Abstract
It is the purpose of this paper to review the main aspects related to multiple test problems. This concerns among others the particularities of multiple tests as for instance the formulation of restrictions to avoid inconsistent decisions and of criteria to control for a multiple type I error rate. In addition, the basic principles for constructing multiple tests are introduced and their properties are summarized. The paper closes with giving a rough idea of further special multiple test problems and their corresponding test procedures.
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References
Alt, R. (1988) Hierarchical test problems and the closure principle. In: P. Bauer, G. Hommel, E. Sonnermann (eds.):Multiple Hypotheses Testing. Springer-Verlag, Berlin, 163–176
Banik, N., Köhne, K., andBauer, P. (1996) On the power of Fisher's combination test for two stage sampling in the presence of nuisance parameters.Biom. J. 38, 25–37
Bauer, P. (1987) On the assessment of the performance of multiple procedures.Biom. J. 29, 895–906
Bauer, P. (1989) Multistage testing with adaptive designs (with discussion).Biom. und Inf. in Med. und Biol. 20, 130–148.
Bauer, P. (1997) A note on multiple testing procedures in dose finding.Biometrics 53, 1125–1128.
Bauer, P., Hackl, P., Hommel, G., andSonnemann, E. (1986) Multiple testing of pairs of one-sided hypotheses.Metrika 33, 121–127.
Bauer, P., andHackl, P. (1987) Multiple testing in a set of nested hypotheses.Statistics 18, 345–349.
Bauer, P., Pötscher, B.M., andHackl, P. (1988) Model selection by multiple test procedures.Statistics 19, 39–44.
Bauer, P., andKöhne, K. (1994) Evaluation of experiments with adaptive interim analyses.Biometrics 50, 1029–1041.
Bauer, P., andRöhmel, J. (1995) An adaptive method for establishing a dose response relationship.Stat. in Med. 14, 1595–1607.
Begun, J.M., andGabriel, K.R. (1981) Closure of the Newman-Keuls multiple comparison procedure.J. Amer. Statist. Assoc. 76, 241–245.
Benjamini, Y., andHochberg, Y. (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing.J. Roy. Statist. Soc., Ser. B 57, 289–300.
Bofinger, E. (1987) Step-down procedures for comparison with a control.Austral. J. Statist. 29, 348–364
Budde, M., andBauer, P. (1989) Multiple test procedures in clinical dose finding studies.J. Amer. Statist. Assoc. 84, 792–796.
Cheng, P.H., andMeng, C.Y.K. (1995) A new formula for tail probabilities of Dunnett'sT with unequal sample sizes.Commun. Statist., Theory Methods 24, 523–532.
Cox, D.R. (1965) A remark on multiple comparison methods.Technometrics 2 149–156.
Dixon, D.O., andDuncan, D.B. (1975) Minimum Bayes riskt-intervals for multiple comparisons.J. Amer. Statist. Assoc. 70, 822–831.
Duncan, D.B. (1961) Bayes rules for a common multiple comparisons problem and related Student-t problems.Ann. Math. Statist. 32, 1013–1033.
Duncan, D.B. (1965) A Bayesian approach to multiple comparisons.Technometrics 7, 171–222.
Duncan, D.B. (1975)t tests and intervals for comparisons suggested by the data.Biometrics 31, 339–359.
Duncan, D.B., andGodbold, J.H. (1979) Approximatek-ratiot tests for differences between unequally replicated treatments.Biometrics 35, 749–756.
Dunnett, C.W. (1955) A multiple comparison procedure for comparing several treatments with a control.J. Amer. Statist. Assoc. 50, 1096–1121
Dunnett, C.W. (1964) New tables for multiple comparisons with a control.Biometrics 20, 482–491.
Dunnett, C.W., andTamhane, A.C. (1992) A step-up multiple test procedure.J. Amer. Statist. Assoc. 87, 162–170.
Efron, B. (1979) Bootstrap methods: Another look at the jackknife.Ann. Statist. 7, 1–26.
Finner, H. (1988a)Multiple Spannweitentests. Dissertation, University of Trier, Department of Mathematics
Finner, H. (1988b) Abgeschlossene multiple Spannweitentests. In: P. Bauer, G. Hommel, E. Sonnemann (eds.):Multiple Hypotheses Testing. Springer-Verlag, Berlin, 10–32.
