存活分析常使用Cox 比例風險模型，來解釋共變量與存活時間的關係；不過在某些資料中，並非所有的共變量皆符合比例風險。在比例風險假設未滿足的情況下，Cox 模型的估計結果會存在偏誤。為了處理此情形，以加乘法模型來配適，將不符比例風險的共變量放置於加法效應風險。然而，現有的計數方法Cox-Aalen 模型，僅能處理符合比例風險的乘法效應，不符合者的設定為加法效應，且需假設時間相關。本研究以完整概似函數方法的Cox-Aalen 模型，進行時間獨立的共變量效應分析。另外，對於不符合比例風險假設的情況，使用AFT-Aalen 模型，亦即乘法效應的準線風險基於加速失效風險。使用核平滑方法建構AFT-Aalen 模型的概似函數，並藉此以Nelder-Mead 方法進行估計。以模擬資料對模型做效力的評估，並將其用於皮膚黑色素瘤資料的效應估計。

The Cox proportional hazards model is popular for survival analysis to explain the relations between survival time and covariates. However, if the proportional hazard (PH) assumption is not satisfied by some covariates of data, the estimation of PH model would
be biased. To solve these situation, we fit the additive-multiplicative hazards model, which puts those covariates not satisfying proportional hazards into additive terms. The existing methods for additive-multiplicative hazards (AMH) models like Cox-Aalen model by counting process can only describe the covariates satisfying proportional hazards in multiplicative submodels, and the remains in additive terms must be assumed to have time-varying regression coefficient. The study use the full likelihood Cox-Aalen model to analyse the time-independent effects of covariates. On the other hand, we use accelerated
failure time to construct AFT-Aalen model when PH assumption fails. We build likelihood functions for AFT-Aalen model by kernel smoothing method and estimate the
parameters by Nelder-Mead method. We evaluate the performance of our AMH models through simulation study and apply them to Melanoma data.

Aalen, O. O. (1980). A model for non-parametric regression analysis of counting process. In Mathematical statictics and probability theory. Lecture Notes in Statistics 2, 1-25. New York: Springer.
Cox, D. R. (1972). Regression models and life tables. Journal of the Royal Statistics Society series B, 34, 187-220.
Etezadi-Amoli, J., and Ciampi, A. (1987) Extended Hazard Regression for Censored Survival Data with Covariates: A Spline Approximation for the Baseline Hazard Function.
Biometrics, 43, 181-192.
Henningsen, A., and Toomet, O. (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics, 26(3), 443-458.
Jones, M. C. (1990), The Performance of Kernel Density Functions in Kernel Distribution Function Estimation, Statistics and Probability Letters, 9, 129–132.
Jones, M. C., and Sheather, S. J. (1991), Using Non-Stochastic Terms to Advantage in Kernel-Based Estimation of Integrated Squared Density Derivatives, Statistics and
Probability Letters, 11, 511–514.
Lin, D. Y., and Ying, Z. (1995). Semiparameter analysis of general addictivemultiplicative hazard models for counting process. The Annal of Statistic, 23, 1712-1734.
Martinussen, T., and Scheike, T. H. (2002). A flexible addictive multiplicative hazard model. Biometrika, 89, 283-298.
Martinussen, T., and Scheike, T. H. (2006). Dynamic Regression Models for Survival Data, New York: Springer.
McKeague, I. W., and Sasieni, P. D. (1994). A partly parametric addictive risk model. Biometrika, 81, 501-14.
Murphy, S. A., and Sen, P. K. (1991). Time-dependent coefficients in a Cox-type regression model. Stochastic Process and their Applications, 39, 153-180.
Nelder, J. A., and Mead, R. (1965). A Simplex Method for Function Minimization. The Computer Journal, 7, 308–313.
Scheike, T. H., and Zhang, M. J. (2002). An Addictive-Multiplicative Aalen-Cox Regression Model. Board of the Foundation of the Scandinavian Journal of Statistic, 29, 75-88.
Therneau, T. M., and Grambsh, P. M. (2000). Extending the Cox Model, New York: Springer.
Tseng, Y. K., Su, Y. R., Mao, M. and Wang, J. L. (2015). An extend hazard model with longitudinal covariates. Biometrika, 102 ,135-150.
Tseng, Y. K., and Shu, K. (2011). Efficient Estimation for a Semiparametric Extended Hazards Model. Communications in Statistics—Simulation and Computation®, 40, 258-273.
Wikipedia contributors (2021, May 11). Melanoma. In Wikipedia, The Free Encyclopedia. Available from: https://en.wikipedia.org/w/index.php?title=Melanoma&oldid=1022566772 [Online; accessed 13-May-2021]
Zeng, D., and Lin, D. Y. (2007). Efficient Estimation for the Accelerated Failure Time Model. Journal of the American Statistical Association, 102, 1387-1396.