FDR- and FWE-controlling methods using data-driven weights

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1 FDR- and FWE-controlling ethods using data-driven weights LIVIO FINOS Center for Modelling Coputing and Statistics, University of Ferrara via N.Machiavelli 35, 44 FERRARA - Italy livio.finos@unife.it LUIGI SALMASO Departent of Manageent and Engineering, University of Padova Str. S. Nicola 3, 36 VICENZA - Italy Abstract Weighted ethods are an iportant feature of ultiplicity control ethods. The weights ust usually be chosen a priori, on the basis of experiental hypotheses. Under soe conditions, however, they can be chosen aking use of inforation fro the data (therefore a posteriori) while aintaining ultiplicity control. In this paper we provide: ) a review of weighted ethods for FWE (both paraetric and nonparaetric) and FDR control; 2) a review of data-driven weighted ethods for FWE control; 3) a new proposal for weighted FDR control (data-driven weights) under independence aong variables; 4) under any type of dependence; 5) a siulation study that assesses the perforance of procedure of point 4 under various conditions. Key words: A priori ordered hypotheses; FDR; FWE; Multiplicity control; Weighted procedures. Introduction In probles dealing with thousands (and soeties hundreds of thousands) of variables, the standard Type I error rate criterion used to evaluate tests becoes less iportant as hundreds of significances ight easily be, as a atter of fact, Type I errors. On the other hand, attepts to rigorously control the failywise type I error rate (FWE) are typically excessively conservative. For exaple, if the Bonferroni ethod is used to control the FWE with tests, then a test will have to be significant at the α/ level for it to be declared real. The two coplaints ost often ade about this ethod are that (i) the procedure s dependence on sees arbitrary, and related to that, (ii) the ethod is exceedingly conservative for large values of. Hol s (979) step-down approach is a slight iproveent on the siple Bonferroni ethod, but gains little in ters of power when is large. The Hol s ethod rejects the hypothesis corresponding to the ost significant

2 (sallest p-value) test if the p-value is less than α/; if this hypothesis is rejected, then the second sallest p-value is copared to α/( ), and so on. It is usually called Bonferroni-Hol Procedure (BHP). Benjaini and Hochberg s false discovery rate (FDR) controlling ethod (995) has been proposed as an alternative to FWE-controlling ethods. To a certain extent, FDR solves probles (i) and (ii) though it ay becoe exceedingly conservative for large. FDR-controlling procedures do not generally control the FWE, thus they allow soe fraction of detected significances to be in error. Weighted ethods are useful when soe H i hypotheses are deeed ore iportant than others - e.g. in clinical trials the various patient end points ight be ranked a priori, and the testing procedure designed to give ore power to the ore iportant hypotheses. The siplest weighted ultiple testing procedure, discussed for exaple in Rosenthal and Rubin (983), is to reject H i if p i w i α, where the weights w i lie in the siplex w i ; Σw i =, and p i is the p-value of the i-th test, i =,...,. The choice of w i ay be based purely on the a priori iportance of the hypotheses or, to optiise power, based on prior inforation (Spjøtvoll, 972; Westfall et al., 2). Giving greater weight to the ore significant hypotheses is generally considered data snooping, and such ethods inflate (ultivariate) type I error rates. However, when properly chosen, weights can be taken fro the concurrent data set so as to iprove power without coproising significance levels. In this paper we consider a class of weighted FWE-controlling procedures and a class of weighted FDRcontrolling procedures which exploit inforation fro the data and, under soe conditions, iprove perforances in ters of power. All proofs of the proposed new theores are given in the Appendix. 2 Weighted FWE and FDR ethods In order to discuss the error rates we would like to control, we get R i = if H i (H i being a true or false null hypothesis) is rejected and otherwise, and we get V i = if a true H i is (erroneously) rejected, and otherwise. The failywise error rate can be defined as FWE = P( V i > ), which is the probability of aking at least one erroneous rejection. The false discovery rate, which is the expected proportion of erroneously rejected hypotheses aong the rejected ones, can be defined as FDR= E( V i / R i ), where V i / R i is defined as when R i =. It is easy to see that FDR FWE. Note also that when all hypotheses are true FDR = FWE (Benjaini and Hochberg, 997; Benjaini and Yekutieli, 2). 2. Weighted FWE ethods When we require FWE control, there are at least two weighted procedures based on Bonferroni s inequality to be considered. The first is Hol s (979) weighted procedure (known as the Weighted Hol s Procedure - WHP), the second, proposed in Benjaini and Hochberg (997), ight be called Weighted Benjaini- 2

