Adaptive procedures for directional false discovery rate control

In multiple hypothesis testing, it is well known that adaptive procedures can enhance power via incorporating information about the number of true nulls present. Under independence, we establish that two adaptive false discovery rate (FDR) methods, upon augmenting sign declarations, also offer directional false discovery rate (FDR$_\text{dir}$) control in the strong sense. Such FDR$_\text{dir}$ controlling properties are appealing because adaptive procedures have the greatest potential to reap substantial gain in power when the underlying parameter configurations contain little to no true nulls, which are precisely settings where the FDR$_\text{dir}$ is an arguably more meaningful error rate to be controlled than the FDR.


Introduction
Consider independent observations z 1 , . . ., z m P R, where each "z-value" z i is a noisy measurement of an effect parameter θ i P R. We suppose m is quite large, and will use the notational shorthand rms " t1, . . ., mu in the sequel.For testing the multiple point null hypotheses H i : θ i " 0, i P rms, (1.1) Benjamini and Hochberg (1995) proposed their now-celebrated BH procedure to control the false discovery rate (FDR), ff , below a target level q P p0, 1q.However, many statisticians, such as Tukey (1962Tukey ( , 1991) ) and Gelman and Tuerlinckx (2000), consider testing the point nulls in (1.1) futile, because the effects in reality, however small, are rarely exactly zero.Instead, they argue that one should test the direction/sign of the effect by declaring either θ i ą 0 or θ i ă 0 as a discovery, or making no declaration about θ i at all if there is insufficient evidence to support either direction.Under this new paradigm, a generic discovery procedure consists of two components: (i) R Ď t1, . . ., mu, the set of rejected indices for which sign declarations (discoveries) are made, and (ii) y sgn i , the positive or negative sign declared for each i P R. (Note that We denote such a procedure, or its associated decisions, by py sgn i q iPR .Its error rate analogous to the FDR is the directional false discovery rate (FDR dir ), defined as and its power can be measured by the expected number of true discoveries (ETD) ETD where for any x P R, sgnpxq " 1px ą 0q´1px ă 0q.This paper treats the control of FDR dir for the sign discoveries of the θ i 's; we don't exclude the possibility that some effect parameters can indeed be zero, so a false discovery amounts to declaring θ i ă 0 when the truth is θ i ě 0, or vice versa.1 Methods for controlling the FDR dir are surprisingly scant for the simple testing problem above.To our best knowledge, under the independence among z 1 , . . ., z m and some standard assumptions (Assumption 1 and 2 below), the only known procedure that can provably control the FDR dir under a target level q P p0, 1q in the strong sense, i.e. irrespective of the configuration of θ i 's, is what we call the directional Benjamini and Hochberg (BH dir ) procedure proposed in Benjamini and Yekutieli (2005, Definition 6).The BH dir procedure first decides on the set of rejected indices among rms by applying the standard BH procedure at level q to the two-sided p-values constructed from z 1 , . . ., z m , and then declares the sign of each rejected θ i as sgnpz i q.Guo and Romano (2015, Procedure 6) proposed another procedure almost identical to the BH dir , except that the screening BH step is applied at level 2q instead of q.This latter procedure, however, can only control the FDR dir under its intended target level q when all θ 1 , . . ., θ m are nonzero.Guo and Romano (2015, Procedures 7-9), Zhao and Fung (2018) and Guo, Sarkar and Peddada (2010) consider extensions of the current problem that involve either specific patterns of dependence or multidimensional directional decisions.
We will expand the repertoire of available methods for FDR dir inference with strong theoretical guarantee.In the FDR literature, it is known that adaptive methods (Benjamini andHochberg, 2000, Benjamini, Krieger andYekutieli, 2006) that incorporate a data-driven estimate of the proportion of true nulls into their procedures have the potential to improve upon the power offered by the vanilla BH procedure.Using martingale arguments, we prove that under independence, two adaptive methods can also provide strong FDR dir control upon the augmentation of sign declarations.The first is Storey, Taylor and Siegmund (2004)'s adaptive FDR procedure, which can be seen as an adaptive variant of the BH procedure (Section 2).The second is a specific procedure belonging to a more recent line of methods driven by a technique called "data masking" first introduced in Lei and Fithian (2018) (Section 3).We also numerically demonstrate their competitive power performances in Section 4. Adaptive procedures that can offer FDR dir guarantees are particularly important for settings whose underlying parameter configurations contain little to no true nulls.Arguably, if most θ i 's in question are non-zero, the FDR dir is more meaningful as an error measure compared to the FDR because querying about their signs matters more.Moreover, such "non-sparse-signal" settings are precisely those in which adaptive procedures can reap substantial gain in power; see Storey, Taylor and Siegmund (2004, Section 3.1), where their adaptive FDR procedure demonstrates greater improvements in power over the BH procedure as π decreases.

Notation and assumptions
For a, b P R, we let a ^b " minpa, bq and a _ b " maxpa, bq.For any two subsets A, B Ă rms, A Ĺ pĽqB means A is a strict subset (superset) of B. U p¨; a, bq denotes a uniform density on the interval ra, bs Ď R. E θ r¨s means a (frequentist) expectation with respect to fixed values of θ 1 , . . ., θ m .For each i P rms and given θ i " θ, F i,θ p¨q denotes the distribution function of z i with density f i,θ p¨q " F 1 i,θ p¨q ą 0 with respect to the Lebesgue measure on R (so F i,θ p¨q is implicitly assumed to be smooth and strictly increasing).Φp¨q and ϕp¨q denote the standard normal distribution and density functions, respectively.Additionally, the following assumptions will be made for the two main theoretical results in this paper (Theorems 2.1 and 3.1): Assumption 1.The null distribution of z i is known and symmetric around zero, i.e.F i,0 p´zq " 1 ´Fi,0 pzq and f i,0 p´zq " f i,0 pzq for any z P R.
Assumption 2. The family of densities tf i,θ p¨qu θPR satisfies the monotone likelihood ratio (MLR) property, i.e. for any given θ ă θ ˚and z ă z These are not necessarily the weakest assumptions for our theorems to hold, but are standard enough so as not to distract from the key ideas of the proofs.Essentially, Assumption 2 guarantees that z i becomes "stochastically larger" as θ i increases, and two examples satisfying Assumption 2 are the normal distributions N pθ, σ 2 i q for a fixed variance σ 2 i and the noncentral t-distributions N CT pθ, v i q for a fixed degree v i (Kruskal, 1954, Section 3).Moreover, the symmetry condition on F i,0 in Assumption 1 is not crucial; it is included primarily to streamline our presentation.If this condition isn't satisfied, instead of z i , one can alternatively consider the transformed statistic Φ ´1pF i,0 pz i qq whose density has the form As a function in z, (1.4) boils down to the symmetric density function ϕpzq when θ i " 0. The density (1.4) also maintains the MLR property, provided that the base density f i,θ p¨q satisfies the MLR property.

