Name: Chen

Firstname: Jie

Title: Biometrician

Institution: Merck & Co., Inc.

Street: 770 Sumneytown Pike, WP37C-305

City: West Point

Zip-Code: 19486

Country: USA

Phone: (215)6529285

Fax: (215)652-3355


Authors: Jie Chen and Sanat K. Sarkar Merck Research Laboratories and Temple University

Title: A Bayesian Approach to Stepwise Simultaneous Testing

Abstract: We propose a Bayesian procedure for simultaneous testing of \textit{k} null hypotheses. This procedure starts with testing the intersection of all the null hypotheses by comparing the corresponding Bayes factor to a constant. If this intersection hypothesis is accepted the procedure stops by declaring all the hypotheses to be true; otherwise, it declares the hypothesis with the smallest posterior probability to be false and continues to the next step where the intersection of the alternatives to the rejected hypotheses with the remaining null hypotheses is tested using its Bayes factor. The procedure continues until an acceptance occurs or all the hypotheses are rejected. The hierarchical model structure is considered in general case. The stepwise Bayes factors are derived in a variety of situations, involving point null hypotheses as well as those arising in one-sided testing problems. Finally, the procedure is illustrated for testing the equality of normal means with known and unknown equal variance and the equality of normal variances as well as binomial proportions. The advantages and disadvantages of the procedure are discussed.\\ \\ Key Words: Hierarchical priors; Posterior probabilities of hypotheses; Bayes factor; Point null hypothesis; One-sided hypothesis.

