# Biometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika.

Size: px
Start display at page:

## Transcription

2 Biometrika (1988), 75, 2, pp Printed in Great Britain A stagewise rejective multiple test procedure based on a modified Bonferroni test BY G. HOMMEL Institutfiir Medizinische Statistik und Dokumentation, University of Mainz, D-6500 Mainz, Federal Republic of Germany SUMMARY Simes (1986) has proposed a modified Bonferroni procedure for the test of an overall hypothesis which is the combination of n individual hypotheses. In contrast to the classical Bonferroni procedure, it is not obvious how statements about individual hypotheses are to be made for this procedure. In the present paper a multiple test procedure allowing statements on individual hypotheses is proposed. It is based on the principle of closed test procedures (Marcus, Peritz & Gabriel, 1976) and controls the multiple level a. Some key words: Closed test procedure; Control of multiple level; Modified Bonferroni procedure; Multiple test procedure; Quasi-coherence. 1. INTRODUCTION When n hypotheses HI,..., Hn with associated test statistics T1,..., Tn are to be tested, one can make use of the corresponding p-values PI,-.., Pn. A first step to aim for an overall statement can be based on a test of the overall hypothesis Ho= ri{hi: i = 1,..., n}. Application of the Bonferroni inequality leads to a very simple level a test of Ho: reject Ho, if P(,) - a/ n, where P(,) is the smallest one of the p-values. A disadvantage of this procedure is that it may be very conservative, in particular, if the test statistics are highly correlated; moreover, it is often inappropriate to use only the smallest p-value. Another level a test which might avoid this disadvantage is based on Riiger's (1978) inequality: reject Ho, if P(k) - ka/n, where P(k) is the kth smallest of the p-values; here k (2< k s n) has to be determined before performing the n tests. If one wishes to avoid the problem of choosing k in advance, one can combine the Bonferroni test and all (n - 1) possible Riiger tests and obtain the following level a test of Ho (Hommel, 1983): reject Ho, if P(k) < ka/(ncn) for at least one k (1 - k - n), where Cn = /n. A very similar test of Ho which is less conservative because of omitting the constant Cn has been proposed by Simes (1986): reject Ho, if P(k)jka/n for at least one k (1sk n). Since the inequalities of Bonferroni, Riiger and Hommel are all strict, there will be constellations of dependencies among the test statistics where the test of Ho has exactly the level a; it seems, however, that these situations are rather pathological. In practical applications, the corresponding tests of Ho can be expected to be conservative. As Simes pointed out, his procedure does not always lead to a level a test of Ho; nevertheless, he suggested by a simulation study that the level of his procedure is less than or equal to a for a large family of multivariate distributions of (T1,..., T,,), and he proved that the level is exactly equal to a if the test statistics are independent. Therefore, in such cases application of Simes's procedure is recommended since it is strictly more powerful than each of the other three procedures. When, by any of these procedures, Ho has been rejected, the question remains which of the individual hypotheses Hi (i = 1,..., n) should be rejected. An answer is easy for the Bonferroni procedure, where one can reject all Hi with Pi S a/n. For the other procedures, however, it is not quite clear which of the Hi should be rejected. Simes has proposed for his procedure to reject