Directional control with Storey, Taylor and Siegmund (2004)'s adaptive procedure
Compute the two-sided p-values p i " 2F i,0 p´|z i |q, i P rms (2.1) from the null distributions F i,0 , and for any t P r0, 1s, let Rptq " ti : p i ď tu be the set of rejected indices i defined by p i ď t; Algorithm 1 is a sign-augmented version of Storey, Taylor and Siegmund (2004)'s adaptive procedure for FDR dir control, which we call the "STS dir " for short.
Rpt λ q q is precisely the rejection set produced by the adaptive procedure in Storey, Taylor and Siegmund (2004, Theorem 3) that was proved to offer strong FDR control for testing the point nulls in (1.1), under the assumptions that the null p-values are independent and uniformly distributed.Roughly speaking, compared to the BH procedure, which essentially uses as an estimate for the FDR incurred by rejecting any p i below a given threshold t, Storey, Taylor and Siegmund (2004)'s FDR estimate z FDR λ ptq adjusts (2.2) by the factor πpλq.This factor serves as a conservative estimate for the unknown null proportion π P r0, 1s in (1.3).If π is close to zero, i.e. there are very few true nulls in the problem, Storey, Taylor and Siegmund (2004)'s procedure could produce substantially more rejections compared to the BH procedure.Our first result is that, by simply augmenting the rejections with sign declarations as in the output of Algorithm 1, the procedure also offers strong FDR dir control under independence; we note that by the construction of the p-values in (2.1) and Assumption 1, it must be the case that sgnpz i q ‰ 0 for any i P Rpt λ q q: Theorem 2.1 (FDR dir control of STS dir ).Under Assumptions 1 and 2, as well as the independence among z 1 , . . ., z m , Algorithm 1 controls the FDR dir at level q P p0, 1q, i.e. by letting Sptq " ti : sgnpθ i q ‰ sgnpz i q and i P Rptqu.
The proof of Theorem 2.1 in Appendix A.1 relies on the optional stopping time theorem for supermartingales, which extends the original arguments by Storey, Taylor and Siegmund (2004, Theorem 3) based on martingales.The choice of λ entails a bias-variance trade-off in the estimation of π, and Storey, Taylor and Siegmund (2004) fixes λ " 0.5 in their simulations.Alternatively, similarly to Storey, Taylor and Siegmund (2004, Section 6), we provided a method in Appendix A.2 for automatically selecting λ based on the observed data z 1 , . . ., z m to achieve a balance between bias and variance.Since Theorem 2.1 is only valid on the premise of a fixed λ, using the provided method of automatically selecting λ may not guarantee FDR dir control.Regardless, our simulations in Section 4 demonstrate that this data-driven choice of λ typically leads to empirically robust control of FDR dir .

Directional control with data masking: ZDIRECT
Early FDR methods, such as Storey, Taylor and Siegmund (2004)'s adaptive procedure covered in Section 2, process data in a pre-determined manner to decide on the set of rejected hypotheses.In contrast to the conventional belief that the design of a testing procedure should not be affected by the observed data's patterns to avoid data snooping, Lei and Fithian (2018) recently showed that, as long as the data are initially "masked" as a trade-off, one can iteratively interact with the gradually revealed data in a legitimate way to devise valid FDR procedures.The flexibility of this masking technique not only allows one to adapt to information about the null proportion in (1.3) implied by the data, but also side information provided by suitable external covariates that are present; see (Chao and Fithian, 2021, Lei, Ramdas and Fithian, 2021, Leung and Sun, 2022, Tian, Liang and Li, 2021, Yurko et al., 2020) for a series of follow-up works, including the ZAP (finite) algorithm proposed by one of the present authors (Leung and Sun, 2022, Algorithm 2).
Motivated by the construct of ZAP, we propose a masking algorithm that is augmented with sign declarations for the rejected θ i 's, and is proven to provide strong FDR dir control.To facilitate our presentation, for each i P rms, we first define the probability integral transform of z i under the null that takes values in the unit interval p0, 1q, as well as its reflection q u i " p0.5 ´ui qIpu i ď 0.5q `p1.5 ´ui qIpu i ą 0.5q (3.2) about the midpoint 0.25 of the left sub-interval p0, 0.5q, or about the midpoint at 0.75 of the right sub-interval p0.5, 1q, depending on whether u i ď 0.5 or u ą 0.5.Moreover, we let which, for any given i, is the value between u i and q u i closer to the endpoints of the unit interval.In particular, u 1 i only provides partial knowledge about the value of u i : If u 1 i ď 0.5, u i may be either u 1 i or 0.5 ´u1 i ; if u 1 i ą 0.5, u i may be either u 1 i or 1.5 ´u1 i .With these preparations, we are now in a position to describe our iterative testing procedure for the directional inference of θ 1 , . . ., θ m with a target FDR dir level q P p0, 1q, whose steps are listed out in Algorithm 2; we call it "ZDIRECT" as it interacts with the z-values via their one-to-one transformations u i 's for FDR dir control.Essentially, the algorithm iterates through steps t " 0, 1, . . . to construct a strictly decreasing sequence of subsets rms " M 0 Ľ M 1 Ľ M 2 ¨¨b ased on the data, and from each M t , it forms the candidate "acceptance" and "rejection" sets A t " ti : i P M t and u i P p0.25, 0.75qu and R t " ti : i P M t and u i P p0, 0.25s Y r0.75, 1qu; the algorithm terminates at step t " mintt : z FDR dir ptq ď qu as soon as the FDR dir estimate falls below q, and declares the sign discoveries psgnpz i qq iPR t .Since R t Ă ti : u i P p0, 0.25s Y r0.75, 1qu, it must be the case that sgnpz i q ‰ 0 for any i P R t, under Assumption 1.Moreover, the sets M t are shrunk in such a way that these two conditions must be respected (line 6 in Algorithm 2): (C1) M t`1 must be constructed based only on the partial data tũ i,t u iPrms available at step t, where for each i P rms, we define which may reveal the true value of u i depending on whether u i P M t .Essentially, any human/computer routine who shrinks M t to M t`1 can only know that the true value of u i is one of two possibilities if i is still in the "masked" set M t .(C2) M t`1 must be a strict subset of M t to ensure the algorithm does terminate.
Algorithm 2: The ZDIRECT procedure at target FDR dir level q P p0, 1q Data: z 1 , . . ., zm Input: FDR dir target q P p0, 1q, the initial set M 0 " rms ; 1 for t= 0,1 . . ., do 2 Find the candidate "acceptance set" At " ti : i P Mt and u i P p0.25, 0.75qu; Theorem 3.1, which is proven in Appendix B.2, states that ZDIRECT provides strong FDR dir control under the assumptions in this paper: Theorem 3.1 (FDR dir control of ZDIRECT).Under Assumptions 1 and 2, as well as the independence between z 1 , . . ., z m , Algorithm 2 controls the FDR dir at level q P p0, 1q.Specifically, if the algorithm terminates at step t " mintt : Apart from the final sign declarations, Algorithm 2 inherits much of its structure from the ZAP (finite) algorithm in Leung and Sun (2022), but is situated in the more general "without-threshold" framework (Lei and Fithian, 2018, Section 6.1) that does not explicitly involve any thresholding functions.In fact, a quantity like z FDR dir ptq has been used as an FDR estimate in Leung and Sun (2022).To also make sense of it as a suitable FDR dir estimate for the sign discoveries of psgnpz i qq iPRt , note that an i P R t constitutes a false discovery if either u i P p0, 0.25s and sgnpθ i q ě 0, (3.6) or u i P r0.75, 1q and sgnpθ i q ď 0.
(3.7)By the continuity of F i,0 , each u i is uniformly distributed when θ i " 0. Suppose we are in an error-prone scenario where all θ i 's are either exactly or very close to zero, so that all u i 's stochastically behave much like uniform random variables.
For a given i P M t , the event tu i P p0, 0.25su and tu i P p0.25, 0.5su should be approximately equally likely, so the set size |ti : i P M t and u i P p0.25, 0.5su| is a reasonable estimate of the number of false discoveries by way of (3.6).Analogously, |ti : i P M t and u i P r0.5, 0.75qu| serves as an estimate of the number of false discoveries by way of (3.7).As such, |A t | makes sense as an estimate of the number of false discoveries in R t , where the additive "1" in the numerator of z FDR dir ptq is a theoretical adjustment factor to make it conservative enough.In fact, this concept is akin to how the FDR estimate for the knockoff filter for variable selection in linear regressions (Barber et al., 2015) can also serve as an FDR dir estimate when sign declarations are augmented (Barber et al., 2019).
However, our specific methodology for updating M t , as described next, considerably differs from existing data masking algorithms.Notably, we rely solely on z 1 , . . ., z m as the available data for our problem, without harnessing external covariate information.This poses a greater challenge for boosting power, but our simulations in Section 4 demonstrate that ZDIRECT remains competitive in terms of power for FDR dir control when compared to other existing methods.

