Name: Aoshima

Firstname: Makoto

Title: Professor

Institution: University of Tsukuba, Institute of Mathematics

Street: 1-1-1 Tennoudai

City: Tsukuba

Zip-Code: Ibaraki 305-8571

Country: Japan

Phone: +81-298-51-6078

Fax: +81-298-53-6501

Email: aoshima@math.tsukuba.ac.jp

Authors: Makoto Aoshima

Title: Sample Size Determination for Multiple Comparisons with Components of a Linear Function of Mean Vectors

Abstract: This article considers sample size determination for designing multiple comparisons with respect to correlated $p\ (\ge2)$ components $(\xi_1,...,\xi_p)$ of a linear function of mean vectors $\xi=\sum_{i=1}^k b_i\mu_i$ from $N_p(\mu_i, \Sigma_i),\ i=1,...,k$. Let us suppose that there are several remedies to be compared with each other, say $k=2$, and those effects are observed at $p$ points of time series, the user would be typically interested in the direction and the magnitude of differences - which points are more significant differences, and by how much - with respect to $p$ correlated time components $(\xi_1,...,\xi_p)$ of $\mu_1-\mu_2\ (b_1=1, b_2=-1)$. We propose a two-stage procedure to determine the sample size so that multiple comparisons confidence intervals will cover the true parameters and be sufficiently narrow with a guaranteed high accuracy. {\bf Tukey`s method} of all pairwise multiple comparisons (MCA), {\bf Hsu`s method} of multiple compari! sons with the best (MCB), and {\bf Dunnett`s method} of multiple comparisons with a control (MCC) are considered in this context. An advantage of the proposed procedure is to guarantee a high probability of simultaneous confidence intervals for treatment contrasts as well as maintaining a prespecified width when $\Sigma_i$`s are unknown but spherical models.

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