH},q}\ge t^*\), and \(\widehat{\mathop{\mathrm{FDR}}}(t^*)\stackrel{p}{\to} G(t^*)/F(t^*)<q\) by (13 ). This shows that \(\mathop{\mathrm{FDP}}(\mathcal{R}_{\operatorname{BH},q},\mathcal{H}_0)\le q+o_P(1)\), from which it also follows that \(\mathop{\mathrm{FDR}}(\mathcal{R}_{\operatorname{BH},q},\mathcal{H}_0,P)\le q+o(1)\) by dominated convergence.
As noted in the main text, procedures that attempt to incorporate estimates \(\hat{\pi}\) of \(\#\mathcal{H}_0/m\) will in not, in general, lead finite sample or asymptotic FDR control under weak dependence conditions on the \(p\)-values. As a counterexample, consider the procedure \(\mathcal{R}_{\hat{\pi},\beta(\cdot),q}\) with \(\beta(t)=t\) that uses the estimate \(\hat{\pi}=(\sum_{i=1}^m I(p_i>\lambda)+1)/((1-\lambda) m)\) of \(\pi_0\) for some \(\lambda\in(0,1)\) as in [21] and suppose that \(\#\mathcal{H}_0/m\to \pi_0\) where \(q<\pi_0<\lambda\). Suppose that \(p_i\) is independent over \(i\) and follows a uniform distribution on \([0,\pi_0]\) for \(i\in\mathcal{H}_0\), and that \(p_i\) follows some other distribution with support contained in \([0,\pi_0)\) for \(i\not\in\mathcal{H}_0\). Then (12 ) and (13 ) hold. However, note that the cutoff \(\hat{t}_{\hat{\pi},\beta(\cdot),q}\) in (10 ) will be equal to \(1\) so long as \(q\ge \hat{\pi} m \cdot 1/R(1)=\hat{\pi}\), which holds for large enough \(m\) with probability one since \(\hat{\pi}=1/((1-\lambda) m)\to 0\) with probability one. Since the FDP is equal to \(\#\mathcal{H}_0/m\) on the event that \(\hat{t}_{\hat{\pi},\beta(\cdot),q}=1\), this implies that the FDP is equal to \(\#\mathcal{H}_0/m\to \pi_0>q\) with probability one for large enough \(m\), so that the FDR is equal to \(\#\mathcal{H}_0/m>q\) for large enough \(m\). Note that the FDP is also equal to \(\#\mathcal{H}_0/m\) on the event that \(\hat{t}_{\hat{\pi},\beta(\cdot),q}\ge \lambda\), since the \(p\)-values are all supported on \([0,\lambda)\). Thus, the same argument goes through for the modification of the procedure in which \(\hat{t}_{\hat{\pi},\beta(\cdot),q}\) is replaced by \(\lambda\) when \(\hat{t}_{\hat{\pi},\beta(\cdot),q}>\lambda\).
Note also that the oracle version of this procedure (\(\mathcal{R}_{\pi_0,\beta(\cdot),q}\) with \(\beta(t)=t\) and \(\pi\) set to the true asymptotic null proportion \(\pi_0\)) also fails to control FDR asymptotically. For this procedure, the cutoff \(\hat{t}_{\pi_0,\beta(\cdot),q}\) is equal to \(1\) so long as \(q\ge \pi_0\), which holds by assumption. Thus, this oracle procedure has FDP equal to \(\pi_0\) with probability one, which means that it has FDR equal to \(\pi_0\) for all \(m\).
email: timothy.armstrong@usc.edu. Support from National Science Foundation Grant SES-2049765 is gratefully acknowledged.↩︎
For results with fixed \(m\), [21] consider a modification in which \(\hat{t}_{\pi,\beta(\cdot),q}\) is replaced by \(\lambda\) if \(\hat{t}_{\pi,\beta(\cdot),q}>\lambda\).↩︎
I thank an anonymous referee for pointing this out.↩︎