Shrinking the masked sets M t
To achieve power, we aim to shrink M t in accordance with conditions (C1)-(C2) in such a way that Algorithm 2 can mimic the optimal discovery procedure (ODP) under a Bayesian formulation2 of the problem, which imposes the additional assumption that the effects are random and independently generated by a common prior distribution, i.e.Gpθq,i P rms,(3.8)for an unknown distribution function Gp¨q.For a given target level q P p0, 1q, the ODP, denoted by py sgn ODP i q iPR ODP , is defined to be the procedure with the properties that FDR dir " py sgn declare θ i ą 0 P pθ i ą 0|z i q ă P pθ i ă 0|z i q declare θ i ă 0 P pθ i ą 0|z i q " P pθ i ă 0|z i q declare either θ i ą 0 or θ i ă 0 Table 1 Optimal sign declaration strategy for a given i.
for any procedure py sgn i q iPR with FDR dir " py sgn i q iPR ı ď q, where the expectation operator defining all the FDR dir and ETD quantities just mentioned is with respect to both the randomness of the data tz i u iPrms and that of the parameters tθ i u iPrms under (3.8) (i.e.different from the frequentist E θ r¨s operator).In other words, the ODP is the best procedure in terms of maximizing ETD, among all that can control FDR dir under a target level.
In order to shrink M t in a manner that Algorithm 2 can mimic the ODP, we have to better understand the latter's operational characteristics; to that end, we shall first intuitively grasp what the best course of action for a directional decision maker should be under the Bayesian assumption (3.8).If s/he were obliged to unambivalently declare a non-zero sign for a specific θ i based on z i , the optimal strategy is clearly the one outlined in Table 1 based on the posterior probabilities P pθ i ă 0|z i q and P pθ i ą 0|z i q, whose associated probability of making a false discovery can be calculated as and must be no larger than the probability of false discovery made by any other strategy.In the literature, the quantity in (3.9) is known as the local false sign rate for i (Stephens, 2017, p.279), and a smaller lfsr i suggests higher confidence in the sign declaration for θ i prescribed by Table 1.Now, if s/he were to make non-zero sign declarations for the largest possible subset of parameters from tθ i u iPrms with FDR dir control in mind, the intuition would be to prioritize making sign declarations for those i's with the smallest local false sign rates, each using Table 1's strategy.This is in fact what the ODP does, as stated in Theorem 3.2 below.We note that the ODP is not implementable in practice as the underlying prior Gp¨q is, by assumption, unknown for computing the local false sign rates.
Theorem 3.2 (Operational characteristics of the ODP under Bayesian formulation).Assume the prior in (3.8), and that conditional on tθ i u iPrms , z 1 , . . ., z m are independent with respective distributions F 1,θ1 , . . ., F m,θm .For a given level q P p0, 1q, the optimal discovery procedure py sgn ODP i q iPR ODP must be such that (i) lfsr i ď lfsr j for any i P R ODP and j P rmszR ODP , and (ii) For each i P R ODP , y sgn is declared in accordance with Table 1.
The proof of Theorem 3.2 is in Appendix B.3, which extends the arguments in Heller and Rosset (2021, Theorem 2.1) on the optimal FDR procedure for the point null testing problem in (1.1).While the ODP cannot be operationalized in practice, one can make ZDIRECT mimic its characteristic that indices with the smallest local false sign rates get rejected first: At each step t, we aim to get rid of exactly one element from the masked set M t that has potentially the largest local false sign rate, since only elements remaining in the next M t`1 may be rejected.In what follows, let which convert u 1 i , q u i and r u i,t back onto their original z-value scale.Specifically, we estimate the local false sign rates as proposed in Stephens (2017), where w " pw ´K , . . ., w ´1, w 0 , w 1 , . . ., w K q are mixing probabilities that sum to 1, δ 0 p¨q denotes the delta function at zero, and h k 's are uniform densities of the forms for predetermined endpoints a 1 , . . ., a K ą 0 and a ´1, . . ., a ´K ă 0. The loglikelihood of (3.12) with respect to the partial data tz i,t u iPrms at step t can then be computed as where l k,i,t are the likelihoods of the mixture components of the forms (3.14)When f i,θ is from the family of normal distributions N pθ, σ 2 i q, the quantities in (3.14) have closed-form expressions; if f i,θ is from the family of noncentral t-distributions N CT pθ, ν i q, methods for approximating the quantities in (3.14) are discussed in Appendix B.1.From (3.13), the density of Ĝt is taken as ĝt p¨q " gp¨; ŵt q, where ŵt solves the penalized maximum likelihood estimation: Above, the last term is a Dirichlet penalty with tuning parameters λ k ą 0. Remarks.Adaptivity is implicitly built into our algorithm, since the null probability w 0 in our modeling density (3.12) is the Bayesian analogue of the frequentist null proportion in (1.3).By striving to mimic the operational characteristic of the ODP, it also allows adaptivity to other features, such as the asymmetry in the distribution of the z i 's; in the FDR literature, it is well known that local false discovery rate approach based on z-values can further boosts testing power by leveraging distributional asymmetry (Storey, Dai andLeek, 2007, Sun andCai, 2007), and the same discussion can carry over to FDR dir control with local false sign rates.We stress that although (3.12) may well be misspecified as a density for the hypothetical prior Gp¨q, strong frequentist FDR dir control is guaranteed by (3.5) in Theorem 3.1.Moreover, our choice of it as a working model carries two main advantages: (a) Speed: The iterative updates of M t for most existing FDR data-masking algorithms are computationally expensive, as they usually employ beta mixture models that require the EM algorithm (Dempster, Laird and Rubin, 1977) for estimation.On the contrary, as explained in Stephens (2017, Supplementary material), an optimization problem with the form in (3.15) is convex and can be solved by fast and reliable interior point methods.(b) Appropriate flexibility: Stephens (2017) argues for the plausibility of unimodality in many real applications since most effects are close to zero while larger effects are decreasingly likely; by increasing the number of components K and expanding the supporting intervals defined by a k , (3.12) can approximate any unimodal distribution about zero (Feller, 1971, p.158).On the other hand, as heuristically discussed in Leung and Sun (2022, Section 5), having a too-flexible model could overfit the partial data tz i,t u iPrms , which one is constrained to work with to adhere to data masking, only to underfit the original data tz i,t u iPrms .Stephens (2017, Section 3.1.4)also advocates for unimodality as a form of "regularization" because it prevents density estimates from concentrating in small pockets.Our simulation re-sults in Section 4 show that ZDIRECT can still be competitive against other practical methods even when the unimodality is misspecified.