3 384 G. HOMMEL in an exploratory sense the individual hypotheses H(l),..., H(j), wherej = max {k: P(k) - ka/ n}, H(i) being the hypotheses corresponding to P(i) for i = 1,...,j. However, this procedure is not always satisfactory. Suppose that the test statistics Ti (i= 1,..., n) are independent, that m individual hypotheses are true, and that the other (n - m) hypotheses Hi are false to such an extent that pr (Pi - a/n) is nearly equal to 1. Then the probability of rejecting at least one of the m true Hi is nearly equal to 1-{1-(n-m+1)a/n}I'. If, for example, a=005, n=100 and m =50, then the probability of committing a type I error is 0 725, and it tends to 1 for m =2n and n -oo. In the following, multiple test procedures are proposed which are based on the described tests of the overall hypothesis and keep the probability of committing a type I error less than or equal to a. 2. CLOSED TEST PROCEDURES We apply the following modification of the principle of 'closed test procedures' (Marcus et al., 1976; Sonnemann, 1982). THEOREM (Hommel, 1986). Let there be given, for n > 1, n individual hypotheses H1,..., Hn, and define HI = n{hi: i E I} for all I c K, where K is the set of all nonempty subsets of {1,..., n}. Assume that there exists for each I c K a level a test based on a test statistic T,. Reject HI if it is rejected by T, and if all HJ with J D I, J c K, are rejected by TJ, too. Then this multiple test procedure controls the multiple level a; that is the probability of committing any type I error when testing all HI, I E K, is at most a irrespective of which of the HI are true. Since every HI is the intersection of the individual hypotheses Hi, i c I, it can be interpreted as a possibly 'small' overall hypothesis. Hence a level a test of HI which is only based on the p-values Pi, i E I, can be found as described in? 1. If one chooses Bonferroni tests for testing HI with the decision rule 'reject HI if there is at least one Pi, i c I, with Pi - a/lil', then the application of the Theorem leads directly to Holm's (1979) sequentially rejective procedure which is an improvement of the classical multiple Bonferroni procedure. Test strategies arising when the theorem is applied to overall tests based on the inequalities of Riiger or of Hommel, are described by Hommel (1986). When all overall tests are tests as proposed by Simes, the arising multiple test procedure can be presented by the flow chart of Hommel (1986, Fig. 3) with the choice 8kj = kalj. The decisions for the individual hypotheses can be performed in the following simpler way: compute j = max {i c{1,..., n}: P(n-i+k)> ka/ i for k = 1,..., i}. If the maximum does not exist, reject all Hi (i = 1,..., n), otherwise reject all Hi with Pi - acxj. It follows that this procedure controls the multiple level a provided each of Simes's tests for HI is a level a test. In particular, the multiple level a is kept if the n tests are independent. 3. USE OF LOGICAL RELATIONS AMONG THE HYPOTHESES Shaffer (1986) gives the following improvement of Holm's (1979) general procedure. Let, for a given system of hypotheses HI,..., Hn, S be the set of all j c {1,..., n} such that it can occur that exactly j of the n hypotheses are true and the remaining (n -j) are false. Define ti = max {j c S: j - n - i + 1} for i = 1,..., n. Then the stagewise rejective procedure using the stepwise significance bounds a/ti instead of Holm's bounds a/(n - i+ 1) controls the multiple level a. As an example, consider all n = 10 pairwise comparisons of 5 distributions. Then S= {1, 2,3,4,6, 1O} (Shaffer, 1986, Table 2), and ti= 10, t2= t3= t4= t5=6, t6= t7=4, t8=3, tg=2, t1o= 1. An analogous improvement can be found for the multiple Simes procedure provided each of Simes's tests for HI is a level a test. For this improved procedure, one has to compute j= max {i E S: P(n-i?k) > ka! i for k = 1,..., i}; then the decisions for the individual hypotheses can be taken as described for the general procedure.

4 Miscellanea NUMERICAL EXAMPLE Assume that in a study with n =10 statistical tests the p-values are ordered as following: PI = , P2= 00074, P3= , P4= 0*0106, P5s 00121, P6=0 0218, P7= , P8= , P9 = , Plo = Let a = 0-05 be chosen as the multiple level. In this example, because of P1 - a/10 and P2> a/9, Holm's procedure rejects HI as the only individual hypothesis. If the general multiple Simes procedure is applied, one obtains j = 5, and therefore all Hi with Pi - a/5; that is H1, H2, H3 are rejected. If it is known that the 10 tests are all pairwise comparison tests of 5 distributions, because of 5 E S one obtains j = 4. Therefore all Hi with Pi - a/4, that is H,..., H5, are rejected by the improved procedure. In order to ensure that the multiple level a for this procedure is kept, it is sufficient that Simes's test for each HI is a level a test. A simulation study was performed for the case that each pairwise comparison test is based on a test statistic Ti= XI -XmI/2, where XI, Xm (1, 1< m <5) are independently N(0, 1)-distributed, which is fulfilled asymptotically for many types of pairwise comparisons. Simulations were carried out at an IBM/AT personal computer using the SAS function RANNOR for generating random normal variables. From the results in Table 1, the level a = 0 05 is exceeded for no type of hypothesis, and the type I error rates of Simes's tests are slightly higher than those of the Bonferroni tests. Table 1. Type I error rates* for Simes's, S, and Bonferroni, B, overall testsfor all types of intersection hypotheses HI, for 10 pairwise comparisons of 5 distribution parameters il, /2, /13, /14, I15; III = number of individual hypotheses implied by HI 'Typical' hypothesis III S B 'Typical' hypothesis III S B 1 = 2= 3 = 4= L Hk = t2= /11 = /12 = /13 = = 2 and 3= t t 1= 2= 3 and 4= l= t 0 05t * Based on simulations each; estimated standard error t Exact error rates. 5. DiscussION The proposed multiple test procedure is strictly not less powerful than Holm's procedure as well as all other procedures mentioned by Hommel (1986); in many cases, it seems to be considerably more powerful. The computations needed for testing the individual hypotheses are very simple. If decisions for all HI, I E K, are to be taken, it is recommended to use a computer program based on Hommel (1986, Fig. 3). This can be performed also for a large n, since the computational time is proportional to n2. An important logical property of multiple test procedures is coherence (Gabriel, 1969); i.e. if a hypothesis is retained, all its implications also have to be retained. As Hommel (1986) pointed out, general multiple test procedures, as the Bonferroni or Holm's procedure, need not be coherent, but they should be quasi-coherent; i.e. if HI = r{hi: i c I} is retained, all HJ with J c I are retained. Since the theorem is applied, the proposed procedure is quasi-coherent. In? 3 it is shown how logical dependencies in a given system of hypotheses can lead to an improvement of the procedure. Another question is how one can make use of stochastical dependencies between the test statistics. A solution of this problem seems to be more difficult; on the other hand, the overall tests according to Simes are much more flexible against different structures of stochastical dependence than, for example, Bonferroni overall tests. ACKNOWLEDGEMENT The author would like to thank Peter Bauer for helpful comments and suggestions.

