本篇論文主要考慮成群資料的存活分析，其特點為群內個體間具有相關性，並假定群內個體具有相同但無法觀測到的事故傾向。首先，探討事故傾向服從任一連續分配時混合Weibull迴歸模式的特性，接著，推導出事故傾向服從血Inverse Gaussian吧時之混合Weibull模式，並介紹參數的估計問題。然後，推導出群內個體是否獨立之分數檢定統計量，以分別就兩種最常見的存活資料型態一完整型態與右設限型態：檢定模式中事故傾向的效應是否存在。最後，並以實例說明分數檢定之程序。

In this paper, we study survival analysis for grouped data, where the within group correlations are considered. It is also assumed that individuals within the same group share a common but unobservable random frailty. First, we discuss the properties of the Weibull regression model mixed by any continuous distribution. Next, we derive an Inverse Gaussan mixture of Weibull regression model, and discuss the estimation problem. Then, we derive the score test for testing independence between components within the same group, where the two most common cases are discussed the complete data case and the right censoring case. Finally, the testing procedures are illustrated by two examples.

Aalen, O. O. (1994). "Effects of frailty in survival analysis." Statistical Method in Medical Research 3, 227-243.
Bernardo, J. M. (1976). "Algorithm AS 103: Psi (Digamma) function." Applied Statistics 25, 315-317.
Clayton, D. (1978). "A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence." Biometrika 65,141-151.
Commenges, D. and Andersen, P. K. (1995). "Score test of homogeneity for survival data." Lifetime Data Analysis 1,145-156.
Crowder, M. J., Kimber, A. C., Smith, R. L. and Sweeting, T. J. (1991). Statistical Analysis of Reliability Data. Chapman & Hall, London.
Crowder, M. J. and Kimber, A. C. (1997). "A score test for the multivariate Burr and other weibull mixture distributions." Scandinavian Journal of Statistics, Theory and Applications 24, 419-432.
Hougaard, P. (1986a). "Survival models for heterogeneous populations derived from stable distributions." Biometrika 73, 387-396.
Hougaard, P. (1986b). "A class of multivariate failure time distributions." Biometrika 73, 671-678.
Kimber, A. C. (1996). "A weibull-based score test for heterogeneity." Lifetime Data Analysis 2, 63-71.
Lawless, J. F. (1982). Statistical Models and Methods for Lifetime Data. John Wiley & Sons, New York.
Lee, M.-L. T. (1985). "Dependence by total positivity." The Annals of Probability 13, 572-582.
Moore, R. J. (1982). "Algorithm AS 187: Derivatives of the incomplete gamma integral." Applied Statistics 31, 330-335.
Pierce, D. A. (1982). "The asymptotic effect of substituting estimators for parameters in certain types of statistics." The Annals of Statistics 10, 475-478.
Sahu, S. K., Dey, D. K., Aslanidou, H. and Sinha, D. (1997). "A weibull regression model with gamma frailties for multivariate survival data." Lifetime Data Analysis 3, 123-137.
Schneider, B. E. (1978). "Algorithm AS 121: Trigamma function." Applied Statistics 27, 97-99.
Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York.
Tweedie, M. C. K. (1957). "Statistical properties of inverse gaussian distributions. I." The Annals of Mathematical Statistics 28, 362-377.
Venables, W. N. and Ripley, B. D. (1994). Modern Applied Statistics with S-Plus. Springer-Verlag, New York.
Whitmore, G. A. and Lee, M.-L. T. (1991). "A multivariate survival distribution generated by an inverse gaussian mixture of exponentials." Technometrics 33, 39-50.