The use of an identity in Anderson for multivariate multiple testing. (English) Zbl 1137.62038
Summary: Consider the model where there are \(I\) independent, multivariate normal treatment populations with \(p\times1\) mean vectors \(\mu_i\), \(i=1,\dots,I\), and covariance matrix \(\Sigma\). Independently the \((I+1)\)st population corresponds to a control and it too is multivariate normal with mean vector \(\mu_{I+1}\) and covariance matrix \(\Sigma\). Now consider the following two multiple testing problems.
Problem 1: Test \(H_{ij}: \mu_{ij}- \mu_{(I-1)j}= 0\) vs \(K_{ij}: \mu_{ij}- \mu_{(I-1)j}\neq 0\), \(i=1,\dots, I\); \(j=1,\dots,p\).
Problem 2: Test \(H_i:\mu_i- \mu_{(I+1)}=0\) vs \(K_i:\mu_i- \mu_{(I+1)}\neq0\).
For each problem an identity of T. W. Anderson [An introduction to multivariate statistical analysis. 2nd ed., New York: Wiley (1984; Zbl 0651.62041)] is used to derive families of likelihood ratio statistics that can be used to determine step-down multiple testing procedures with desirable properties.
Problem 1: Test \(H_{ij}: \mu_{ij}- \mu_{(I-1)j}= 0\) vs \(K_{ij}: \mu_{ij}- \mu_{(I-1)j}\neq 0\), \(i=1,\dots, I\); \(j=1,\dots,p\).
Problem 2: Test \(H_i:\mu_i- \mu_{(I+1)}=0\) vs \(K_i:\mu_i- \mu_{(I+1)}\neq0\).
For each problem an identity of T. W. Anderson [An introduction to multivariate statistical analysis. 2nd ed., New York: Wiley (1984; Zbl 0651.62041)] is used to derive families of likelihood ratio statistics that can be used to determine step-down multiple testing procedures with desirable properties.
MSC:
| 62H15 | Hypothesis testing in multivariate analysis |
| 62J15 | Paired and multiple comparisons; multiple testing |
Keywords:
false discovery rate; FDR; maximum-likelihood ratio down; M-LRD; maximum residual down; MRD; multivariate treatments vs controlCitations:
Zbl 0651.62041References:
| [1] | Anderson, T. W., An Introduction to Multivariate Statistical Analysis (1984), Wiley: Wiley New York · Zbl 0651.62041 |
| [2] | Cohen, A., Sackrowitz, H.B., Xu, M., 2008. A new multiple testing method in the dependent case, submitted for publication.; Cohen, A., Sackrowitz, H.B., Xu, M., 2008. A new multiple testing method in the dependent case, submitted for publication. · Zbl 1161.62040 |
| [3] | Dudoit, S.; Shaffer, J. P.; Boldrick, J. C., Multiple hypothesis testing in microarray experiments, Statist. Sci., 18, 71-103 (2003) · Zbl 1048.62099 |
| [4] | Dunnett, C. W., A multiple comparison procedure for comparing several treatments with a control, J. Amer. Statist. Assoc., 50, 1096-1121 (1955) · Zbl 0066.12603 |
| [5] | Hochberg, Y.; Tamhane, A. C., Multiple Comparison Procedure (1987), Wiley: Wiley New York · Zbl 0731.62125 |
| [6] | Imada, T.; Douke, H., Step-down procedure comparing several treatments with a control based on multivariate normal response, Biometrical J., 49, 18-29 (2007) · Zbl 1442.62426 |
| [7] | Lehmann, E. L.; Romano, J. P., Testing Statistical Hypotheses (2005), Springer: Springer Berlin · Zbl 1076.62018 |
| [8] | Nakamura, T.; Imada, T., Multiple comparison procedure of Dunnett’s type for multivariate normal means, J. Japanese Soc. Comput. Statist., 18, 21-32 (2005) · Zbl 1261.62069 |
| [9] | Scheffé, H., The Analysis of Variance (1959), Wiley: Wiley New York · Zbl 0086.34603 |
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