Simulation setups
We simulate m " 1000 independent z i " N pθ i , 1q values where θ 1 , . . ., θ m are independently generated from a mixture density of the form gpθq " wδ 0 pθq `p1 ´wqg 1 pθq, (4.1) where δ 0 p¨q is the delta function at zero for the nulls, and g 1 pθq is an "alternative" density which is itself a mixture of two normal components of the form g 1 pθq " p1 ´vqϕpθ `ξq `vϕpθ ´ξq.
The simulation parameters controlling different aspects are chosen as follows.
Letting w " 0.8 renders a setting with approximately 80% of the θ i 's equal to 0, and taking w " 0 renders a setting with all θ i 's being non-zero.How the different combinations of w, ξ and v change the shape of gpθq is illustrated in Figure 4.1; note that many of these gpθq are evidently not unimodal.

Methods compared
We compare the following seven methods for FDR dir control: (a) BH dir : The directional BH procedure proposed by Benjamini and Yekutieli (2005).We note that the validity of BH dir is originally proved under the assumption that the family tF i,θ p¨qu θPR is stochastically increasing, which is implied by the MLR property in Assumption 2 (Lehmann, Romano and Casella, 2005, Lemma 3.4.2piiq).(b) LFSR: A computationally simpler substitute for the ODP based on the oracle lfsr i 's in (3.9); inspired by Sun and Cai (2007)'s earlier work on optimal FDR control.Its implementation details are deferred to Appendix C.1.The implementation of the ODP described in Theorem 3.2 entails solving a complex infinite integer problem to compute a rejection threshold for the lfsr i 's (Heller and Rosset, 2021).In comparison, LFSR is simpler to compute whilst often having comparable power.Like the ODP, LFSR offers FDR dir control under the Bayesian formulation in (3.8), but it can only serve as a hypothetical benchmark for other methods since it also requires oracle knowledge of the true generating prior in (4.1).(c) ASH ("adaptive shrinkage", Stephens (2017)): A procedure that is almost the same as LFSR in its implementation, except that the oracle gp¨q from (4.1) is replaced by an estimated gp¨; ŵq based on the unimodal model in (3.12),where ŵ is a penalized maximum likelihood estimate of w with respect to the full data tz i u iPrms obtained by the R package ashr; all tuning parameters involved are chosen to be the default described in Stephens (2017, Supplementary information).This procedure may risk violating the desired FDR dir target if the unimodal density (3.12) is too far from the true prior density of the θ's.(d) GR (Guo and Romano (2015, Procedure 6)): The FDR dir testing procedure mentioned in Section 1 that provides (frequentist) control of the FDR dir under the target level q when all the θ i 's are non-zero, i.e. π " 0; see Guo and Romano (2015, Theorem 5 and its proof).(e) STS dir : Algorithm 1 by setting λ " 0.5.(f) aSTS dir : The STS dir procedure, except with an automatic (data-driven) choice for λ as described in Appendix A.2.It has no proven theoretical control of the FDR dir .(g) ZDIRECT: Algorithm 2, by initializing M 1 " ti : u 1 i ď 0.2 or u 1 i ě 0.8u based on the tũ i,0 u m i"1 " tũ 1 i u m i"1 , which respects condition (C1).Thereafter, M t is updated as in Section 3.1, where the optimization in (3.15) is performed using a solver in the R package Rmosek (ApS, 2022).The points a k are picked to give a large and dense grid; in particular, for the positive supports, the minimum and maximum are set as a 1 " 10 ´1 and a K " 2 a max i z 12 i ´1, with the rest set as a k`1 " ?2a k ď a K based on the multiplicative factor ? 2 (and so, K is implicitly determined).The negative supports are set by taking a ´1 " ´a1 , . . ., a ´K " ´aK .This grid follows the recommendation of Stephens (2017) except a K is determined with z 1 i 's instead of z i to observe the masking condition (C1).Moreover, we set λ k " 0.8 for all k " ´K, . . ., 0, . . ., K.This further regularizes the estimation by encouraging sparsity in the estimates of the mixing proportions w, and provides consistently good performance.Lastly, to speed up the algorithm, ĝt p¨q is only re-estimated by (3.15) for every rm{200s steps, i.e., the same ĝt p¨q is used rm{200s times to update the candidate rejection and acceptance sets before the algorithm terminates.