5 386 G. HOMMEL REFERENCES GABRIEL, K. R. (1969). Simultaneous test procedures-some theory of multiple comparisons. Ann. Math. Statist. 40, HOLM, S. (1979). A simple sequentially rejective multiple test procedure. Scand. J. Statist. 6, HOMMEL, G. (1983). Tests of the overall hypothesis for arbitrary dependence structures. Biom. J. 25, HOMMEL, G. (1986). Multiple test procedures for arbitrary dependence structures. Metrika 33, MARCUS, R., PERITZ, E. & GABRIEL, K. R. (1976). On closed testing procedures with special reference to ordered analysis of variance. Biometrika 63, RUGER, B. (1978). Das maximale Signifikanzniveau des Tests "Lehne Ho ab, wenn k unter n gegebenen Tests zur Ablehnung fuihren". Metrika 25, SHAFFER, J. P. (1986). Modified sequentially rejective multiple test procedures. J. Am. Statist. Assoc. 81, SIMES, R. J. (1986). An improved Bonferroni procedure for multiple tests of significance. Biometrika 73, SONNEMANN, E. (1982). Allgemeine Losungen multipler Testprobleme. EDV in Med. u. Biol. 13, [Received July Revised October 1987]

### Biometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika.

Biometrika Trust An Improved Bonferroni Procedure for Multiple Tests of Significance Author(s): R. J. Simes Source: Biometrika, Vol. 73, No. 3 (Dec., 1986), pp. 751-754 Published by: Biometrika Trust Stable

### ON STEPWISE CONTROL OF THE GENERALIZED FAMILYWISE ERROR RATE. By Wenge Guo and M. Bhaskara Rao

ON STEPWISE CONTROL OF THE GENERALIZED FAMILYWISE ERROR RATE By Wenge Guo and M. Bhaskara Rao National Institute of Environmental Health Sciences and University of Cincinnati A classical approach for dealing

### On adaptive procedures controlling the familywise error rate

, pp. 3 On adaptive procedures controlling the familywise error rate By SANAT K. SARKAR Temple University, Philadelphia, PA 922, USA sanat@temple.edu Summary This paper considers the problem of developing

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

On the Bound for a Pair of Consecutive Quartic Residues of a Prime Author(s): R. G. Bierstedt and W. H. Mills Source: Proceedings of the American Mathematical Society, Vol. 14, No. 4 (Aug., 1963), pp.

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Selecting the Better Bernoulli Treatment Using a Matched Samples Design Author(s): Ajit C. Tamhane Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 42, No. 1 (1980), pp.

### Biometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika.

Biometrika Trust Discrete Sequential Boundaries for Clinical Trials Author(s): K. K. Gordon Lan and David L. DeMets Reviewed work(s): Source: Biometrika, Vol. 70, No. 3 (Dec., 1983), pp. 659-663 Published

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

On Runs of Residues Author(s): D. H. Lehmer and Emma Lehmer Source: Proceedings of the American Mathematical Society, Vol. 13, No. 1 (Feb., 1962), pp. 102-106 Published by: American Mathematical Society

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Modalities in Ackermann's "Rigorous Implication" Author(s): Alan Ross Anderson and Nuel D. Belnap, Jr. Source: The Journal of Symbolic Logic, Vol. 24, No. 2 (Jun., 1959), pp. 107-111 Published by: Association

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

A Generic Property of the Bounded Syzygy Solutions Author(s): Florin N. Diacu Source: Proceedings of the American Mathematical Society, Vol. 116, No. 3 (Nov., 1992), pp. 809-812 Published by: American

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Merging of Opinions with Increasing Information Author(s): David Blackwell and Lester Dubins Source: The Annals of Mathematical Statistics, Vol. 33, No. 3 (Sep., 1962), pp. 882-886 Published by: Institute

### On weighted Hochberg procedures

Biometrika (2008), 95, 2,pp. 279 294 C 2008 Biometrika Trust Printed in Great Britain doi: 10.1093/biomet/asn018 On weighted Hochberg procedures BY AJIT C. TAMHANE Department of Industrial Engineering

### A class of improved hybrid Hochberg Hommel type step-up multiple test procedures

Biometrika (2014), 101,4,pp. 899 911 doi: 10.1093/biomet/asu032 Printed in Great Britain Advance Access publication 24 October 2014 A class of improved hybrid Hochberg Hommel type step-up multiple test

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Uncountably Many Inequivalent Analytic Actions of a Compact Group on Rn Author(s): R. S. Palais and R. W. Richardson, Jr. Source: Proceedings of the American Mathematical Society, Vol. 14, No. 3 (Jun.,

### A Note on a Method for the Analysis of Significances en masse. Paul Seeger. Technometrics, Vol. 10, No. 3. (Aug., 1968), pp