References: Bayarri, M. J., and Berger, J. O. (2000), ¡§P Values for Composite Null Models,¡¨ Journal of the American Statistical Association, 95, 1127-1142. Berger, J. O. (1985), Statistical Decision Theory and Bayesian Analysis, (2nd ed.), Springer¡VVerlag, New York. Berger, J. O., Boukai, B., and Wang, Y. (1997), ¡§Unified Frequentist and Bayesian Test-ing of a Precise Hypotheses,¡¨ Statistical Science, 12, 133-160. Berger, J. O., Brown, L. D., and Wolper, R. L. (1994), ¡§A Unified Conditional Frequen-tist and Bayesian Test for Fixed and Sequential Simple Hypothesis Testing,¡¨ The Annals of Statistics, 22, 1787-1807. Berger, J. O., and Deely, J. (1988), ¡§A Bayesian Approach to Ranking and Selection of Related Means with Alternatives to Analysis-of-Variance Methodology,¡¨ Journal of the American Statistical Association, 83, 364-373. Berger, J. O., and Delampady, M. (1987), ¡§Testing Precise Hypothesis,¡¨ Statistical Sci-ence, 2, 317-352. Berger, J. O. and Selke, T. (1987), ¡§Testing a Point Null Hypothesis: the Irreconcilability of p Values and Evidence,¡¨ Journal of the American Statistical Association, 82, 112-122. Berger, J. O. and Pericchi, L. R. (2001), ¡§Objective Bayesian Methods for Model Selec-tion: Introduction and Comparison,¡¨ in Model Selection, eds. P. Lahiri, Institute of Mathematical Statistics Lecture Notes, Monograph Series volume 38. Berry, D. A., and Hochberg, Y. (1999), ¡§Bayesian Perspectives on Multiple Compari-sons,¡¨ Journal of Statistical Planning and Inference, 82, 215-227. Box, G. E. P., and Tiao, G. C. (1973), Bayesian Inference in Statistical Analysis, John Wiley and Sons, Inc., New York. Breslow, N. (1990), ¡§Biostatistics and Bayes,¡¨ Statistical Science, 5, 269-298. Dominici, F. (1998), ¡§Testing Simultaneous Hypotheses in Pharmaceutical Trials: A Bayesian Approach,¡¨ Journal of Biopharmaceutical Statistics, 8, 283-297. Duncan, D. B. (1965), ¡§A Bayesian Approach to Multiple Comparisons,¡¨ Technometrics, 7, 171-222. Dunnett, C. W. and Tamhane, A. C. (1991), ¡§Step-down Multiple Tests for Comparing Treatments with a Control in Unbalanced One-way Layouts,¡¨ Statistics in Medicine, 11, 1057-1063. Dunnett, C. W. and Tamhane, A. C. (1992), ¡§A Step-up Multiple Test Procedure,¡¨ Jour-nal of the American Statistical Association, 87, 162-170. Goodman, S. N. (1999), ¡§Toward Evidence-Based Medical Statistics. 1: The P Value Fallacy, 2: The Bayes Factor,¡¨ Annals Internal Medicine, 130, 995-1013. Gopalan, R. and Berry, D. A. (1998), ¡§Bayesian Multiple Comparisons Using Dirichlet Process Priors,¡¨ Journal of the American Statistical Association, 93, 1130-1139. Hochberg, Y., and Tamhane, A. C. (1987), Multiple Comparison Procedures, John Wiley and Sons, Inc., New York. Hochberg, Y., and Westfall, P. H. (2000), ¡§On Some Multiplicity Problems and Multiple Comparison Procedures in Biostatistics,¡¨ in Handbook of Statistics, Vol. 18, eds. P. K. Sen and C. R. Rao, pp. 75-113. Hsu, J. C. (1996), Multiple Comparisons: Theory and Methods, Chapman & Hall /CRC, Washington, D. C. Kass, R. E., and Raftery, A. E. (1995), ¡§Bayes Factors,¡¨ Journal of the American Statisti-cal Association, 90, 773-795. Marden, J. I. (2000), ¡§Hypothesis Testing: From p Values to Bayes Factors,¡¨ Journal of the American Statistical Association, 95, 1316-1320. Matthews, R. A. J. (2001), ¡§Why Should Clinicians Care About Bayesian Methods?¡¨ Journal of Statistical Planning and Inference, 94, 43-58. Santis, F. D., and Spezzaferri, F. (1998), ¡§Bayes Factors and Hierarchical Models,¡¨ Jour-nal of Statistical Planning and Inference, 74, 323-342. Schervish, M. J. (1995), Theory of Statistics, Springer-Verlag New York, Inc. Shaffer, J. P. (1999), ¡§A Semi-Bayesian Study of Duncan¡¦s Bayesian Multiple Compari-son Procedures,¡¨ Journal of Statistical Planning and Inference, 82, 197-213. Spiegelhalter, D. J., and Smith, A. F. M. (1982), ¡§Bayes Factors for Linear and Log-Linear Models with Vague Prior Information,¡¨ Journal of Royal Statistical Society B, 44, 377-387. Sun, D., and Kim, S. W. (1999), ¡§Intrinsic Priors for Testing Ordered Exponential Means,¡¨ ISDS Technical Report, Duke University, NC. Tamhane, A. C., and Dunnett, C. W. (1999), ¡§Stepwise Multiple Test Procedures with Biometric Applications,¡¨ Journal of Statistical Planning and Inference, 82, 55-68. Tanhane, A. C., and Gopal, G. V. S. (1993), ¡§A Bayesian Approach to Comparing Treatments with a Control,¡¨ Multiple Comparisons, Selection, and Applications in Biometry, ed. Hoppe, F. M., Marcel Dekker, Inc., New York. Tamhane, A. C., Liu, W., and Dunnett, C. W. (1998), ¡§A Generalized Step-up-down Multiple Test Procedure,¡¨ The Canadian Journal of Statistics, 26, 55-68. Westfall, P. H., Johnson, W. O., and Utts, J. M. (1997), ¡§A Bayesian Perspective on the Bonferroni Adjustment,¡¨ Biometrika, 84, 419-427. Westfall, P. H., Tobias, R. D., Rom, D., Wolfinger, R. D., and Hochberg, Y. (1999), Multiple Comparisons and Multiple Tests Using the SAS„¥ System, SAS Institute Inc., Cary, North Carolina.