摘要

本文主要考慮群集樣本(clustered samples)的存活分析，而每一群集中又分為兩種組別(groups)。假定同群集同組別內的個體共享相同但不可觀測的隨機脆弱性(frailty)，因此面臨的是雙變量脆弱性變數的多變量存活資料。首先，驗證雙變量脆弱性對雙變量對數存活時間及雙變量存活時間之相關係數所造成的影響。接著，假定雙變量脆弱性服從雙變量對數常態分配，條件存活時間模式為韋伯迴歸模式，我們利用EM法則，推導出雙變量脆弱性之多變量存活模式中母數的估計方法。

關鍵詞：雙變量脆弱性，Weibull迴歸模式，對數常態分配，EM法則

Abstract

Consider survival analysis for clustered samples, where each cluster contains two groups. Assume that individuals within the same cluster and the same group share a common but unobservable random frailty. Hence, the focus of this work is on bivariate frailty model in analysis of multivariate survival data. First, we derive expressions for the correlation between the two survival times to show how the bivariate frailty affects these correlation coefficients. Then, the bivariate log-normal distribution is used to model the bivariate frailty. We modified EM algorithm to estimate the parameters for the Weibull regression model with bivariate log-normal frailty.

Key words：bivariate frailty, Weibull regression model, log-normal distribution, EM algorithm.

參考文獻
[1] Aalen, O. O., (1988). "Heterogeneity in Survival Analysis",
Statistics in medicine, vol. 7, p. 1121-1137.
[2] Aalen, O. O., (1992). "Modeling Heterogeneity in Survival
Analysis by the compound Poisson distribution", Ann. Appl. Prob.,
vol. 2, p. 951- 972.
[3] Clayton, D. G., (1978). "A Model for Association in Bivariate Life
Tables and Its Application in Epidemiological Studies of Familial
Tendency in Chronic Disease Incidence", Biometrika, vol. 65, p.
141-151.
[4] Clayton, D. G., and Cuzick, J., (1985). "Multivariate Associations of
The Proportional Hazards Model", Journal of the Royal Statistical
Society, Ser. A vol. 148, p. 82-108.
[5] Clayton, D. G., (1991). "A Monte Carlo Method for Binary
Inference in Frailty Models", Biometrics, vol. 47, p. 467-485.
[6] Gail, M. H., Wieand, S. and Piantados, S., (1984). "Biased
Estimates of Treatment Effect in Randomized Experiments with
Nonlinear Regression and Omitted Covariates", Biometrika, vol. 71,
p. 431-444.
[7] Gilks, W. R., Best, N. G., Tan, K. K. C., (1995). "Adaptive Rejection
Metropolis Sampling within Gibbs Sampling", Applied Statistics,
vol. 44., p.455-472.
[8] Hougaard, P., (1986). "Survival Models for Heterogeneous
Populations Derived from Stable Distributions", Bimoetrika, vol. 73,
p. 387-396.
[9] Hougaard, P., (1986). "A Class of Multivariate Failure Time
Distributions ", Bimoetrika, vol. 73, p. 671-678.
[10] Huster, W. J., Brookmeyer, R., and Self,,S. G., (1989). "Modeling
Paired Survival Data with Covariates", Biometrics, vol. 45, p. 145-
156.
[11] Klein, J. P., and Moeschberger, M. I., (1988). "Bounds on Net
Survival Probabilities for Dependent Competing Risks",
Biometrics, vol. 44 ,p. 529-538.
[12] Lancaster, T., (1990). The Econometrics Analysis of Transition Data.
CUP, Cambridge.
[13] Lindley, D. V., and Singpurwalla, N. D., (1986). "Multivariate
Distributions for the Life Lengths of Components of a System
Sharing a Common Environment", Journal of Applied Probability,
vol. 23, p. 418-431.
[14] Mcgilchrist, A., and Aisbett, C. W., (1991). "Regression with Frailty
in Survival Analysis", Biometrics, vol. 47, p. 461-466.
[15] Pickles, A., and Crouchley, R., (1995). "A Comparison of Frailty
Models for Multivariate Survival Data", Statistics in Medicine, vol.
14, p. 1447-1461.
[16] Wei, L. J., Lin, D. Y., and Weissfeld, L., (1989). "Regression
Analysis of Multivariate Incomplete Failure Time Data by
Modeling Marginal Distributions", Journal of the American
Statistical Association, vol. 84, p. 1065-1073.
[17] Wei, G. C. G., Tanner, M. A., (1990). "A Monte Carlo
Implementation of the EM algorithm and the Poor Man's Data
Augmentation Algorithms", Journal of the American Statistical
Association, vol. 85, p. 699-704.
[18] Xue, X., (1995). Analysis of Survival Data under Heterogeneity:
Univariate and Bivariate Frailty Models. Unpublished Ph.D. Thesis,
School of Hygiene and Public Health, Johns Hopkins University.
[19] 陳麗霞, (民84). "脆弱性Weibull迴歸模式之貝氏推論".國科
會計畫, NSC-84-2415-H-004-006.