More Specific Signal Detection in Functional Magnetic Resonance Imaging by False Discovery Rate Control for Hierarchically Structured Systems of Hypotheses

Signal detection in functional magnetic resonance imaging (fMRI) inherently involves the problem of testing a large number of hypotheses. A popular strategy to address this multiplicity is the control of the false discovery rate (FDR). In this work we consider the case where prior knowledge is available to partition the set of all hypotheses into disjoint subsets or families, e. g., by a-priori knowledge on the functionality of certain regions of interest. If the proportion of true null hypotheses differs between families, this structural information can be used to increase statistical power. We propose a two-stage multiple test procedure which first excludes those families from the analysis for which there is no strong evidence for containing true alternatives. We show control of the family-wise error rate at this first stage of testing. Then, at the second stage, we proceed to test the hypotheses within each non-excluded family and obtain asymptotic control of the FDR within each family at this second stage. Our main mathematical result is that this two-stage strategy implies asymptotic control of the FDR with respect to all hypotheses. In simulations we demonstrate the increased power of this new procedure in comparison with established procedures in situations with highly unbalanced families. Finally, we apply the proposed method to simulated and to real fMRI data.

If the m hypotheses are structured in disjoint families H 1 , . . . , H k with |H | = m for 1 ≤ k ≤ m, a multiple test ϕ (m ) is applied within each family, and we set ϕ (m) = (ϕ (m 1 ) , . . . , ϕ (m k ) ) , we define the global FDR of ϕ (m) by In the sequel, all considered multiple test procedures are such that the quantities in (A.1) -(A.4) actually only depend on the joint distribution of the (random) p-values p 1 , . . . , p m , and one may assume that (Ω, F ) = ([0, 1] m , B([0, 1] m )) without loss of generality.
and the corresponding critical value function is given by r −1 see Finner et al. [2009]. The critical values induced by this critical value function are the ones given in Definition 3.
Let ϑ * = ϑ * (m N1 , . . . , m Nk ) denote a parameter value such that for every family H , 1 ≤ ≤ k, the m N p-values corresponding to true null hypotheses are uniformly distributed on [0, 1] and jointly stochastically independent, and that the remaining (m − m N ) p-values corresponding to false null hypotheses are almost surely equal to zero. Such a parameter value is commonly referred to as a Dirac-uniform configuration, see, e. g., Section 2.2.2 of Dickhaus [2014] and references therein.
Finally, we introduce the following assumption regarding the type I error behavior of ϕ HO with respect to the parameter ϑ of the statistical model. H . This means that FDR ϑ (ϕ HO (m ) ) ≤ FDR ϑ * (ϕ HO (m ) ) for all ϑ which are such that exactly m N null hypotheses are true in family H , 1 ≤ ≤ k.
Assumption A.1 is a standard assumption in FDR theory; see, among others, Blanchard et al. [2014] and Bodnar and Dickhaus [2014] and references therein.
Proof. The global FDR computes as Let t m ∈ [0, 1] denote the random crossing point between r and the ecdf of the p-valuesF m , characterizing the rejection rule of ϕ (m) . This allows for the representation R m /m = r(t m ) = F m , (t m ) and V m = m N F Nm , (t m ). This means that the right-hand side of (A.5) equals .
(A.6) An argumentation analogous to the one in the proof of Theorem 4.5 in Gontscharuk [2010] allows us to find an asymptotic non random upper bound for q N F Nm (t m )/r(t m ). According to (5) in Definition 5, we can choose a δ > 0 and m large enough such that sup t∈[0,1] |F Nm (t) − F N (t)| ≤ δ. Then it holds that By design of the function r α , it holds that q N t q N /r α (t q N ) = min{α, q N }. Thus, it holds that the right-hand side of (A.6) can for eventually all large m be bounded from above by Since δ can be chosen arbitrarily small, this entails lim sup m→∞ gFDR ϑ (ϕ (m) ) ≤ α.