A Note on a Method for the Analysis of Significances en masse Paul Seeger Technometrics, Vol. 10, No. 3. (Aug., 1968), pp. 586-593. Stable URL: http://links.jstor.org/sici?sici=0040-1706%28196808%2910%3a3%3c586%3aanoamf%3e2.0.co%3b2-j

### Applying the Benjamini Hochberg procedure to a set of generalized p-values

U.U.D.M. Report 20:22 Applying the Benjamini Hochberg procedure to a set of generalized p-values Fredrik Jonsson Department of Mathematics Uppsala University Applying the Benjamini Hochberg procedure

### Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Biometrika Trust Robust Regression via Discriminant Analysis Author(s): A. C. Atkinson and D. R. Cox Source: Biometrika, Vol. 64, No. 1 (Apr., 1977), pp. 15-19 Published by: Oxford University Press on

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Fermat's Little Theorem: Proofs That Fermat Might Have Used Author(s): Bob Burn Source: The Mathematical Gazette, Vol. 86, No. 507 (Nov., 2002), pp. 415-422 Published by: The Mathematical Association Stable

### International Biometric Society is collaborating with JSTOR to digitize, preserve and extend access to Biometrics.

400: A Method for Combining Non-Independent, One-Sided Tests of Significance Author(s): Morton B. Brown Reviewed work(s): Source: Biometrics, Vol. 31, No. 4 (Dec., 1975), pp. 987-992 Published by: International

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

The Variance of the Product of Random Variables Author(s): Leo A. Goodman Source: Journal of the American Statistical Association, Vol. 57, No. 297 (Mar., 1962), pp. 54-60 Published by: American Statistical

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

The Indices of Torsion-Free Subgroups of Fuchsian Groups Author(s): R. G. Burns and Donald Solitar Source: Proceedings of the American Mathematical Society, Vol. 89, No. 3 (Nov., 1983), pp. 414-418 Published

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

!"#\$%%&"%&"'()"*+,&%\$-,./',%+"0()%\$)-"&%\$"!\$1,'\$/\$2"#3/+)"4%-/,+5!6'(%\$7589":6;/3"

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

6625 Author(s): Nicholas Strauss, Jeffrey Shallit, Don Zagier Source: The American Mathematical Monthly, Vol. 99, No. 1 (Jan., 1992), pp. 66-69 Published by: Mathematical Association of America Stable

### Mind Association. Oxford University Press and Mind Association are collaborating with JSTOR to digitize, preserve and extend access to Mind.

Mind Association Response to Colyvan Author(s): Joseph Melia Source: Mind, New Series, Vol. 111, No. 441 (Jan., 2002), pp. 75-79 Published by: Oxford University Press on behalf of the Mind Association

### Detection of Influential Observation in Linear Regression. R. Dennis Cook. Technometrics, Vol. 19, No. 1. (Feb., 1977), pp

Detection of Influential Observation in Linear Regression R. Dennis Cook Technometrics, Vol. 19, No. 1. (Feb., 1977), pp. 15-18. Stable URL: http://links.jstor.org/sici?sici=0040-1706%28197702%2919%3a1%3c15%3adoioil%3e2.0.co%3b2-8

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Measuring the Speed and Altitude of an Aircraft Using Similar Triangles Author(s): Hassan Sedaghat Source: SIAM Review, Vol. 33, No. 4 (Dec., 1991), pp. 650-654 Published by: Society for Industrial and

### Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Biometrika Trust Some Simple Approximate Tests for Poisson Variates Author(s): D. R. Cox Source: Biometrika, Vol. 40, No. 3/4 (Dec., 1953), pp. 354-360 Published by: Oxford University Press on behalf of

### A note on tree gatekeeping procedures in clinical trials

STATISTICS IN MEDICINE Statist. Med. 2008; 06:1 6 [Version: 2002/09/18 v1.11] A note on tree gatekeeping procedures in clinical trials Alex Dmitrienko 1, Ajit C. Tamhane 2, Lingyun Liu 2, Brian L. Wiens

### Testing hypotheses in order

Biometrika (2008), 95, 1,pp. 248 252 C 2008 Biometrika Trust Printed in Great Britain doi: 10.1093/biomet/asm085 Advance Access publication 24 January 2008 Testing hypotheses in order BY PAUL R. ROSENBAUM

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

On the Probability of Covering the Circle by Rom Arcs Author(s): F. W. Huffer L. A. Shepp Source: Journal of Applied Probability, Vol. 24, No. 2 (Jun., 1987), pp. 422-429 Published by: Applied Probability

### Journal of Applied Probability, Vol. 13, No. 3. (Sep., 1976), pp

Buffon's Problem with a Long Needle Persi Diaconis Journal of Applied Probability, Vol. 13, No. 3. (Sep., 1976), pp. 614-618. Stable URL: http://links.jstor.org/sici?sici=0021-9002%28197609%2913%3a3%3c614%3abpwaln%3e2.0.co%3b2-p

### An Application of the Closed Testing Principle to Enhance One-Sided Confidence Regions for a Multivariate Location Parameter

