Name: Hirotsu

Firstname: Chihiro

Title: Professor

Institution: Meisei University

Street: 2-1-1 Hodokubo Hino-City

City: Tokyo

Zip-Code: 191-8506

Country: Japan

Phone: +81 425917142

Fax: +81 425918181

Email: hirotsu@ge.meisei-u.ac.jp

Authors: Chihiro Hirotsu and Eri Ohta

Title: Comparing subjects according to their time profiles based on repeated measurements

Abstract: Suppose we have repeated measurements of blood pressure at every 30 minutes for 24 hours on several subjects. Those measurements are known normally to go down slightly in the night and a too large decline is abnormal as well as a flat or an upward tendency in the night. Then it is of great interest to classify those subjects according to their time profiles such as flat, convex or concave. In Hirotsu (1991) a procedure has been proposed to classify subjects according to their time profiles of cholesterol measurements obtained every 1 month for 6 months after a treatment for cholesterolemia. In that case we are interested in distinguishing the monotone tendencies of time profiles such as flat, up- or downward, and could assume the independence of measurement errors. In this case, however, the measurements normally come back approximately to the same level as the initial value so that the previous procedure assuming monotone changes cannot be applied. Further we cannot assume the independence of errors since the interval of measurements is 30 minutes and some amount of serial correlation is inevitable. The basic model we assume is a two-way layout without replication, $$y_{ij}=\mu_{ij}+\varepsilon_{ij}, \hspace{2mm}, i=1, \ldots, a; \hspace{1mm} j=1, \ldots, b,$$ where we assume a normal distribution for the $\varepsilon_{ij}$ with a serial correlation along the columns. The null hypothesis of the parallelism of the time profiles is given by $$H_0: \mu_{ij}=\bar{\mu}_{i \cdot} +\bar{\mu}_{\cdot j} -\bar{\bar{\mu}}_{\cdot \cdot},$$ where we employ the usual dot and bar notation. Therefore this is a problem of analyzing interaction effects. In the presentation we introduce an appropriate ordered alternative for interaction to express the interest here and propose a multiple comparisons procedure to classify subjects according to their time profiles such as convex, flat or concave. A real example of classifying 147 subjects is also given.

References: Hirotsu, C. (1991). An approach to comparing treatments based on repeated measures. Biometrika 78, 583-594.