3 Hochberg s Procedure (WBHP). Both of the above procedures control the unweighted FWE. The natural weighted extension of FWE (P[ V i > ]) which we call WFWE is P[ w i V i > ]. These procedures control the WFWE as well since the latter is equal to the forer for any set of weights. We shall now briefly describe the two procedures. 2.. Hol s weighted procedure (979) Let Pi = P i /w i and order P()... P (). Let H (i), w (i), correspond to P (i). Reject H(i) for i =,2,3,... as long as P (i) α. k=i w (k) Hol s unweighted procedure is the above procedure with equal weights which all have to be equal to one. If the weighted procedure called for the rejection of H i when P(i) k=i w (k) = P (i) (+ k=i+ w (k) /w (i)) is saller than a constant α, then a larger w (i) iplies greater power for rejecting that hypothesis. The weights w (i) coe into play also in the definition of the ordered vector P(i) ; therefore w (i) affects the probability of rejecting H (i) given that H (j) j < i has been rejected and it also affects the order of the tested hypothesis Benjaini-Hochberg s weighted procedure (997) Benjaini and Hochberg (997) discuss a procedure which controls the WFWE by the ordered P i, P (i)... P (). Let H (i), w (i) ( i= w (i) = ), correspond to P (i). Reject H (i) for i =,2,3,... as long as P (i) w (i) k=i w (k) α. Proof of the FWE control is found in Benjaini and Hochberg (997, theore 2). Again, Hol s unweighted procedure is the sae as the above procedure with equal weights which all have to be equal to one. Note that this procedure respects the natural ordering of p-values (not corrected for ultiplicity). Therefore, w i affects the probability of rejecting H i given that H j j < i has been rejected, but it does not affect the order of the tested hypothesis. 2.2 Weighted FDR ethods For ease of notation assue that the first o hypotheses tested are in fact true and = are false. The false discovery rate is ( i= E(Q) = E V ) i i= R () i 3

4 which is the expected proportion of the falsely rejected hypotheses aong the rejected ones. When weighting is desired, the FDR can be generalized as follows. Let Q(w) be Q(w) = { È i= wivi È i= wiri i= w ir i > otherwise, then the weighted false discovery rate (WFDR) is defined as E(Q(w)). Note that if soe of the weights are, and the others are all equal, then the WFDR is identical to the FDR for the proble restricted to testing hypotheses with positive weights. Under the intersection null hypothesis that all tested hypotheses are true, WFDR = WFWE. It is again easy to show that WFDR WFWE, so a WFDR controlling procedure is potentially ore powerful than a WFWE controlling procedure Weighted FDR ethods (Benjaini and Hochberg, 997) for independent variables Consider the following procedure: Let k be the largest j satisfying (2) j i= P (j) w (i) q, (3) then reject H (),...,H (k) (q being the desired proportion of false rejections). Theore 2. (Benjaini and Hochberg, 997; theore 4) Procedure i M w i (3) always controls the FDR at level q (M being the set of indexes that correspond to true null hypotheses) for independent test statistics. In this procedure, the weights incorporated into the error rate are suitably accuulated to for the procedural weights. It is iportant to reject a hypothesis with a high weight as it considerably increases the weight of the total discoveries. However it also increases the weight of the errors. Essentially we are incorporating the sae weights into the loss fro errors w i V i and the gain fro rejections w i R i Weighted FDR ethods for dependent variables Benjaini and Yekutieli (2) provide a correction constant that guarantees FDR control under every type of dependence between variables. In this section we extend previous results fro the literature to the weighted case for dependent variables. Nothing that this situation is not the ain task of the paper, hence this paragraph is self-contained with respect of the rest of the paper. È Theore 2.2 When procedure (3) is perfored with j= (/Σ h jw h )q taking the place of q, it always controls the FDR at level È i M w i q (M being 4