An Application of the Closed Testing Principle to Enhance One-Sided Confidence Regions for a Multivariate Location Parameter Universität Bern Institut für mathematische Statistik und Versicherungslehre

### The American Mathematical Monthly, Vol. 100, No. 8. (Oct., 1993), pp

A Visual Explanation of Jensen's Inequality Tristan Needham The American Mathematical Monthly, Vol. 100, No. 8. (Oct., 1993), pp. 768-771. Stable URL: http://links.jstor.org/sici?sici=0002-9890%28199310%29100%3a8%3c768%3aaveoji%3e2.0.co%3b2-8

### The American Mathematical Monthly, Vol. 104, No. 8. (Oct., 1997), pp

Newman's Short Proof of the Prime Number Theorem D. Zagier The American Mathematical Monthly, Vol. 14, No. 8. (Oct., 1997), pp. 75-78. Stable URL: http://links.jstor.org/sici?sici=2-989%2819971%2914%3a8%3c75%3anspotp%3e2..co%3b2-c

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

There are no \$Q\$-Points in Laver's Model for the Borel Conjecture Author(s): Arnold W. Miller Source: Proceedings of the American Mathematical Society, Vol. 78, No. 1 (Jan., 1980), pp. 103-106 Published

### Hochberg Multiple Test Procedure Under Negative Dependence

Hochberg Multiple Test Procedure Under Negative Dependence Ajit C. Tamhane Northwestern University Joint work with Jiangtao Gou (Northwestern University) IMPACT Symposium, Cary (NC), November 20, 2014

### Journal of the American Mathematical Society, Vol. 2, No. 2. (Apr., 1989), pp

An Estimate for Character Sums Nicholas M. Katz Journal of the American Mathematical Society, Vol. 2, No. 2. (Apr., 1989), pp. 197-200. Stable URL: http://links.jstor.org/sici?sici=0894-0347%28198904%292%3a2%3c197%3aaefcs%3e2.0.co%3b2-l

### Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Regression Analysis when there is Prior Information about Supplementary Variables Author(s): D. R. Cox Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 22, No. 1 (1960),

### A Mixture Gatekeeping Procedure Based on the Hommel Test for Clinical Trial Applications

A Mixture Gatekeeping Procedure Based on the Hommel Test for Clinical Trial Applications Thomas Brechenmacher (Dainippon Sumitomo Pharma Co., Ltd.) Jane Xu (Sunovion Pharmaceuticals Inc.) Alex Dmitrienko

### Adaptive, graph based multiple testing procedures and a uniform improvement of Bonferroni type tests.

1/35 Adaptive, graph based multiple testing procedures and a uniform improvement of Bonferroni type tests. Martin Posch Center for Medical Statistics, Informatics and Intelligent Systems Medical University

### Variance of Lipschitz Functions and an Isoperimetric Problem for a Class of Product Measures

Variance of Lipschitz Functions and an Isoperimetric Problem for a Class of Product Measures Sergei G. Bobkov; Christian Houdré Bernoulli, Vol. 2, No. 3. (Sep., 1996), pp. 249-255. Stable URL: http://links.jstor.org/sici?sici=1350-7265%28199609%292%3a3%3c249%3avolfaa%3e2.0.co%3b2-i

### MULTISTAGE AND MIXTURE PARALLEL GATEKEEPING PROCEDURES IN CLINICAL TRIALS

Journal of Biopharmaceutical Statistics, 21: 726 747, 2011 Copyright Taylor & Francis Group, LLC ISSN: 1054-3406 print/1520-5711 online DOI: 10.1080/10543406.2011.551333 MULTISTAGE AND MIXTURE PARALLEL

### On the Asymptotic Power of Tests for Independence in Contingency Tables from Stratified Samples

On the Asymptotic Power of Tests for Independence in Contingency Tables from Stratified Samples Gad Nathan Journal of the American Statistical Association, Vol. 67, No. 340. (Dec., 1972), pp. 917-920.

### Proceedings of the American Mathematical Society, Vol. 17, No. 5. (Oct., 1966), pp

On the Existence and Uniqueness of the Real Logarithm of a Matrix Walter J. Culver Proceedings of the American Mathematical Society, Vol. 17, No. 5. (Oct., 1966), pp. 1146-1151. Stable URL: http://links.jstor.org/sici?sici=0002-9939%28196610%2917%3a5%3c1146%3aoteauo%3e2.0.co%3b2-b

### Mathematics of Computation, Vol. 17, No. 83. (Jul., 1963), pp

Abscissas and Weight Coefficients for Lobatto Quadrature H. H. Michels Mathematics of Computation, Vol. 17, No. 83. (Jul., 1963), pp. 237-244. Stable URL: http://links.jstor.org/sici?sici=0025-5718%28196307%2917%3a83%3c237%3aaawcfl%3e2.0.co%3b2-7

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

A Model for Teaching Mitosis & Meiosis Author(s): James E. Mickle Source: The American Biology Teacher, Vol. 52, No. 8 (Nov. - Dec., 1990), pp. 500-503 Published by: National Association of Biology Teachers