Results
The empirical FDR dir and power of the different methods implemented for the target FDR dir level q " 0.1 are evaluated with 1000 sets of repeatedly generated tz i , θ i u iPrms .The results are shown in Figure 4.2, where the power is shown as the true positive rate, defined as E " ř iPR 1psgnpµiq" y sgn i q| 1_|R| ı for a generic procedure py sgn i q iPR , which some consider to be more illustrative than the ETD.The following observations can be made: (a) Throughout, the only methods that can visibly control FDR dir under the target q " 0.1 in all settings are LSFR, BH dir , ZDIRECT and STS dir , precisely the ones with theoretical guarantees.However, LSFR is not implementable in practice and only has FDR dir guarantee under the Bayesian formulation in (3.8).While ZDIRECT and STS dir still lag considerably behind in power compared to LFSR in settings with small w (or small π, as a frequentist analogue), their power advantage over BH dir becomes substantial as w approaches 0. Across the board, aSTS dir , which is calibrated with a data-driven λ and has no theoretical guarantee, displays slightly better power than STS dir , but also violates the FDR dir target ever so slightly for large ξ when w " 0.8.(b) ASH is the one practical method that overall matches LFSR closest in power, but severely violates the desired FDR dir level in some settings when w P t0.2, 0.5u.This is not surprising because the unimodal working model in (3.12), which ASH is based on, is misspecified for many of the multimodal data generating gp¨q in Figure 4.1.(c) GR visibly violates the FDR dir target when w P t0.5, 0.8u, and severely so for w " 0.8.When w " 0, the only case where GR can provably control the FDR dir , GR's power is at best comparable to STS dir and ZDIRECT when the signal size ξ is small, but inferior to them when ξ is large.(d) ZDIRECT's power is considerably better than STS dir when v " 1, the most asymmetric setting for g.As discussed in the remarks of Section 3.1, by attempting to mimic the operational characteristics of the ODP via estimating the lfsr i quantities in (3.10), ZDIRECT has the potential to leverage asymmetry in the distribution of the z-values to boost testing power, just like the ODP does.This is even more remarkable, considering that the working model (3.12) is obviously misspecified for the true θ-generating prior in (4.1), which attests to the practical usefulness of Stephens (2017)'s unimodal assumption when combined with ZDIRECT's data masking mechanism to ensure strong FDR dir control (Theorem 3.1).Empirical directional false discovery rate and true positive rates of the seven compared methods for the simulations in Section 4; each method was implemented at a target FDR dir level q " 0.1 (black horizontal lines).

Discussion
We have proved that, under independence and upon augmenting sign declarations, Storey, Taylor and Siegmund ( 2004)'s adaptive procedure and ZDIRECT, a particular implementation of the recently introduced line of adaptive "data masking" algorithms, can offer FDR dir control in the strong sense.These results are particularly important when the parameter configurations contain little to no true nulls because adaptive procedures precisely reap the most power benefit in such scenarios.Moreover, under "non-sparse-signal" settings, FDR dir is arguably a more meaningful error rate to be controlled than the FDR.Both methods require tuning parameters; in our experience, the simple choice of λ " 0.5 for STS dir and λ k " 0.8 for ZDIRECT have consistently given us competitive power performance, even for some less instructive simulation setups considered in earlier versions of this paper.For ZDIRECT, while other working models that may further boost the power can be deployed, we find the current implementation based on Stephens ( 2017)'s unimodal model to be attractive, as the optimization under the hood is numerically very stable and fast.While our theory doesn't cover settings where the z-values can be dependent, additional simulation results in this vein are included in Appendix C.2.An interesting dual problem to sign declarations is to construct, for each i in a data-dependent selected subset R Ď rms, a confidence interval CI i Ď R such that (a) each CI i is sign-determining, i.e.CI i Ď p´8, 0s or CI i Ď p0, 8q, and (b) the false coverage rate (FCR) E " |ti : is controlled under a desired level q P p0, 1q.
This paradigm of inference has been recently suggested in Weinstein and Yekutieli (2020), and they proposed a first procedure that constructs such selective sign-determining confidence intervals.The adjective "selective" indicates that the set R is also chosen based on the same data tz i u m i"1 the CI i 's are constructed with.Note, STS dir and ZDIRECT do correspond to such procedures that construct trivially long intervals for each i in their rejection sets.It will be a challenging but meaningful task to devise non-trivial selective sign-determining confidence intervals where the target set R is chosen in a more adaptive manner akin to STS dir and ZDIRECT, to offer more powerful alternative procedures to Weinstein and Yekutieli (2020, Definition 2)'s procedure.