Finner, H. (1988c) Multiple Tests und Fehler III. Art. In: P. Bauer, G. Hommel, E. Sonnemann (eds.):Multiple Hypotheses Testing, Springer-Verlag, Berlin, 144–153.
Finner, H. (1990) On the modifiedS-method and directional errors.Commun. Statist., Theory Methods 19, 41–53.
Finner, H., Hayter, A.J., and Roters, M. (1993) On the joint distribution function of order statistics with reference to step-up multiple test procedures.Forschungsbericht Nr. 93-19, Mathematik/Informatik, University of Trier
Fisher, R.A. (1935)The Design of Experiments. Oliver & Boyd, Edingburgh, London
Gabriel, K.R. (1969) Simultaneous test procedures—some theory of multiple comparisons.Ann. Math. Statist. 40, 224–250.
Gabriel, K.R. (1978) A simple method of multiple comparisons of means.J. Amer. Statist. Assoc. 73, 724–729.
Gather, U., Pawlitschko, J., andPigeot, I. (1997) Unbiasedness of multiple tests.Scand. J. Statist. 23, 117–127.
Gather, U., Pawlitschko, J., andPigeot, I. (1997) A note on invariance of multiple tests.Statist. Neerl. 51, 366–372.
Hawkins, D.M. (1980)Identification of Outliers. Chapman & Hall, London.
Hochberg, Y. (1974) Some conservative generalization of theT-method in simultaneous inference.J. Mult. Anal. 4, 224–234.
Hochberg, Y. (1988) A sharper Bonferroni procedure for multiple tests of significance.Biometrika 75, 800–802.
Hochberg, Y., Tamhane, A.C. (1987)Multiple Comparison Procedures. John Wiley & Sons, New York
Holm, S. (1979) A simple sequentially rejective multiple test procedure.Scand. J. Statist. 6, 65–70.
Holm, S. (1985) Multiple test unbiasedness. In: M. Iosifescu, S. Grigorescu, T. Postelnicu (eds.):Proceedings of the Seventh Conference on Probability Theory. Editura Academiei, Bucuresti; VNU Science Press, Utrecht, 183–193.
Hommel, G. (1986) Multiple test procedures for arbitrary dependence structures.Metrika 33, 321–336.
Hommel, G. (1988) A stagewise rejective multiple test procedure based on a modified Bonferroni test.Biometrika 75, 383–386.
Hommel, G. (1989) A comparison of two modified Bonferroni procedures.Biometrika 76, 624–625
Hommel, G., andBernhard, G. (1992) Multiple hypotheses testing. In: J. Antoch (ed.):Computational Aspects of Model Choice. Physica-Verlag, Heidelberg, 211–235
Hommel, G., andBernhard, G. (1994) A multiple test procedure for nested systems of hypotheses. In: P. Dirschedl, R. Ostermann (eds):Computational Statistics. Physica-Verlag, Heidelberg, 419–433.
Hommel, G., andHoffmann, T. (1988) Controlled uncertainty. In: P. Bauer, G. Hommel, E. Sonnemann (eds.):Multiple Hypotheses Testing. Springer-Verlag, Berlin, 154–161
Hoover, D.R. (1991) Simultaneous comparisons of multiple treatments to two (or more) controls.Biom. J. 33, 913–921
Horn, M., Vollandt, R. (1995)Multiple Tests und Auswahlverfahren. Gustav Fischer Verlag, Stuttgart
Hothorn, L.A., Neuhäuser, M., andKoch, H.-F. (1997) Analysis of randomized dose-finding-studies: closure test modifications based on multiple contrast tests.Biom. J. 39, 467–479.
Hsu, J.C. (1984) Ranking and selection and multiple comparisons with the best. In: T.J. Santner, A.C. Tamhane (eds.):Design of Experiments: Ranking and Selection. Marcel Dekker, New York.
Hsu, J.C. (1985) A note on multiple comparisons with the best. In:45th Session of the International Statistical Institute, Book 2, 445–446.