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

On the Hahn-Banach Extension Property Author(s): Jonathan M. Borwein Source: Proceedings of the American Mathematical Society, Vol. 86, No. 1, (Sep., 1982), pp. 42-46 Published by: American Mathematical

### Mixtures of multiple testing procedures for gatekeeping applications in clinical trials

Research Article Received 29 January 2010, Accepted 26 May 2010 Published online 18 April 2011 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/sim.4008 Mixtures of multiple testing procedures

### The College Mathematics Journal, Vol. 16, No. 2. (Mar., 1985), pp

On Rearrangements of the Alternating Harmonic Series Fon Brown; L. O. Cannon; Joe Elich; David G. Wright The College Mathematics Journal, Vol. 16, No. 2. (Mar., 1985), pp. 135-138. Stable URL: http://links.jstor.org/sici?sici=0746-8342%28198503%2916%3a2%3c135%3aorotah%3e2.0.co%3b2-q

### The College Mathematics Journal, Vol. 24, No. 4. (Sep., 1993), pp

Taylor Polynomial Approximations in Polar Coordinates Sheldon P. Gordon The College Mathematics Journal, Vol. 24, No. 4. (Sep., 1993), pp. 325-330. Stable URL: http://links.jstor.org/sici?sici=0746-8342%28199309%2924%3a4%3c325%3atpaipc%3e2.0.co%3b2-m

### Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Biometrika Trust Some Remarks on Overdispersion Author(s): D. R. Cox Source: Biometrika, Vol. 70, No. 1 (Apr., 1983), pp. 269-274 Published by: Oxford University Press on behalf of Biometrika Trust Stable

### Control of Directional Errors in Fixed Sequence Multiple Testing

Control of Directional Errors in Fixed Sequence Multiple Testing Anjana Grandhi Department of Mathematical Sciences New Jersey Institute of Technology Newark, NJ 07102-1982 Wenge Guo Department of Mathematical

### Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

The Interpretation of Interaction in Contingency Tables Author(s): E. H. Simpson Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 13, No. 2 (1951), pp. 238-241 Published

### An Improved Approximate Formula for Calculating Sample Sizes for Comparing Two Binomial Distributions

An Improved Approximate Formula for Calculating Sample Sizes for Comparing Two Binomial Distributions J. T. Casagrande; M. C. Pike; P. G. Smith Biometrics, Vol. 34, No. 3. (Sep., 1978), pp. 483-486. Stable

### Lecture 6 April

Stats 300C: Theory of Statistics Spring 2017 Lecture 6 April 14 2017 Prof. Emmanuel Candes Scribe: S. Wager, E. Candes 1 Outline Agenda: From global testing to multiple testing 1. Testing the global null

### Covering Lemmas and an Application to Nodal Geometry on Riemannian Manifolds

Covering Lemmas and an Application to Nodal Geometry on Riemannian Manifolds Guozhen Lu Proceedings of the American Mathematical Society, Vol. 117, No. 4. (Apr., 1993), pp. 971-978. Stable URL: http://links.jstor.org/sici?sici=0002-9939%28199304%29117%3a4%3c971%3aclaaat%3e2.0.co%3b2-t

### Consonance and the Closure Method in Multiple Testing. Institute for Empirical Research in Economics University of Zurich

Institute for Empirical Research in Economics University of Zurich Working Paper Series ISSN 1424-0459 Working Paper No. 446 Consonance and the Closure Method in Multiple Testing Joseph P. Romano, Azeem

### The College Mathematics Journal, Vol. 21, No. 4. (Sep., 1990), pp

Tabular Integration by Parts David Horowitz The College Mathematics Journal, Vol. 21, No. 4. (Sep., 1990), pp. 307-311. Stable URL: http://links.jstor.org/sici?sici=0746-8342%28199009%2921%3a4%3c307%3atibp%3e2.0.co%3b2-o

### GOTEBORG UNIVERSITY. Department of Statistics

GOTEBORG UNIVERSITY Department of Statistics RESEARCH REPORT 1994:5 ISSN 0349-8034 COMPARING POWER AND MULTIPLE SIGNIFICANCE LEVEL FOR STEP UP AND STEP DOWN MULTIPLE TEST PROCEDURES FOR CORRELATED ESTIMATES

### Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

American Society for Quality A Note on the Graphical Analysis of Multidimensional Contingency Tables Author(s): D. R. Cox and Elizabeth Lauh Source: Technometrics, Vol. 9, No. 3 (Aug., 1967), pp. 481-488

### 11] Index Number Which Shall Meet Certain of Fisher's Tests 397

Necessary and Sufficient Conditions Regarding the Form of an Index Number which Shall Meet Certain of Fisher's Tests Author(s): Ragnar Frisch Reviewed work(s): Source: Journal of the American Statistical

### ON TWO RESULTS IN MULTIPLE TESTING

ON TWO RESULTS IN MULTIPLE TESTING By Sanat K. Sarkar 1, Pranab K. Sen and Helmut Finner Temple University, University of North Carolina at Chapel Hill and University of Duesseldorf Two known results in