A.1. Proof of Theorem 2.1
We shall first state four intermediate results, which allow us to extend the arguments in Storey, Taylor and Siegmund (2004, Section 4.3) to prove the FDR dir controlling properties of STS dir under Assumptions 1 and 2.
Proof of Lemma A.1.To prove statement (a), let θ i ă 0 and let z 0 , z 1 P R such that z 0 ă z 1 .By Assumption 2, we have that Multiplying both sides by f i,0 pz 1 q and integrating over z 1 from z 0 to 8 yields " f i,θi pz 0 q f i,0 pz 0 q ȷ r1 ´Fi,0 pz 0 qs ě r1 ´Fi,θi pz 0 qs , which proves statement (a).The proof of statement (b) follows analogously to that of (a).
In the lemma below, the probability operator P θi"0 p¨q emphasizes the law is driven by a value of θ i equal to zero; the operators P θiă0 p¨q and P θią0 p¨q have similar meanings.
Lemma A.2.Under Assumptions 1 and 2, for 0 ă s ď t ď 1, Proof of Lemma A.2.For any x P p0, 1s, we first rewrite piq follows from (A.1) since P θi"0 pp i ď s|p i ď t, z i ‰ 0q " P θ i "0 ppiďs,zi‰0q piiq is obvious when s " t; when s ă t, by applying the mean value theorem on 1´F i,θ p´F ´1 i,0 px{2qq x as a function in x in light of (A.2), there exists c P ps, tq such that, for y " F ´1 i,0 pc{2q, ´" P θ i ă0 ppiďs,ziě0q ı r1 ´Fi,0 p´yqs ´r1 ´Fi,θi p´yqqs where the last equality follows from the symmetry of F i,0 in Assumption 1.Since θ i ă 0, by applying Lemma A.1paq to the numerator in (A.4), we get that which is equivalent to P θ i ă0 ppiďt,ziě0q ď s t , and piiq is proved.The proof for piiiq is analogous to that of piiq, using the mean value theorem on F i,θ i pF ´1 i,0 px{2qq x , (A.3) and Lemma A.1pbq.
Lemma A.3.Let Sptq be defined as in Theorem 2.1.Under Assumptions 1 and 2, as well as the independence among z 1 , . . ., z m , |Sptq|{t for 0 ď t ă 1 is a supermartingale with time running backward, with respect to the filtration Proof of Lemma A.3.Since |Spsq| can be written as |Spsq| " it is not difficult to observe from statements piq-piiiq of Lemma A.2 that |Spsq| given F t is stochastically dominated by the Binomialp|Sptq|, s{tq distribution.
Lemma A.4.If Y " Binomialpn, λq, then for any λ P p0, 1q and n P N, where D n pλq is strictly increasing in λ.
Proof of Lemma A.4.By direct computation, which proves the expectation result.Taking the derivative of D n pλq with respect to λ yields Let the numerator of D 1 n pλq be denoted as 9 D 1 n pλq " λ n pnλ ´n ´1q `1.To prove that D n pλq is strictly increasing in λ, we will show that 9 D 1 n pλq ą 0 by induction.Suppose 9 D 1 N pλq ą 0 is true for some fixed N P N. Then where the first inequality stems from the inductive condition 9 D 1 N pλq ą 0. Since 9 D 1 1 pλq " pλ´1q 2 ą 0, it follows that 9 D 1 n pλq ą 0 for any λ P p0, 1q and n P N. Now we can begin the proof of Theorem 2.1.First, we can write If z FDR λ pλq ě q, then t λ q ď λ.Hence, |Rpt λ q q| _ 1 ě π0 pλqt λ q m{q and so where the last step follows by Lemma A.3 and the optional stopping theorem since t λ q is a stopping time with respect to F t with time running backward.If z FDR λ pλq ă q, then t λ q " λ and so Hence, By taking s " λ and t " 1 in statements piq-piiiq of Lemma A.2, in light of the equality in (A.5), it is not difficult to see that |Spλq| given |Sp1q| is stochastically dominated by the Binomialp|Sp1q|, λq distribution.Hence, by Lemma A.4. Combining (A.6) with 1 ´Eθ rλ |Sp1q| s ď 1 ´λE θ r|Sp1q|s ď 1, a consequence of Jensen's inequality, Theorem 2.1 is proved.

A.2. Automatic λ selection procedure
Two inputs are required for this procedure: B, the number of bootstrap samples; and Λ, a set of candidate values for λ.Our recommendations are B " 1000 and Λ " t0.05, 0.10, . . ., 0.95u.The procedure is summarized in the following algorithm.
The above algorithm is nearly identical to that of Storey, Taylor and Siegmund (2004)'s automatic λ selection algorithm (Section 6), except that Storey, Taylor and Siegmund (2004) omit the additive factor "1" in the numerator of all the estimators for π involved, but we retain it to robustify the FDR dir controlling property of the resulting procedure.Regardless, the intuition behind is the same, i.e. choose a λ which minimizes an estimated mean square error.

B.1. Computation of the component loglikelihoods
We discuss computations of the likelihoods in (3.14) when f i,θ belongs to the normal family N pθ, σ 2 i q or the noncentral t-distributional family N CT pθ, ν i q. (i) N pθ, σ i q: In this case, each l k,i,t in (3.14) has the explicit analytic form (ii) N CT pθ, ν i q: Without sophisticated numerical integration methods, it may be hard to obtain good numerical values for the quantities in (3.14).However, approximation methods can be potentially leveraged; in what follows we assume the common use case where ν i " ν for all i P rms.In a variancestabilizing manner, Laubscher (1960, Section 2) suggests that, if z i is a noncentral t-distributed random variable with noncentrality parameter θ and degree ν ě 4, by bijectively transforming z i to the variable where α " αpνq and β " βpνq are positive numbers depending only on ν, ξ i is approximately distributed as N pγ, 1q for the mean which is also strictly increasing in θ.Hence, by also letting q ξ i " α sinh ´1pβ q z i q, one can alternatively implement ZDIRECT by replacing the component likelihoods in (3.14) with In doing so, we have essentially imposed the prior gp¨, wq in (3.12) on γ instead of θ, but it doesn't change things in the grand scheme as it still approximates a unimodal density about zero on θ; note that sinh ´1p0q " 0. Importantly, we are still working with the partial data tz i,t u iPrms so strong FDR dir control is guaranteed by virtue of Theorem 3.1.Other such strategies may also be explored, possibly based on ideas from Kraemer and Paik (1979) and other references therein.