Hsu, J.C. (1996)Multiple Comparisons: Theory and Methods. Chapman & Hall, London
Keuls, M. (1952) The use of the “studentized range” in connection with an analysis of variance.Euphytica 1, 112–122
Lehmacher, W. (1988a) Multiples Testen bei Verlaufskurvenanalysen—T 2—Tests und Folgeanalysen mitt-Tests. In: H.-K. Selbmann, K. Dietz (eds.):Medizinische Informationsverarbeitung und Epidemiologie im Dienste der Gesundheit. Proceedings der 32. Jahrestagung der GMDS, Tübingen, 1987. Springer-Verlag, Heidelberg, 93–96.
Lehmacher, W. (1988b) Analyse vonK Stichproben von Verlaufskurven. In: P. Bauer, G. Hommel, E. Sonnemann (eds.):Multiple Hypotheses Testing. Springer-Verlag, Berlin, 33–47
Hehmacher, W. (1989) Schrittweises Testen a priori geordneter Hypothesen mit Kontrolle des experimentweisen Niveaus.Unpublished manuscript
Lehmacher, W., Wassmer, G., andReitmeir, P. (1991) Procedures for two-sample comparisons with multiple endopoints controlling the experimentwise error rate.Biometrics 47, 511–521
Lehmann, E.L. (1957a) A theory of some multiple decision problems. I..Ann. Math. Statist. 28, 1–25
Lehmann, E.L. (1957b) A theory of some multiple decision problems. II.Ann. Math. Statist. 28, 547–572
Liu, W. (1996) Multiple tests of a non-hierarchical finite family of hypotheses.J. Roy. Statist. Soc., Ser. B 58, 455–461
Marcus, R., Peritz, E., andGabriel, K.R. (1976) On closed testing procedures with special reference to ordered analysis of variance.Biometrika 63, 655–660
Maurer, W., Hothorn, L.A., andLehmacher, W. (1995) Multiple comparisons in drug clinical trials and preclinical assays: a-priori ordered hypotheses. In: J. Vollmar (ed.):Biometrie in der chemisch-pharmazeutischen Industrie. Gustav Fischer Verlag, Stuttgart
Maurer, W., andMellein, B. (1988) On new multiple tests based on independentp-values and the assessment of their power. In: P. Bauer, G. Hommel, E. Sonnemann (eds.):Multiple Hypotheses Testing. Springer-Verlag, Berlin, 48–66
Miller, R.G. (1981)Simultaneous Statistical Inference, 2nd ed. Springer-Verlag, New York
Naik, U.D. (1975) Some selection rules for comparingp processes with a standard.Commun. Statist. Theory and Methods 4, 519–535
Newman, D. (1939) The distribution of the range in samples from a normal population, expressed in terms of an independent estimate of standard deviation.Biometrika 31, 20–30
O'Brien, P.C. (1984) Procedures for comparing samples with multiple endpoints.Biometrics 40 1079–1087
Pawlitschko, J. (1992)Multiples Testen unter besonderer Berücksichtigung des Abschlußprinzips. Diploma Thesis, University of Dortmund, Department of Statistics
Pearce, S.C. (1983) The monstrous regiment of mathematicians.The Statistician 32, 375–378
Pigeot, I. (1993)Multiple Testtheorie in der Ausreißererkennung. Habilitationsschrift, University of Dortmund, Department of Statistics
Pigeot, I., andGather, U. (1994) Identifikation von Ausreißern als multiples Test-problem. In: S.J. Pöppl, H.-G. Lipinski, T. Mansky (eds.):Medizinische Informatik. Ein integrierender Teil arztunterstützender Technologien. Proceedings der 38. Jahrestagung der GMDS, Lübeck, 1993. MMV Medizin Verlag, München, 474–477
Pocock, S.J., Geller, N.L., andTsiatis, A.A. (1987) The analysis of multiple endpoints in clinical trials.Biometrics 43, 487–498
Rom, D.M. (1990) A sequentially rejective test procedure based on a modified Bonferroni inequality.Biometrika 77, 663–665
Rom, D.M., andHolland, B. (1995) A new closed multiple testing procedure for hierarchical families of hypotheses.J. Statist. Planning Inf. 46, 265–275
Rosner, B. (1975) On the detection of many outliers.Technometrics 17, 221–227
Roy, S.N. (1953) On a heuristic method of test construction and its use in multivariate analysis.Ann. Math. Statist. 24, 220–238
Rüger, B. (1978) Das maximale Signifikanzniveau des Tests “LehneH 0 ab, wennk untern Tests zur Ablehnung führen”.Metrika 25, 171–178
Savin, N.E. (1984) Multiple hypothesis testing. In: Z. Griliches, M.D. Intriligator (eds.):Handbook of Econometrics, Vol. 2. Elsevier Science Publishers B.V., Amsterdam, 827–879
Scheffé, H. (1953) A method for judging all contrasts in the analysis of variance.Biometrika 40, 87–104.