5 the set of indexes that correspond to true null hypotheses). This result ay increase the range of probles for which a powerful procedure with proven FDR control can be offered. Obviously the adjustent by j= Σ h j w h is very often unnecessary, and yields a procedure that is too conservative. Indeed the control is given at level no correction whatsoever is necessary until ( )È h M w h j= Σ h j w h ( )È h M w h j= Σ h j w h q and. Given that in any real experiental cases the proportion of active variables is low, this condition is usually respected and the correction is not therefore necessary. In the ain theore of the work of Benjaini and Yekutieli (2, theore.2), the authors prove that a procedure without this correction still controls the (unweighted) FDR in failies with Positive Regression Dependency on each eleent fro a Subset M, or PRDS on M. Recall that a set D is called increasing if x D and y x, iplies that y D as well. Property PRDS: For any increasing set D, and for each i M, P(X D X i = x) is non-decreasing in x. Hence, the structure of the dependency assued ay be different for the set of the true hypotheses with respect to the set of the false hypotheses. Background on these concepts is clearly presented in Eaton (986), suppleented by Holland and Rosenbau (986). The PRDS property is a relaxed for of the positive regression dependency property (Sarkar, 969) and of MTP 2 (Sarkar, 998) as well. Hence the PRDS is a very weak condition of dependence between variables which often holds for the real data. The control of the FDR over PRDS variables can be easly extended to the case of weighted FDR procedures as follows Theore 2.3 If the joint distribution of the test statistics is PRDS on the subset M of test statistics corresponding Èto true null hypotheses, the procedure i M w i given by (3) controls the FDR at level q. Proof is straightforward if one adapts the proof given in Benjaini and Yekutieli (2) to the weighted procedure. In the sae way we adapt theore 2.2 fro theore.3 of the sae authors. Hint: define v as the su of weights of true null hypotheses and s as the weights of false ones. In the sae way, it is trivial atter to prove the control for discrete and one-sided test statistics (see theores 5.2 and 5.3 fro Benjaini and Yekutieli, 2). 3 Data-driven weighted procedures We show a way to define data-driven weights that gathers inforation on the distribution of the hypotheses under the null or under the alternative. After- 5

6 wards, we show how these weights allows for FWE or FDR control for paraetric and nonparaetric tests. 3. Data-driven weights We consider paraetric and nonparaetric versions of the one saple and twosaple tests. Extension to C independent saples is straightforward. The paraetric one-saple case is characterized by a saple of n i.i.d. ultivariate noral observation vectors of diension x j = µ = x j. x j N (µ,σ), j =,...,n, (4) µ.. µ,σ = σ... σ σ... σ and we wish to test the partial hypotheses H i : µ i = (i =,...,) under strong control of the failywise type I error. The covariance atrix Σ is arbitrary and positive seidefinite. In a nonparaetric setting, we assue that the n i.i.d. -diensional saple vectors x j (j =,...,n) have a density f(x) which is syetrical to location vector µ = (µ,...,µ ), f(µ + x) = f(µ x). Dependences aong variables are not specified but are generally assued to exist. In the case of two independent saples fro two -diensional noral populations with the sae covariance atrix x jc = x jc. x jc N (µ c,σ),c =,2; j =,...,n c, n = n + n 2, (5) we consider the usual two-saple t statistic in order to test partial null hypotheses H i : µ i = µ 2i (i =,...,). In the nonparaetric approach, we siply assue that under the null hypothesis the two saples have the sae distribution: f(x i ) = f(x i2 ) i =,...,. One way to obtain inforation about the presence of possible alternative hypotheses is given by the use of the total variance calculated on the entire saple without considering the classification into groups. Indeed the total su of squares SST i (i =,...,) is given by su of between groups su of squares 6