### On Generalized Fixed Sequence Procedures for Controlling the FWER

Research Article Received XXXX (www.interscience.wiley.com) DOI: 10.1002/sim.0000 On Generalized Fixed Sequence Procedures for Controlling the FWER Zhiying Qiu, a Wenge Guo b and Gavin Lynch c Testing

### Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Some Applications of Exponential Ordered Scores Author(s): D. R. Cox Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 26, No. 1 (1964), pp. 103-110 Published by: Wiley

### E. DROR, W. G. DWYER AND D. M. KAN

Self Homotopy Equivalences of Postnikov Conjugates Author(s): E. Dror, W. G. Dwyer, D. M. Kan Reviewed work(s): Source: Proceedings of the American Mathematical Society, Vol. 74, No. 1 (Apr., 1979), pp.

### Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

A Note on the Efficiency of Least-Squares Estimates Author(s): D. R. Cox and D. V. Hinkley Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 30, No. 2 (1968), pp. 284-289

### IMPROVING TWO RESULTS IN MULTIPLE TESTING

IMPROVING TWO RESULTS IN MULTIPLE TESTING By Sanat K. Sarkar 1, Pranab K. Sen and Helmut Finner Temple University, University of North Carolina at Chapel Hill and University of Duesseldorf October 11,

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

The Relationship Between the Zeros of Best Approximations and Differentiability Author(s): Peter B. Borwein Source: Proceedings of the American Mathematical Society, Vol. 92, No. 4, (Dec., 1984), pp. 528-532

### The Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica.

A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision Author(s): Kenneth O. May Source: Econometrica, Vol. 20, No. 4 (Oct., 1952), pp. 680-684 Published by: The Econometric

### Annals of Mathematics

Annals of Mathematics Asymptotically Neutral Distributions of Electrons and Polynomial Approximation Author(s): Jacob Korevaar Source: The Annals of Mathematics, Second Series, Vol. 80, No. 3 (Nov., 1964),

### Philosophy of Science, Vol. 43, No. 4. (Dec., 1976), pp

Many Worlds Are Better than None Stanley Kerr Philosophy of Science, Vol. 43, No. 4. (Dec., 1976), pp. 578-582. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28197612%2943%3a4%3c578%3amwabtn%3e2.0.co%3b2-c

### Philosophy of Science Association

Philosophy of Science Association Why Bohm's Theory Solves the Measurement Problem Author(s): Tim Maudlin Source: Philosophy of Science, Vol. 62, No. 3 (Sep., 1995), pp. 479-483 Published by: The University

### Proceedings of the American Mathematical Society, Vol. 88, No. 1. (May, 1983), pp

Approximate Identities and H1(R) Akihito Uchiyama; J. Michael Wilson Proceedings of the American Mathematical Society, Vol. 88, No. 1. (May, 1983), pp. 53-58. Stable URL: http://links.jstor.org/sici?sici=0002-9939%28198305%2988%3a1%3c53%3aaia%3e2.0.co%3b2-4

### arxiv: v4 [math.st] 21 Apr 2018

Combining p-values via averaging arxiv:1212.4966v4 [math.st] 21 Apr 2018 Vladimir Vovk Ruodu Wang April 24, 2018 Abstract This paper proposes general methods for the problem of multiple testing of a single

### Proceedings of the American Mathematical Society, Vol. 57, No. 2. (Jun., 1976), pp

Elliptic Curves and Dedekind Domains Michael Rosen Proceedings of the American Mathematical Society, Vol. 57, No. 2. (Jun., 1976), pp. 197-201. Stable URL: http://links.jstor.org/sici?sici=0002-9939%28197606%2957%3a2%3c197%3aecadd%3e2.0.co%3b2-7

### Multistage Tests of Multiple Hypotheses

Communications in Statistics Theory and Methods, 39: 1597 167, 21 Copyright Taylor & Francis Group, LLC ISSN: 361-926 print/1532-415x online DOI: 1.18/3619282592852 Multistage Tests of Multiple Hypotheses

### Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

On the Estimation of the Intensity Function of a Stationary Point Process Author(s): D. R. Cox Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 27, No. 2 (1965), pp. 332-337

### American Journal of Mathematics, Vol. 109, No. 1. (Feb., 1987), pp

A Simple Algorithm for Cyclic Vectors Nicholas M. Katz American Journal of Mathematics, Vol. 109, No. 1. (Feb., 1987), pp. 65-70. Stable URL: http://links.jstor.org/sici?sici=0002-9327%28198702%29109%3a1%3c65%3aasafcv%3e2.0.co%3b2-a

### Annals of Mathematics

Annals of Mathematics The Multiplier Problem for the Ball Author(s): Charles Fefferman Source: The Annals of Mathematics, Second Series, Vol. 94, No. 2 (Sep., 1971), pp. 330-336 Published by: Annals of

### Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

A Simple Non-Desarguesian Plane Geometry Author(s): Forest Ray Moulton Source: Transactions of the American Mathematical Society, Vol. 3, No. 2 (Apr., 1902), pp. 192-195 Published by: American Mathematical

### The International Journal of Biostatistics

The International Journal of Biostatistics Volume 7, Issue 1 2011 Article 12 Consonance and the Closure Method in Multiple Testing Joseph P. Romano, Stanford University Azeem Shaikh, University of Chicago

### The Review of Economic Studies, Ltd.

The Review of Economic Studies, Ltd. Oxford University Press http://www.jstor.org/stable/2297086. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at.