Theorem A.2 (Statistical properties of the procedure ϕ HO ). Assume that the assumptions from above are fulfilled. Then, the proposed procedure ϕ HO defined by Algorithm 2 controls the FWER at the stage of the families at level α. Furthermore, the global FDR of ϕ HO and the FDR of ϕ HO within each family are asymptotically bounded by α.
Proof. Recall that the family H is selected at the first stage of analysis if and only if the corresponding conjunction p-value p u /m does not exceed α/κ. Since κ > k, the Bonferroni inequality yields the first assertion.
In order to show asymptotic control of the global FDR, assume first that q N < 1 for all 1 ≤ ≤ k. We notice that every hypothesis which is rejected by ϕ HO (m ) would also be rejected by ϕ AORC u ,(m ) alone, where ϕ AORC u ,(m ) denotes the SUD test which is applied in family H in the second stage of ϕ HO (m ) , 1 ≤ ≤ k. This follows from the fact that κ and hence, u , are fixed constants and the rejection rule of ϕ HO (m ) involves the additional condition regarding p u /m . Hence, R m (ϕ HO (m ) ) ≤ R m (ϕ AORC u ,(m ) ). Under ϑ * (cf. Assumption A.1) and by construction of r α , we have, by setting t q N = 1 for q N < α, that R m (ϕ AORC u ,(m ) )/m → r α (t q N ) almost surely, cf. Corollary 5.1.(i) of Finner et al. [2009]. We conclude that lim sup m →∞ R m (ϕ HO (m ) )/m ≤ r α (t q N ) for all ϑ ∈ Θ. On the other hand, consider for each 1 ≤ ≤ k such that H has been selected at the first stage of analysis the following chain of inequalities: Thus, if the family H is rejected, the SUD procedure ϕ AORC u ,(m ) will reject at least u hypotheses within H . Notice that, by definition of u , we have that u /m ≥ κ −1 . We conclude that, in each selected family H , lim inf m →∞ R m (ϕ HO (m ) )/m > 0. Thus, Theorem A.1 can be applied with k replaced by |{1 ≤ ≤ k : H has been rejected}| in this case.
However, if for some ∈ {1, . . . , k} we have q N = 1, we can find a number m which is large enough such that P ϑ (p u /m ≤ α/κ) ≤ α/κ due to the assumed validity of the conjunction pvalue. Hence, letting x denote all observed data, straightforward calculation yields for ϑ, which is such that q N = 1, that which completes the argumentation.
Asymptotic FDR control within each family can be established as follows. If a family H is not rejected, we have R m (ϕ HO (m ) ) = V m (ϕ HO (m ) ) = 0. On the other hand, in each selected family H , it holds V m (ϕ HO (m ) ) ≤ V m (ϕ AORC u ,(m ) ) by the same argumentation as for R m (ϕ HO (m ) ). Under the LFC ϑ * , this also entails that u ,(m ) ) ∨ 1 almost surely, because the structure of an SUD test yields that, as soon as V m (ϕ HO (m ) ) ≥ 1, we have R m (ϕ HO (m ) ) = V m (ϕ HO (m ) ) + (m − m N ), and the mapping x → x/(x + a) is isotone in x > 0 for a ≥ 0. Since ϕ AORC u ,(m ) asymptotically controls the FDR under ϑ * , this implies the assertion.

A.2 The tuning parameter κ
Here, we report results of a power study regarding the tuning parameter κ. The study was done in two setups for the normal means problem with effect size µ * and variance 1, analogous to the simulations in "Computer simulations regarding the power of ϕ HO ". Our theoretical investigations indicate that we can expect the power of the procedure ϕ HO within one selected family H (in our case of size m = 2,000) to depend on the ratio of true null hypotheses q N within the family. To this end, we considered a balanced and a highly unbalanced case by setting q N ∈ {0.5, 0.99}. In both cases the power of ϕ HO has been estimated as a function of µ * ∈ [0, 5], and we let the parameter κ range from 1 to 10,000,000 on a log 10 scale.
The plots in S1 Fig. indicate that small values of κ lead to a high specificity in case of a large value of q N , while large values of κ lead to a good sensitivity in case of a moderate value of q N . This is line with the recommendation that κ should be chosen according to the amount of signals within a family which is considered relevant.