7 SSB i and within groups SSW i as follows: SST i = n c (x ijc x i ) 2 = c=,2 j= = n c (x ijc x ic ) 2 + n c (x ic x i ) 2 = c=,2 j= = SSW i + SSB i c=,2 where x ic = n c j= x icj/n c ; c =,2 and x i indicates the overall ean for variable i. The ean of total variance estiated by MST i = SST i /(n ) (i =,...,) is equal to: E(MST i ) = E(SSW i /(n )) + E(SSB i /(n )) = σ ii (n 2)/(n ) + σ ii /(n ) + = σ ii + n/(n )((µ i µ i2 )/2) 2 = σ ii + n/(n )(δ/2) 2 c=,2 n c (µ ic µ i ) 2 /(n ) In this way we highlight that SST i tends to be larger when the variable i is under H. This is especially true when n is sall. In order to understand how the total variance is related to the power of tests with different saple sizes, we keep fixed the value of the t-statistic for increasing n (i.e. the sae p-value when σ ii is known or the asyptotical approxiation to the gaussian distribution holds). Hence δ n = δ / n, being the t-statistic equal to t = δ n n/σii = δ /σii. Now, writing the expected value of MST as a function of n we get: E(MST i ) = σ ii +(δ n /2) 2 n/(n ) = σ ii +(δ/( n2)) 2 n/(n ) = σ ii +(δ/2) 2 /(n ) which is a decreasing function of n. Siilar considerations can be ade in the case of ore than two independent saples defining S i as the total variance of variable i. If we are considering a one-saple test S i becoes S i = n j= x2 ij /n. Note that S i = n j= x2 ij /n = n j= (x ij x i ) 2 /n + x 2 i. Therefore, with this definition we are considering the shift fro zero and the error variability. Using the total variability as for the two saple case, it whold be not sensitive about non null effects. Hence, to define cobining functions which give greater weight to the p- values with large S it is sufficient to consider w i = f(s i ); i =,...,, (6) with f non decreasing onotone function of S i. One particularly interesting case is obtained when weights are zero if S i is below a threshold γ : { Si if S w i = i γ (7) otherwise (i =,...,). 7

8 This is equivalent to excluding the corresponding variables fro the analysis. Also note that for γ = the weights are equal to the total variance, i.e. w i = S i. A choice less sensitive to the agnitude of S i is the following: w i = { if Si γ otherwise (i =,...,). (8) We shall see its usefulness in the siulation study of section FWE-controlling procedures with data-driven weights 3.2. Paraetric approach In the paraetric setting, Läuter, Gli and Kropf (996) noted that linear cobination of weight vectors ay depend on the data through certain quadratic fors, and the resulting (ordinary) t-tests retain their significance levels. Thus, by choosing the vector of weights suitably, the sae procedures ay be used to give ore weight to particular hypotheses selected by the data. Westfall, Kropf and Finos (24) propose using WHP with w i = S r i (r R,) ; i =,...,. This class of functions has the peculiarity of including two well-known procedures as particular cases: - if r =, it is equivalent to Bonferroni-Hol s procedure, and all weights are equal to one. - if r, this corresponds to Kropf and Läuter s procedure (22) which orders the tests based on their variance and sequentially tests the hypotheses of interest without corrections until the first accepted hypothesis is found. In the two extree cases, therefore, the influence of S i is different - in the first case S i is not considered, in the second it is very iportant since it establishes the order of adission of the hypotheses independently of their significance. By using the theory of Spherically Distributed Matrices (Fang and Zhang, 99), it is possible to state that under assuptions (4) and (5) the subset of variables corresponding to the true null hypothesis (defined by X ) is leftspherically distributed and the conditional distribution of X for fixed X X is also left-spherical. Hence, each of the coluns of X is also conditionally left-spherically distributed and the t tests (one-saple or two-saple) exactly aintain their type I error (i.e. their p-values are uniforly distributed). Kropf and Hoel (24), propose the use of the sae kind of weight but with WBHP. Again with r =, we have the usual unweighted Bonferroni-Hol ethod but the procedure does not converge to Kropf and Läuter s paraetric procedure (Kropf and Läuter, 22) when r. The FWE control again akes use of the theory of Spherical Distribution Matrices. 8