### Department of Biostatistics University of North Carolina. Institute of Statistics Mimeo Series No Oct 1997

SOME REMARKS ON SIMES TYPE MULTIPLE TESTS OF SIGNIFICANCE by Pranab K. Sen Department of Biostatistics University of North Carolina Institute of Statistics Mimeo Series No. 2177 Oct 1997 Some Remarks on

### Annals of Mathematics

Annals of Mathematics The Clifford-Klein Space Forms of Indefinite Metric Author(s): Joseph A. Wolf Reviewed work(s): Source: The Annals of Mathematics, Second Series, Vol. 75, No. 1 (Jan., 1962), pp.

### Multiple Testing of General Contrasts: Truncated Closure and the Extended Shaffer-Royen Method

Multiple Testing of General Contrasts: Truncated Closure and the Extended Shaffer-Royen Method Peter H. Westfall, Texas Tech University Randall D. Tobias, SAS Institute Pairwise Comparisons ANOVA, g =10groups,

### Compatible simultaneous lower confidence bounds for the Holm procedure and other Bonferroni based closed tests

Compatible simultaneous lower confidence bounds for the Holm procedure and other Bonferroni based closed tests K. Strassburger 1, F. Bretz 2 1 Institute of Biometrics & Epidemiology German Diabetes Center,

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Review: God Save the Double! Author(s): Mauro Lo Monaco and Sergio Vinciguerra Source: Dance Research: The Journal of the Society for Dance Research Vol. 24 No. 1 (Summer 2006) pp. 66-69 Published by:

### High-Throughput Sequencing Course. Introduction. Introduction. Multiple Testing. Biostatistics and Bioinformatics. Summer 2018

High-Throughput Sequencing Course Multiple Testing Biostatistics and Bioinformatics Summer 2018 Introduction You have previously considered the significance of a single gene Introduction You have previously

### Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

A Quincuncial Projection of the Sphere Author(s): C. S. Peirce Source: American Journal of Mathematics, Vol. 2, No. 4 (Dec., 1879), pp. 394-396 Published by: The Johns Hopkins University Press Stable URL:

### American Society for Quality

American Society for Quality The 2k-p Fractional Factorial Designs Part I Author(s): G. E. P. Box and J. S. Hunter Source: Technometrics, Vol. 3, No. 3 (Aug., 1961), pp. 311-351 Published by: American

### Mathematical Association of America

Mathematical Association of America http://www.jstor.org/stable/2975232. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at. http://www.jstor.org/page/info/about/policies/terms.jsp

### Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Queues with Time-Dependent Arrival Rates: II. The Maximum Queue and the Return to Equilibrium Author(s): G. F. Newell Source: Journal of Applied Probability, Vol. 5, No. 3 (Dec., 1968), pp. 579-590 Published

### False discovery control for multiple tests of association under general dependence

False discovery control for multiple tests of association under general dependence Nicolai Meinshausen Seminar für Statistik ETH Zürich December 2, 2004 Abstract We propose a confidence envelope for false

### Mathematics of Operations Research, Vol. 2, No. 2. (May, 1977), pp

New Finite Pivoting Rules for the Simplex Method Robert G. Bland Mathematics of Operations Research, Vol. 2, No. 2. (May, 1977), pp. 103-107. Stable URL: http://links.jstor.org/sici?sici=0364-765x%28197705%292%3a2%3c103%3anfprft%3e2.0.co%3b2-t

### are equal to zero, where, q = p 1. For each gene j, the pairwise null and alternative hypotheses are,

Page of 8 Suppleentary Materials: A ultiple testing procedure for ulti-diensional pairwise coparisons with application to gene expression studies Anjana Grandhi, Wenge Guo, Shyaal D. Peddada S Notations

### The Periodogram and its Optical Analogy.

The Periodogram and Its Optical Analogy Author(s): Arthur Schuster Reviewed work(s): Source: Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character,

### The Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica.

On the Optimal Character of the (s, S) Policy in Inventory Theory Author(s): A. Dvoretzky, J. Kiefer, J. Wolfowitz Reviewed work(s): Source: Econometrica, Vol. 21, No. 4 (Oct., 1953), pp. 586-596 Published

### Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

A Consistency Result on Thin-Tall Superatomic Boolean Algebras Author(s): Juan Carlos Martínez Source: Proceedings of the American Mathematical Society, Vol. 115, No. 2 (Jun., 1992), pp. 473-477 Published

### econstor Make Your Publications Visible.

econstor Make Your Publications Visible. A Service of Wirtschaft Centre zbwleibniz-informationszentrum Economics Hartung, Joachim; Knapp, Guido Working Paper Confidence intervals for the between group