在臨床實驗中，基於成本與管理的考量，常使用群序檢定以有機會提早結束試驗。當資料為橫斷面（cross-sectional data）型態，且欲檢定兩處理平均數之差異時，常使用的群序檢定方法有Pocock（1977）、O’Brien與Fleming（1979）以及Lan與DeMets（1983）等方法。Lee與DeMet（1991）提出在長期追蹤資料（longitudinal data）中，利用線性混合模式檢定兩處理之線性趨勢。本文延伸Lee與DeMets之方法，推廣其線性混合模式至多項式趨勢型態，提出在多種處理下之群序檢定方法。各階段檢定統計量之聯合分配為多變量卡方分配，文中使用Pocock及O’Brien與Fleming概念，提出二階段與三階段檢定之臨界值，並使用實例說明此檢定程序。

For ethical, economical and administrative considerations, group sequential methods are frequently applied for possibly early determination clinical trials. For cross-sectional data, three common group sequential methods for comparing means between two treatments are proposed by Pocock (1977), O'Brien and Fleming (1979) and Lan and DeMets (1983). For longitudinal data, Lee and DeMets (1991) proposed a sequential comparison for testing the rates of change with linear trend between two treatments.
    In this article, a group sequential method for multi-armed trials is proposed, which is a generalization of Lee and DeMets' method, for testing responses changes with time in polynomial trend. The asymptotic joint distribution of proposed test statistics is a multivariate chi-squared distribution. Boundaries of the proposed methods based on Pocock-type and O'Brien and Fleming-type are provided for practical use. The proposed testing procedure is illustrated by a clinical example.

Contents
1 Introduction 1
2 Historical Review 4
2.1 Classical Group Sequential Method  4
2.1.1 Pocock Procedure 5
2.1.2 O’Brien and Fleming Procedure 6
2.1.3 Lan and DeMets Procedure 7
2.2 Group Sequential Tests for Multi-Armed Trials 7
2.3 Multivariate Chi-Squared Distribution 9
3 Test For Polynomial Trend 14
3.1 Lee and DeMets Procedure 14
3.2 Proposed Test  16
3.3 Numerical Study 18
3.4 An Example 19
4 Conclusions and Remarks 21

List of Figures
1 Plots of bivariate chi-squared distributions with various degrees of freedom and correlations, (a)nu=2, rho=0.1 (b)nu=2, rho=0.5 (c)nu=6,rho=0.5 (d)nu=6,rho=0.9 11
2 Plot of Empirical Density Values against Ordered Values of
f(q1, q2) 19

List of Tables
1 Boundaries of Pocock-type and O’Brien-Fleming-type repeated chi^2 test for two stages 23
2 Boundaries of Pocock-type and O’Brien-Fleming-type repeated chi^2 test for three stages and the correlations ((rho_1,rho_2,rho_3)=(0.7,0.5,0.3) 26

Follmann, D. A., Proschan, M. A. and Geller, N. L. (1994). Monitoring pairwise comparisons in multi-armed clinical trials, Biometrics, 50: 325–336.
Hewett, J. E. and Tsutakawa, R. K. (1972). Two-stage chi-square goodness-of-fit test, Journal of the American Statistical Association, 67: 395–401.
Jennison, C. and Turnbull, B. W. (1991). Exact calculations for sequential t, chi^2 and F tests, Biometrika, 78: 133–141.
Jennison, C. and Turnbull, B. W. (2000). Group Sequential Methods: Applications to Clinical Trials, Chapman & Hall/CRC, Boca Raton.
Jenson, D. R. (1970). The joint distribution of quadratic forms and related distributions, Australian Journal of Statistics, 12: 13–22.
Kibble, W. F. (1941). A two-variate gamma type distribution, Sankhya,5: 137–150.
Kim, K. and DeMets, D. L. (1987). Design and analysis of group sequential tests based on the type I error spending rate function, Biometrika, 74: 149–154.
Krishnamoorthy, A. S. and Parthasarathy, M. (1951). A multivariate gammatype distribution, The Annals of Mathematical Statistics, 4: 549–557.
Lan, K. K. G. and DeMets, D. L. (1983). Discrete sequential boundaries for clinical trials, Biometrika, 70: 659–663.
Lee, J. W. and DeMets, D. L. (1991). Sequential comparison of changes with repeated measurement data, Journal of the American Statistical Association, 86: 757–762.
Miliken, G. A. and Johnson, D. E. (1993). Analysis of Messy Data, Vol. 1: Designed Experiments, Chapman & Hall/CRC, Boca Raton.
O’Brien, P. C. and Fleming, T. R. (1979). A multiple testing procedure for clinical trials, Biometrics, 35: 549–556.
Pocock, S. J. (1977). Group sequential methods in the design and analysis of clinical trials, Biometrika, 64: 191–199.
Proschan, M. A., Follmann, D. A. and Geller, N. L. (1994). Monitoring multi-armed trials, Statistics in Medicine, 13: 1411–1452.
Royen, T. (1991). Expansions for the multiviriate chi-square distribution, Journal of Multivariate Analysis, 38: 213–232.