9 3.2.2 Nonparaetric approach For the one-saple proble the (nonparaetric) rank-based counterpart of WHP and WBHP uses the edians of the absolute values of the original observations instead of the total variances and the p-values fro the one-saple Wilcoxon tests instead of those fro the t tests. For the two-saple proble they ake use of the epirical interquartile range of the pooled saple of n = n + n 2 observations and the p-value fro the two-saple Wilcoxon test (WilcoxonMannWhitney U test). Kropf et al. (24) proved that WHP based on rank tests controls FWE. Kropf and Hoel (24) do the sae for the WBHP. The proofs for the failywise type I error control are very siilar to those for its paraetric counterpart. A second nonparaetric approach is offered by Finos and Salaso (25) and is based on perutation tests. The proof of the FWE control is based on the invariance principle: if w i is defined on the basis of a generic eleent of orbit X /X (i.e. the perutation saple space), the procedure controls the FWE because the distribution of the test statistic is independent of the values of w i. There is an interesting connection between the principle of invariance and the theory of left-spherical distribution. In both cases we obtain tests that, conditional to the weights based on the variance, have a unifor distribution under the null hypothesis. Finos e Salaso also show that, in the conditional fraework, any statistic that evaluates the dispersion of data depending on a generic eleent of the perutation space, is an unbiased ultiple test. Possible choices for defining the weights include using functions of the coefficient of variability (w i = f(cv ) = f(s i /x i )) or functions of the first-third quartile range (w i = f(iqr) = f(q 3i Q i ), where Q 3i and Q i are the third and first quartile for the i-th variable). This last choice sees to be particularly useful when there are heavy-tailed variables, giving little weight to the large values produced by the rando errors and ore weight to the fixed effects. Finally, these results can be extended fro the class of Bonferroni-like cobining function to the broader class of non paraetric cobining functions defined by Pesarin (2). 3.3 FDR controlling procedures with data-driven weights In order to prove that the weights chosen as in (6) guarantee FDR control, we shall ake use of the theory of spherically distributed atrices and follow the ideas provided in Benjaini and Hochberg (997), we point out the steps of the proof where it is necessary to refer to the theory of left-spherically distributed atrices to guarantee that the p-values corresponding to the true hypotheses H i follow a unifor distribution conditional to w i. Lea For any independent p-values corresponding to the true null hypotheses, any set of = p-values, corresponding to the false null hypotheses, any set of weights w i, i= w i = defined as in (6) and any constant q, the ultiple testing procedure defined by (3) satisfies the 9

10 inequality E(Q(w) P + = p,...,p = p ) i= w i q. (9) Theore 3. If test statistics are independent, then the procedure based on (3) and weights defined as in (6) controls the WFDR at level q. The proof of the theore is straightforward fro the previous lea. For dependent variables, WFDR control is proven by the following theore. È Theore 3.2 When procedure (3) with weights defined as in (6) is perfored with j= (/Σ h jw h ) replacing q, it always controls the FDR at sig- i M w i nifance level less than or equal to q (M being the set of indexes that correspond to true null hypotheses). Theore 3.3 If the joint distribution of the test statistics is PRDS in M, the procedure given by (3) controls the FDR at level weights are defined as in (6). 4 A siulation study È i M w i q also when It has been noted that the classic procedures lose power when there is a high nuber of variables. For this reason this siulation study focuses on cases with a high nuber of variables. We consider the following odel for ultivariate one-saple data: at first, the variances of each of the easureents are assued to be independently generated as σi 2 τ λ/χ 2 λ, i =,...,, where χ2 λ denotes a chi-square distributed rando variable with λ degrees of freedo. The paraeter τ is a nuisance paraeter reflecting overall scale; for convenience it is taken to be equal to in the siulations. The paraeter λ is specified in each siulation; sall λ corresponds to large variance heterogeneity across variables, while λ = corresponds to variance hoogeneity across variables; we take λ = 5, λ = and λ = in the siulations. We also consider the average epirical ratio between the variance of the variances and the variance that would results in the case of hoogeneity across all variables (i.e. λ = ), indicated by ζ. This can be considered an indication of heterogeneity across the variables. Next, conditional to σi 2, the effect sizes δ ji = µ i /σ i ; j =,...,n (see (4)) are assued to be drawn independently fro a ixture N(,σδ 2 ) and single point () distributions. The paraeter σδ 2 is specified in the siulations; larger σδ 2 denotes generally larger alternatives. We take σ2 δ = or 2 in our siulations. The ixing paraeter is denoted by π, with π = P(δ i ), and represents the proportion of variables under H. It varies fro to.5. Finally, conditional to the eans and variances, the observed data vector is assued to coe fro a -diensional ultivariate noral distribution, with i.i.d. coponents.