B.2. Proof of Theorem 3.1
We first quote Lei and Fithian (2018, Lemma 2), a fundamental tool for developing data-masking algorithms.
Lemma B.1.Suppose that, conditionally on the σ-field G ´1, b 1 , . . ., b n are independent Bernoulli random variables with If t is an almost-surely finite stopping time with respect to the filtration pG t q tě0 , then In Lemma B.1, we remark that pG t q tě0 defines a filtration precisely because Proof of Theorem 3.1.The arguments below are inspired by those from Barber et al. (2019) for establishing the FDR dir controlling property of the knockoff filter for variable selection in linear regressions.To begin our proof, write the directional false discovery proportion as FDP dir p tq " |ti : sgnpθ i q ‰ sgnpz i q and i P R tu| 1 _ |R t| " |ti : sgnpθ i q ‰ sgnpz i q and i P R tu| 1 `|A t| 1_|R t| ď q by definition, it suffices to show that For each i P rms, we define the variables b i " 1 ´ui P p0.25, 0.75q where, by the symmetry of f i,0 p¨q from Assumption 1, the latter is equal to `1 if u i is at least as close as q u i to the endpoints of the unit interval r0, 1s, or equal to ´1 otherwise.In particular, since R t " ti : i P M t and u i P p0, 0.25s Y r0.75, 1qu, any i P R t must have its corresponding u i at least as close to the two endpoints of r0, 1s as its reflection q u i P r0.25, 0.75s.As such, it must always be that i.e., an element i can possibly be a discovery only if E i " `1.Moreover, define which will take on the same sign as z i if E i " `1 and z i ‰ 0, as well as the set Ĥ0 " ti : S i ‰ sgnpθ i qu.
Here, Ĥ0 can act like a "random null set" since a false discovery precisely amounts to declaring a non-zero sign for any i P Ĥ0 X ti : E i " `1u, in light of (B.2) being always true.Hence, The last inequality in the preceding display implies that (B.1) can be proved if ı ď 1, which, in turn, we will prove by showing 3) The rest of the proof proceeds by setting the scene to apply Lemma B.1.First, recall M 0 " rms and let G ´1 " σt Ĥ0 u.For t " 0, 1, . . ., define and the filtrations  .4).Second, for any i P prmsz Ĥ0 q X M t , it must also be true that i R C t (by the definition of C t ), which implies that b i belongs to G t ; as such, both |ti : i P prmsz Ĥ0 q X M t and b i " 1u| and |ti : i P prmsz Ĥ0 q X M t and b i " 0u| are measurable with respect to G t .Hence, t is a stopping time with respect to pG t q tě0 , and is almost surely finite because ZDIRECT guarantees to terminate in light of the condition (C2).Lastly, by writing in light of Lemma B.1, one only need to show that P pb i " 1 | G ´1q ě 0.5 for each i P Ĥ0 to wrap up the proof of (B.3).We can break this into three cases; in what follows we also use the operator symbols P θi"0 p¨q, P θią0 p¨q and P θiă0 p¨q defined immediately before Lemma A.2 to emphasize the underlying value of θ i driving the law: • Case 1, θ i ą 0: Since i P Ĥ0 , under θ i ą 0 it must be that S i " ´1 or 0. This can be true with either u i P r0.5, 0.75q or u i P p0, 0.25s, only the former of which can give b i " 1.These two events respectively have probabilities P θią0 pu i P r0.5, 0.75qq " f i,θi pzq f i,0 pzq f i,0 pzqdz and By the MLR property in Assumption 2, P θią0 pu i P r0.5, 0.75qq ě P θią0 pu i P p0, 0.25sq and hence P θią0 pb i " 1 | G ´1q " P θią0 pu i P r0.5, 0.75qq P θią0 pu i P r0.5, 0.75qq `Pθią0 pu i P p0, 0.25sq ě 0.5.
• Case 2, θ i ă 0: The derivations are completely analogous to that of Case 1. • Case 3, θ i " 0: In that case, S i can be `1 or ´1.Since u i is uniformly distributed under θ i " 0, it is easy to see that P θi"0 pb i " 1 | G ´1q " 0.5 (We remark that the arguments above work precisely because G ´1 only provides the meager knowledge of Ĥ0 , without any other knowledge about the data tz i u m i"1 .)

B.3. Proof of Theorem 3.2
For two procedures py sgn p1q i q iPR p1q and py sgn p2q i q iPR p2q , py sgn p2q i q iPR p2q is said to improve upon py sgn p1q i q iPR p1q if ETDrpy sgn p2q i q iPR p2q s ě ETDrpy sgn p1q i q iPR p1q s and FDR dir rpy sgn p2q i q iPR p2q s ď FDR dir rpy sgn p1q i q iPR p1q s.
Let z " pz 1 , . . ., z m q, and let py sgn i q iPR be a certain procedure with FDR dir rpy sgn i q iPR s ď q.We also write R " Rpzq and y sgn i " y sgn i pzq to emphasize that both are functions in z.It suffices to show that the two statements below are true: • Statement 1: If there exists j P rms such that one or both of the disjoint events % z ˇˇˇˇˇj P Rpzq; P pθ j ď 0|z j q ă P pθ j ě 0|z j q; y sgn j pzq " ´1.
and Z p2q j
have non-zero probabilities, the procedure py sgn 1 i q iPR 1 defined by R 1 pzq " Rpzq for all z and, for i P R " R 1 , y sgn

´1
if i " j and z P Z p2q j y sgn i pzq if otherwise improves upon py sgn i q iPR .• Statement 2: If there exist two distinct j, l P rms such that the event Z jl " tz : lfsr j ă lfsr l , l P R and j R Ru has non-zero probability, then it is possible to construct an improved procedure py sgn 1 i q iPR 1 with the property that Suppose both statements can be shown.Then any procedure can be improved by repeatedly applying Statement 2, until we end up with a procedure for which P pZ jl q " 0 for all pj, lq pairs.We can then further improve this procedure by applying Statement 1, and end up with a procedure for which P pZ p1q j q " P pZ p2q j q " 0 for all j, and hence satisfying the conditions piq and piiq in Theorem 3.2; since the ODP cannot be improved, it must have the latter two conditions satisfied.
Proof of Statement 1.We write ETDrpy sgn 1 i q iPR 1 s ´ETDrpy sgn i q iPR s " ż ÿ rP pθ j ą 0|z j q ´P pθ j ă 0|z j qsP pzqdz `żZ p2q j rP pθ j ă 0|z j qq ´P pθ j ą 0|z j qsP pzqdz ą 0, where the second and third equalities come from the fact that py sgn 1 i q iPR 1 and py sgn i q iPR differ only on Z p1q j Y Z p2q j and for j, the fourth equality is from the disjointness of Z p1q j and Z p2q j , and the last equality is from how y sgn 1 j is defined on Z p1q j and Z p2q j as well as the independence across all i " 1, . . ., m.Similarly, FDR dir rpy sgn i q iPR s ´FDR dir rpy sgn so py sgn 1 i q iPR 1 improves upon py sgn i q iPR .
Appendix C: Additional content for Section 4

C.1. Implementation of the LSFR procedure
An exact implementation of the ODP described in Theorem 3.2 involves solving a rather complex infinite integer programming problem (Heller and Rosset, 2021) to determine a threshold for the local false sign rates.As an alternative, in Section 4, LFSR is a similar oracle procedure with an attractively simpler implementation first suggested by Sun and Cai (2007), and it suffices to serve as an oracle benchmark for the power of our compared procedures.Suppose we denote this procedure as py sgn SC i q iPR SC in our notation.
where lfsr p1q ď ¨¨¨ď lfsr pmq are the order statistics of true local false sign rates, and R SC is the empty set if j is not well-defined.The ratio ‰ sgnpθ i q and lfsr i ď lfsr pi 1 q u| i 1 ˇˇtziu iPrms ff of optimally declaring the signs for the subset ti : lfsr i ď lfsr pi 1 q u given the data, which also implies the FDR dir of py sgn is less than q by how j was defined, under the Bayesian formulation (3.8).