Shaffer, J.P. (1980) Control of directional errors with stagewise multiple test procedures.Ann. Stat. 8, 1342–1348
Shaffer, J.P. (1986) Modified sequentially rejective multiple test procedures.J. Amer. Statist. Assoc. 81, 826–831
Simes, R.J. (1986) An improved Bonferroni procedure for multiple tests of significance.Biometrika 73, 751–754
Sonnemann, E. (1981) Tests zum multiplen Niveau α.ROeS-Seminar, Bad Ischl
Sonnemann, E. (1982) Allgemeine Lösungen multipler Testprobleme.EDV in Med. Biol. 13, 120–128
Spjøtvoll, E. (1972) On the optimality of some multiple comparison procedures.Ann. Math. Statist. 43, 398–411
Spjøtvoll, E. (1974) Multiple testing in the analysis of variance.Scand. J. Statist. 1, 97–114
Spjøtvoll, E., andStoline, M.R. (1973) An extension of theT-method of multiple comparisons to include the cases with unequal sample sizes.J. Amer. Statist. Assoc. 68, 975–978
Stefánsson, G., Kim, W.-C., andHsu, J.C. (1988) On confidence sets in multiple comparisons. In: S.S. Gupta, J.O. Berger (eds.):Statistical Decision Theory and Related Topics IV, Vol. 2. Springer, New York, 89–104
Tamhane, A.C., Hochberg, Y., andDunnett, C.W. (1996) Multiple test procedures for dose findings.Biometrics 52, 21–37
Tippett, L.H.G. (1952)The Methods of Statistics.4th ed. Wiley, New York
Troendle, J.F. (1995) A stepwise resampling method of multiple hypothesis testing.J. Amer. Statist. Assoc. 90, 370–378
Tukey, J.W. (1953) The problem of multiple comparisons.Unpublished manuscript
Victor, N. (1982) Exploratory data analysis and clinical research.Meth. Inf. Med. 21, 53–54
Waller, R.A., andDuncan, D.B. (1969) A Bayes rule for the symmetric multiple comparisons problem.J. Amer. Statist. Assoc. 64, 1484–1503
Waller, R.A., andDuncan, D.B. (1972) A corrigendum to “A Bayes rule for the symmetric multiple comparisons problem”.J. Amer. Statist. Assoc.,67, 253–255
Waller, R.A., andDuncan, D.B. (1974) A Bayes rule for the symmetric multiple comparisons problem II.J. Ann. Inst. Statist. Math. 26, 247–264
Wassmer, G. (1997) A technical note on the power determination for Fisher's combination test.Biom. J. 39, 831–838
Wassmer, G. (1998) A comparison of two methods for adaptive interim analyses in clinical trials.Biometrics 54, 696–705
Wassmer, G., Reitmeir, P., Kieser, M., and Lehmacher, W. (1999) Procedures for testing multiple endpoints in clinical trials: an overview. To appear inJ. Statist. Planning Inf.
Welsch, R.E. (1977) Stepwise multiple comparison procedures.J. Amer. Statist. Assoc. 72, 566–575
Westfall, P.H. (1985) Simultaneous small-sample multivariate Bernoulli confidence intervals.Biometrics 41, 1001–1013
Westfall, P.H., andYoung, S.S. (1989)P value adjustments for multiple tests in multivariate binomial models.J. Amer. Statist. Assoc. 84, 780–786
Westfall, P.H., andYoung, S.S. (1993)Resampling-Based Multiple Testing. John Wiley & Sons, New York
Zieliński, W. (1992) Monte Carlo comparison of multiple comparison procedures.Biom. J. 34, 291–296
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Pigeot, I. Basic concepts of multiple tests — A survey. Statistical Papers 41, 3–36 (2000). https://doi.org/10.1007/BF02925674
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DOI: https://doi.org/10.1007/BF02925674