11 Power is taken to be the average fraction of correctly rejected hypotheses in each siulation. Specifically, defining K = {i H i is rejected} and K 2 = {i δ i }, we define Power = E( K K 2 / K 2 K2 > ). Figure and 2 show the power estiates with saple size (n) equal to 5 and respectively, for different proportion of active variables (π), variances of effects σδ 2, variance heterogeneity (λ) and nuber of variables () using weights as defined in (6), (7) and (8) (with γ = 2), using siulated data sets. Aong copeting procedures that control the FWE, we show only the WBHP that generally proves to be ore powerful than the WHP. A detailed report of the power estiates for all the presented procedures and various thresholds is available upon request to the corresponding author. [Figure about here.] [Figure 2 about here.] A discussion of the results is given in the Conclusion section. 5 Conclusions In this paper we have discussed data-driven weighted procedures controlling FWE. We also propose new data-driven weighted procedures controlling FDR in the case of independence between variables and under Positive Regression Dependency on the subset of variables corresponding to null hypotheses (PRDS). These procedures provide good results as long as the variances of the variables are approxiately unifor under H. As heterogeneity grows, the procedures (though reaining correct) becoe less powerful. However the siulation study suggests they can also be applied in cases of high incidence of nonhoogeneity. In particular, the solution proposed in (8) based on a threshold, sees to provide better results in the FWE-controlling WBHP procedure even when there is very high heterogeneity. This is particularly true for siulations with a high nuber of variables (M = ). This is due to the fact that the threshold value reduces the nuber of tested variables and this copensates the loss of power characterizing the Bonferroni-like procedures in high diensional situations. In contrast, the proposal ade in (6) sees to behave better in ters of power in the WFDR-controlling procedure even when the ζ ratio is large. As an exaple, consider the setting n = 5, = and σ δ = 2 (figure, plots on the botto) and consider the case with % of active variables. The power of the FDR-controlling procedure is about 33%. When there is coplete hoogeneity, the power of the WFDR-controlling procedure proposed in (6) (w = S 2 ) is very high (6%). Power does not change when the variance heterogeneity across variables is large. On the contrary, when the lack of hoogeneity is huge then power decreases very rapidly. As a general guideline, the gain in power is higher for saller saple size (see also the foral discussion of section 3.), whereas power sees to be independent fro the nuber of variables involved. Although the cobination of low saple size and high nuber of variables used in the

12 siulation could sees negligible in the real applications, this setting is the ost coon one in brain iaging studies, where the saple size rarely exceeds 8- subjects and the nuber of variables involved are rarely less than.. Also in icroarray studies the saple size and nuber of variables often have siilar settings. The software has been developed in Matlab 7 (Mathwork inc. c ) and is available upon request to the corresponding author. REFERENCES Benjaini, Y. and Hochberg, Y. (995), Controlling the false discovery rate: A new and powerful approach to ultiple testing. J. Roy. Statist. Soc. Ser. B 57, Benjaini, Y., Hochberg, Y. (997), Multiple Hypotheses Testing with Weights. Scandinavian Journal of Statistics, 24: Benjaini, Y., Yekutieli, D. (2), The Control of the False Discovery Rate in Multiple Testing under Dependency. The Annals of Statistics Vol 29, n 4, Fang, K.T., Zhang, Y.T. (99), Generalized Multivariate Analysis, Science Press, Beijing. Finos, L., Pesarin, F., Salaso, L. (23), Cobined tests for controlling ultiplicity Closed Testing procedures. Italian Journal of Applied Statistics 5, Finos, L., Salaso, L. (26), Weighted ethods controlling the ultiplicity when the nuber of variables is uch higher than the nuber of observations. Journal of Nonparaetric Statistics 8, n 2, Golub, T.R., Sloni, D.K., Taayo, P., Huard, C., Gaasenbeek, M., Mesirov, J.P., Coller, H. (999), Molecular classification of cancer: class discovery and class prediction by gene expression onitoring Science 286, Hol S. (979), A siple sequentially rejective ultiple test procedure Scand. J. Statist., 6, Läuter, J. (996), Exact t and F tests for analysing studies with ultiple endpoints. Bioetrics 52, Läuter, J., Gli, E., Kropf, S. (996), New ultivariate tests for data with an inherent structure. Bioetrical Journal 38, Erratu: Bioetrical Journal 4, 5. Kropf, S. and Läuter, J. (22), Multiple test for different sets of variables using a data-driven ordering of hypotheses, with an application to gene expression data. Bioetrical Journal 44, Kropf, S., Hoel, G., (24), 7th Tartu Conference on Multivariate Statistics in 23 and appeared in Acta et Coentationes Universitatis Tartuensis de Matheatica 8 (24), spec. vol., Kropf, S. Läuter, J., Eszlinger M., Krohn K., Paschke R. (24), Nonparaetric Multiple Test Procedures with Data-driven Order of Hypotheses and with Weighted Hypotheses. Journal of Statistical Planning and Inference 25,