C.2. Other simulation results
An R package for our methods is available at https://github.com/ninhtran02/zdirect.We also conducted additional simulation studies to evaluate the performance of different methods under dependent z-values.Specifically, given θ " pθ 1 , . . ., θ m q generated exactly as described in Section 4.1 using the same simulation parameters, we generate z " pz 1 , . . ., z m q with a multivariate normal distribution N pθ, Σq and an autoregressive covariance structure Σ ij " ρ |i´j| for 1 ď i, j, ď m.The following values of ρ are experimented with:

C.2.3. Brief summary
The  Fithian and Lei (2022).It serves as a theoretically valid and more powerful alternative to the widely recognized but very conservative BY procedure (Benjamini and Yekutieli, 2001).We note that dBH requires prior knowledge of the underlying dependence structure, distinguishing it from the BY procedure.dBH dir is a variant of dBH designed specifically for multivariate normal z-values and FDR dir control.It is implemented through the function dBH_mvgauss in the R package dbh, where the gamma parameter is set to 0.9 following the recommendation by Fithian and Lei (2022) for two-sided testing.Its exact FDR dir control for our settings with dependent z-values is established by Fithian and Lei (2022, Corollary 7).
The FDR dir and power of each method under the dependent settings outlined in Appendix C.2.1 and Appendix C.2.2 are similar to those under the independent setting outlined in Section 4. Nevertheless, subtle differences emerge.For strong autoregressive dependence ρ P t´0.8, 0.8u, when w " 0.8 and ξ P t0.5, 1u, the methods STS dir , aSTS dir , BH dir , LFSR, GR and dBH dir exhibited slight FDR dir decreases by approximately 0.01 to 0.02.Conversely, ZDIRECT displayed slight FDR dir increases by approximately 0.01.Despite these increases in FDR dir under strong autoregressive dependence, ZDIRECT consistently maintained empirical control of FDR dir below the designated level of q " 0.10 throughout our additional simulations.
We observed that dBH dir is slightly more conservative in FDR dir control and less powerful than BH dir across our additional simulations.This difference in performance may be attributed to the recommended gamma parameter choice of 0.9 by Fithian and Lei (2022), chosen to reduce the likelihood of obtaining a randomly "pruned" rejection set; see Fithian and Lei (2022, Section 2.2) for an explanation of why it is preferable to avoid the randomized pruning step built into dBH-type methods.This cautious parameter choice may compromise any potential power advantage dBH dir could have over BH dir in the presence of autoregressive dependence.
(b) ξ (signal size parameter): Takes one of the values in t0.5, 1, 1.5, 2, 2.5u.It influences the absolute value of an effect θ i if it is non-zero.(c) v (asymmetry parameter): Takes one of the values in t0.5, 0.75, 1u.It controls the proportion of the alternative θ i 's generated from the positively centered normal component ϕpθ ´ξq; a larger v makes gpθq more asymmetric.

Fig 4 . 1 .
Fig 4.1.Plots of the effect generating density gpθq for every possible combination of w, ξ and v considered in Section 4.
Fig 4.2.Empirical directional false discovery rate and true positive rates of the seven compared methods for the simulations in Section 4; each method was implemented at a target FDR dir level q " 0.1 (black horizontal lines).

Fig C. 1 .Fig C. 2 .Fig C. 3 .
Fig C.1.Empirical directional false discovery rate and true positive rates of the eight compared methods for the simulations in Appendix C.2.1 with ρ " 0.5; each method was implemented at a target FDR dir level q " 0.1 (black horizontal lines).

9 end 10 end Output: Sign
discoveries psgnpz i qq iPR t .
θ pz 1 i qd Ĝt pθq ¸for each i P M t , (3.10) where Ĝt p¨q is an estimate of Gp¨q based on the partial dataset tz i,t u iPrms , or equivalently, tũ i,t u iPrms from (3.4).The index to be unmasked from M t is then ît " arg max iPMt y lfsr i,t ,(3.11)which has the largest estimated local false sign rate; that the estimates in (3.10) are evaluated at the z 1 i 's presumes that any given masked element in M t may come from the rejection set R t .Now we describe how we get Ĝt p¨q based on the partial data.Since the prior Gp¨q is unknown, we will model it to have a unimodal mixture density gp¨; wq " w 0 δ 0 p¨q `ÿ k"´K,...,´1,1,...,K w k h k p¨q (3.12) k ě 1 `1 P G t since Ĥ0 P G ´1 and M t`1 P G t by respecting the condition (C1).By the definitions of A t and R t , we must have that | " |A t X Ĥ0 | `|ti : i R Ĥ0 , i P M t and u i P p0.25, 0.75qu|" |A t X Ĥ0 | `|ti : i P prmsz Ĥ0 q X M t and b i " 1u| and |R t | " |R t X Ĥ0 | `|ti : i R Ĥ0 , i P M t and u i P p0, 0.25s Y r0.75, 1qu| " |R t X Ĥ0 | `|ti : i P prmsz Ĥ0 q X M tand b i " 0u|, one can see that |A t |, |R t | P G t for two reasons: First, |A t X Ĥ0 | and |R t X Ĥ0 | belong to G t because of (B Then the signs y sgn SC " ti : lfsr i ď lfsr pjq u with the index j " jpqq " max # i 1 P rms : performances of different methods are included in Figures C.1 and C.2 for the positively dependent settings in Appendix C.2.1, and Figures C.3 and C.4 for the negatively dependent settings in Appendix C.2.2.Note that, in addition to the existing methods from Section 4.2, we have included an extra method called "dBH dir " in our results.The term "dBH" refers to the dependenceadjusted BH procedure, a recent advancement in FDR testing under arbitrary dependence proposed by rate in multidimensional decisions with applications to microarray studies.Test 27 316-337. FigC.4.Empirical directional false discovery rate and true positive rates of the eight compared methods for the simulations in Appendix C.2.2 with ρ " ´0.8; each method was implemented at a target FDR dir level q " 0.1 (black horizontal lines).ery