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Characterization of Bayes procedures for multiple endpoint problems and inadmissibility of the step-up procedure. (English) Zbl 1066.62010

Summary: The problem of multiple endpoint testing for \(k\) endpoints is treated as a \(2^k\) finite action problem. The loss function chosen is a vector loss function consisting of two components. The two components lead to a vector risk. One component of the vector risk is the false rejection rate (FRR), that is, the expected number of false rejections. The other component is the false acceptance rate (FAR), that is, the expected number of acceptances for which the corresponding null hypothesis is false. This loss function is more stringent than the positive linear combination loss function of E. L. Lehmann [Ann. Moth. Stat. 28, 1–25 (1957; Zbl 0078.33402)] and the authors [see the preceding entry, ibid., 126–144 (2005; Zbl 1066.62008)] in the sense that the class of admissible roles is larger for this vector risk formulation than for the linear combination risk function. In other words, fewer procedures are inadmissible for the vector risk formulation.
The statistical model assumed is that the vector of variables \({\mathbf Z}\) is multivariate normal with mean vector \(\mu\) and known intraclass covariance matrix \(\Sigma\). The endpoint hypotheses are \(H_i:\mu_i=0\) vs \(K_i:\mu_i> 0\), \(i=1,\dots,k\). A characterization of all symmetric Bayes procedures and their limits is obtained. The characterization leads to a complete class theorem. The complete class theorem is used to provide a useful necessary condition for admissibility of a procedure. The main result is that the step-up multiple endpoint procedure is shown to be inadmissible.

MSC:

62C10 Bayesian problems; characterization of Bayes procedures
62C15 Admissibility in statistical decision theory
62C07 Complete class results in statistical decision theory
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[1] Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. Roy. Statist. Soc. Ser. B 57 289–300. · Zbl 0809.62014
[2] Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. Ann. Statist. 29 1165–1188. · Zbl 1041.62061 · doi:10.1214/aos/1013699998
[3] Brown, L. D., Cohen, A. and Strawderman, W. E. (1976). A complete class theorem for strict monotone likelihood ratio with applications. Ann. Statist. 4 712–722. JSTOR: · Zbl 0336.62021 · doi:10.1214/aos/1176343543
[4] Brown, L. D., Johnstone, I. M. and MacGibbon, K. B. (1981). Variation diminishing transformation: A direct approach to total positivity and its statistical applications. J. Amer. Statist. Assoc. 76 824–832. · Zbl 0481.62021 · doi:10.2307/2287577
[5] Cohen, A. and Sackrowitz, H. B. (1984). Decision theory results for vector risks with applications. Statist. Decisions Suppl. 1 159–176. · Zbl 0554.62006
[6] Cohen, A. and Sackrowitz, H. B. (2005). Decision theory results for one-sided multiple comparison procedures. Ann. Statist. 33 126–144. · Zbl 1066.62009 · doi:10.1214/009053604000000968
[7] Dudoit, S., Shaffer, J. P. and Boldrick, J. C. (2003). Multiple hypothesis testing in microarray experiments. Statist. Sci. 18 71–103. · Zbl 1048.62099 · doi:10.1214/ss/1056397487
[8] Ferguson, T. S. (1967). Mathematical Statistics : A Decision Theoretic Approach . Academic Press, New York. · Zbl 0153.47602
[9] Hochberg, Y. (1988). A sharper Bonferroni procedure for multiple tests of significance. Biometrika 75 800–802. · Zbl 0661.62067 · doi:10.1093/biomet/75.4.800
[10] Hochberg, Y. and Tamhane, A. C. (1987). Multiple Comparison Procedures . Wiley, New York. · Zbl 0731.62125
[11] Karlin, S. and Rubin, H. (1956). The theory of decision procedures for distributions with monotone likelihood ratio. Ann. Math. Statist. 27 272–299. · Zbl 0070.37203 · doi:10.1214/aoms/1177728259
[12] Lehmann, E. L. (1957). A theory of some multiple decision problems. I. Ann. Math. Statist. 28 1–25. · Zbl 0078.33402 · doi:10.1214/aoms/1177707034
[13] Marshall, A. W. and Olkin, I. (1979). Inequalities : Theory of Majorization and Its Applications . Academic Press, New York. · Zbl 0437.26007
[14] Matthes, T. K. and Truax, D. R. (1967). Tests of composite hypotheses for the multivariate exponential family. Ann. Math. Statist. 38 681–697. · Zbl 0152.17802 · doi:10.1214/aoms/1177698862
[15] Sarkar, S. (2002). Some results on false discovery rate in stepwise multiple testing procedures. Ann. Statist. 30 239–257. · Zbl 1101.62349 · doi:10.1214/aos/1015362192
[16] Shaffer, J. P. (1995). Multiple hypothesis testing. Annual Review of Psychology 46 561–584.
[17] Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proc. Third Berkeley Symp. Math. Statist. Probab. 1 197–206. Univ. California Press, Berkeley. · Zbl 0073.35602
[18] Van Houwelingen, H. C. and Verbeek, A. (1985). On the construction of monotone symmetric decision rules for distributions with monotone likelihood ratio. Scand. J. Statist. 12 73–81. · Zbl 0647.62014
[19] Weiss, L. (1961). Statistical Decision Theory . McGraw-Hill, New York. · Zbl 0121.35601
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