13 Marcus, R., Peritz, E. and Gabriel, K.R. (976), On closed testing procedures with special reference to ordered analysis of variance. Bioetrika 63, Pesarin, F. (2), Multivariate perutation test with application to biostatistics. Wiley, Chichester. Rosenthal, R. and Rubin, D.B. (983). Enseble-adjusted p-values. Psychological Bulletin 94, Sarkar, T. K., (969). Soe lower bounds of reliability. Tech Report, No. 24 Dept. of Operation Research and Statistics, Stanford University. Sarkar, S. K., (998). Soe probability inequalities for ordered MTP 2 rando variables: A proof of Sies conjecture The Annals of Statistics, 26(2), Spjtvoll, E. (972). On the optiality of soe ultiple coparison procedures. Ann. Math. Statist. 43, Westfall, P.H., Krishen, A. (2). Optially weighted, fixed sequence and gatekeeper ultiple testing procedures Journal of Statistical Planning and Inference 99, Westfall, P.H., Kropf, S., Finos, L. (24). Weighted FWE-controlling ethods in high-diensional situations. Recent Developents in Multiple Coparison Procedures, Institute of Matheatical Statistics Lecture Notes-Monograph Series, Vol. 47, Y. Benjaini, F. Bretz, and S. Sarkar, eds., ACKNOWLEDGEMENT Authors wish to thank two Referees for helpful coents and suggestions. APPENDIX Proof of theore 2.2. Let C (i) k denote the event at which the true null hypothesis i is rejected ( along È with other k (true and/or false) hypotheses. j È i= Now let p ijk = P {P i [ wi q, j i= wi q]} C(i) k ). Note that for any given set of ordered weights w i, k= ( i j p ijk = P {P i [ w i q, Note that, for each i the C (i) k E(Q(w)) = = i M k= i M k= h k w h i M j= k=j = w h i M j= i j w ) i q]} ( kc (i) k ) = qw j. () are disjoint, so the FDR can be expressed as: h k w h ( P (P i k p ijk = j= h j w h h j w h h k w h i M j= k=j p ijk i M j= q ( = 3 j= ) q) C(i) = h k w h h j w h ) Σ h j w h k p ijk = k= p ijk i M w h q ()

14 Q.E.D. Proof of the Lea. When = the result is straightforward. Now, we assue that (9) holds for any. If = : all null hypotheses are false, Q =, and E(Q(w) P = p,...,p + = p + ) = i= w i q. (2) + If > : Let P ( ) denote the p-value corresponding to the largest p-value aong P P. Since these correspond to the true null hypotheses, P ( is ) distributed as the largest of independent U(,) rando variables (this is due to the fact that the conditional distribution of X for fixed X X is leftspherical), and f (p) = P( ) p for p. Let us also define j as the largest j, + j +, satisfying Let p P ( = p, ) p j j i= w i q. (3) + denote the value of the right side of (3) at j. Conditional to p + E(Q(w) P + = p,...,p + = p ) = E(Q(w)) P ( = p,p ) + = p,...,p + = p )f P (p)dp + (4) p E(Q(w)) P ( ) = p,p + = p,...,p + = p )f P (p)dp (5) For p p all j hypotheses are rejected, and Fro (4) we get Q(w) = i= w i j i= w. (6) i i= w i j i= w (p ) i= = w j i i= w i j i i= w i + q(p ) + i= = w i + q(p ) +. (7) In order to evaluate (5) we further condition to the p-value at which P ( ) is achieved, indexed by i, i : 4

