校準(Calibration)為評估模型預測精確度的重要指標，若預測結果為存活時間，常見衡量模型預測校準的指標為布賴爾分數(Brier score)。在過去文獻中，布賴爾分數已應用於未設限資料和右設限資料下的參數模型或半母數存活模型，本研究在區間設限資料下將布賴爾分數推廣至三種半母數存活模型:比例風險(proportional hazards)模型、加速失敗(accelerated failure time)模型和比例勝算(proportional odds)模型，並且藉由布賴爾分數比較不同參數模型與半母數模型的預測準確度，進而選擇較適合的模型。而區間設限下半母數存活模型的參數估計是困難的，所以在比例風險模型和比例勝算模型，本文使用條件牛頓法(conditional Newton-Raphson)和迭代凸次要演算法(iterative convex minorant algorithm)，而在加速失敗模型，使用最大近似伯恩斯坦估計法(Maximum approximate Bernstein likelihood estimation)，來估計區間設限下半母數存活模型的迴歸參數。本文藉由模擬研究觀察區間設限下布賴爾分數在不同模型、樣本數以及區間設限長度下的表現，並將此方法應用於愛滋病資料以及乳癌資料。

Calibration is an important indicator of the predictive accuracy of a model. If the predictive outcome is survival time, the Brier score is widely used as an indicator of calibration. In literature, the Brier score is applied to the parametric model or semiparametric survival model under the uncensored and right censored data. In our study, the Brier score is extended to three types of semiparametric survival models under interval censored: the Cox proportional hazards model, the accelerated failure time model, and the proportional odds model. Consequently, the Brier score may be used for model selection. The parameter estimation of the semiparametric survival model under interval censored is not straightforward. In this study uses the conditional Newton-Raphson method and the iterative convex minorant algorithm proposed for the Cox proportional hazards model and the proportional odds model, and applied the maximum approximate Bernstein likelihood estimation for the accelerated failure time model. The performance of Brier scores under different models, sample sizes, and lengths of interval censored are evaluated by simulation study, and the proposed method is applied to breast cancer data and HIV data.

[1] Anderson-Bergman, C. (2016). An efficient implementation of the EMICM alorithm for the interval censored NPMLE. Journal of Computational and Graphical Statistics, 81, 1-24.

[2] Anderson-Bergman, C. (2017). icenReg: Regression Models for Interval Censored Data in R. Joural of Computational and Graphical Statistics, 81, 1-23.

[3] Bennett, S. (1983). Analysis of survival data by the proportional odds model. Statistics in Medicine, 2, 273-277.

[4] Brier, G. W. (1950). Verification of forecasts expressed in terms of probability. Monthly Weather Review, 78, 1-3.

[5] Cox, D. R. (1972). Regression models and life tables(with Discussion). Journal of Royal statistical Society, Series B, 34, 187-220.

[6] Eeden, C. V. (1958). Testing and Estimating Ordered Parameters of Probability Distributions. PhD. thesis, University of Amsterdam.

[7] Gao, F., Zeng, D., and Lin, D. Y. (2017). Semiparametric Estimation od the Accelerated Failure Time Model with Partly Interval-Censored Data. Biometrics, 73, 1161-1168.

[8] Graf, E., Schmoor, C., Sauerbrei, W., and Schumacher, M. (1999). Assessment and comparsion of prognostic classification schemes for survival data. Statistics in Medicine, 18, 2529-2545.

[9] Guan, Z. (2016). Efficient and robust density estimation using Berstein type polynomials. Journal of Nonparametric Statistics, 28, 250-271.

[10] Guan, Z. (2019). Maximum approximate likelihood estimation in accelerated failure time model for intercal-censored data. arXiv, 1911.07087.

[11] Guan, Z. (2021). Fast nonparametric maximum likelihood density deconvolution using Bernstein polynomial. Statistica Sinica, 31, 891-908.

[12] Heller, G. (2021). The added value of new covariates to the brier score in cox survival models. Lifetime Data Analysis, 27, 1-14.

[13] Huang, J. and Wellner, J. (1997). Interval censored survival data:A review of recent process. Proceedings of the First Seattle Symposium in Biostatistics: Survival Analysis, 123-169.

[14] Jin, Z., Lin, D. Y., Wei, L. J., and Ying, Z. (2003). Rank-based inference for the accelerated failure time model. Biometrika, 90, 341-353.

[15] Kvamme, H. and Borgan, Ø. (2019). The Brier score under administrative censoring: Problems and solutions. arXiv, 1912.08581.

[16] Pan, W. (1999). Extending the iterative convex minorant algorithm to the Cox model for interval-censored data. Journal of Computational and Graphical Statistics,
8, 109-120.

[17] Polak, E. (1971). Computational method in optimization. New York: Academic Press.

[18] Spiegelhalter, D. J. (1986). Probabilistic prediction in patient management and clinical trials. Statistic in Medicine, 5, 421-433.

[19] Thomas, A. G. and Schumacher, M. (2006). Consistent estimation of the expect Brier score in general survival models with right-censored event times. Biometrical Journal, 48, 1029-1040.

[20] Tian, L. and Cai, T. (2006). On the accelerated failure time model for current status and interval censored data. Biometrika, 93, 329-342.

[21] Tsouprou, S. (2015). Measures of discrimination and predictive accuracy for interval censored survival data. Mathematical Institute Master Thesis, Leiden University Medical Center, Nederland.

[22] Wei, L. J. (1992). The accelerated failure time model: A useful alternative to the Cox regression model in survival analysis. Statistics in Medicine, 11, 1871-1879.

[23] Yates, J. F. (1982). External correspondence: Decompositions of the mean probability score. Organizational Behavior and Human Performance, 30, 132-156.

[24] 陳和謙(2020)。區間設限下的存活模型預測準確度。國立中央大學，碩士論文。