15 = i = p E(Q(w) P ( ) = p,p ( ) = P i,p + = p,......,p + = p )f P (p)dp = p E(Q(w) P ( ) = p,p ( ) = P i,p + = p,......,p + = p )f P (p)dp (8) Let us consider both cases: when P j P ( ) = P i = p < p j+ for j > j, or when p < P ( = P ) i = p < p j +. Fro the definition of j and p, no hypothesis can be rejected because of the value of p, p j+,...,p +. Therefore, when all hypotheses true and false are considered together and their p-values are ordered, a hypothesis H (i) can be rejected only if there exists a constant k, i k j, for which k P (k) i= w (i) q. (9) p ( + )p Equivalently, H (i) will be rejected if k satisfies with P (k) p (j ) k i= w (i) (j ) j i= w (i) ( j i= w ) i ( + )p q = w (i) k i= w ( j (i) i= w ) i (j ) ( + )p q. (2) (j ) = j i= w. (2) (i) These weights satisfy w, and j i= w i = j ; since we conditioned to P i = P () = p, the reaning P i /p are distributed as independent U(, ) rando variables (again, this is due to the left-spherical property of X for fixed X X ); p i /p for i = +,...,j are values in the interval (,) corresponding to false null hypotheses. Hence, using (3) to test the j = hypotheses, of which are true, is equivalent to using the procedure with the constant È j i= w (i) (+))p q taking the role of q. Applying now the starting assuption that 9 holds for any induction hypothesis we have that E(Q(w) P ( ) = p,p ( ) = P i,p + = p,...,p + = p ) = i=;i i w i (j ) = ( j ) i= w i ( + )p q where (23) is derived after replacing w with its definition in (2). (22) i= w i w i q (23) ( + )p 5

16 The upper bound in (23) depends on p, but not on a particular j p j < p < p j+. Let us recall that for j the range for p is p < p P j+ and it does depend on i. Therefore, by integrating (23) over (p,], while still conditional to i we get p i= w i w i q p dp = ( + )p Averaging now over i, fro (8) and (24) we get = i = i= w i w i q( (p ) ). (24) ( + ) E(Q(w)) P + = p,...,p + = p )f P (p)dp = p ( ) i= w i w i q( (p ) i= ) = w i + + q( (p ) ) (25) Finally by adding (25) and (7) we get the desired inequality (2) for +. Q.E.D. Proof of theore 3.2. In order to prove that the procedure for dependent variables controls the WFDR, we siply need to note that P i, i =,..., in () and () are under H, hence uniforly distributed conditional to w i. Indeed, X is left-spherically distributed, its conditional distribution for fixed X X is also left-spherical and each of the coluns of X is conditionally leftspherically distributed as well and the result follows. Q.E.D. Proof of theore 3.3. It directly follows fro the fact that only statistics under the null hypothesis have to be PRDS, therefore the sae consideration of proof of theore 3.2 over the left-spherically distributed variables holds. Q.E.D. 6

17 λ=inf λ=5 λ= n=5 = σ = eff ζ=.e+ λ=inf ζ=4.39e+ λ= ζ=.23e+25 λ= n=5 = σ = eff ζ=.e ζ=4.894e ζ=4.767e+22 n=5 = σ =2 eff ζ=.e ζ=9.36e ζ=5.533e+8 n=5 = σ =2 eff ζ=.e+ ζ=.e+2 ζ=.244e+23 FDR WFDR th=2 WFDR w=s 2 BHP WBHP th=2 WBHP w=s 2 Figure : Siulation results for n = 5: Power estiates for different values of π (abscissa axis), and σδ 2 (along the rows), λ (along the coluns). FDR = FDRcontrolling procedure, WFDR = Weighted FDR-controlling procedure, BHP = Bonferroni-Hol FWE-controlling procedure, WBHP = Benjaini-Hochberg weighted FWE-controlling procedure. Moreover th=2 eans threshold γ = 2 (see definition (8)), w = S 2 eans weight w equal to the total variance (see definition (7) with γ = ). 7

18 λ=inf λ=5 λ= n= = σ = eff ζ=.e+ λ=inf ζ=4.39e+ λ= ζ=.23e+25 λ= n= = σ = eff ζ=.e ζ=4.894e ζ=4.767e+22 n= = σ =2 eff ζ=.e ζ=9.36e ζ=5.533e+8 n= = σ =2 eff ζ=.e+ ζ=.e+2 ζ=.244e+23 FDR WFDR th=2 WFDR w=s 2 BHP WBHP th=2 WBHP w=s 2 Figure 2: Siulation results for n = : Power estiates for different values of π (abscissa axis), and σδ 2 (along the rows), λ (along the coluns). FDR = FDRcontrolling procedure, WFDR = Weighted FDR-controlling procedure, BHP = Bonferroni-Hol FWE-controlling procedure, WBHP = Benjaini-Hochberg weighted FWE-controlling procedure. Moreover th=2 eans threshold γ = 2 (see definition (8)), w = S 2 eans weight w equal to the total variance (see definition (7) with γ = ). 8