具隨機轉折點之比例風險模型基於樣條函數的半參數估計：多種轉折點分配與誤設定研究

Abstract

轉折點模型在醫學統計中扮演重要角色，能夠識別共變數效果的結構性變化，協助建立更精確的風險預測模型。傳統的固定轉折點模型假設所有個體共享相同的轉折點，然而此假設忽略了個體間的異質性，在個人化醫療趨勢下顯得不夠貼合實際。因此，本研究在Cox比例風險模型架構下，將原先假設常態分配的隨機轉折點推廣至更靈活的分配假設，包括指數分配、對數常態分配及伽瑪分配。在估計方法上，採用sieve最大概似估計法，結合B-spline函數近似基礎風險函數，克服未知基礎風險函數估計的困難。本研究推導出包含轉折點積分的個體對數概似函數，可透過轉折點共變數的數值拆解為兩個部分的總和。值得注意的是，個體對數概似函數的每個部分僅涉及轉折點的累積分布函數，而非複雜的分布函數積分運算。透過模擬研究設定不同樣本數、設限率和B-spline階數，評估所提出方法的有限樣本統計性質。模擬結果顯示，在轉折點分配正確設定下，參數估計表現良好。當轉折點分配錯誤設定時，共變數效果與轉折點效果的估計仍維持良好表現，但隨機轉折點的變異數估計表現不佳。本研究並將所提出的方法應用於德國乳癌研究的真實資料進行驗證說明。

Change-point models play an important role in medical statistics by identifying structural changes in covariate effects and helping to establish more accurate risk prediction models. Traditional fixed change-point models assume that all individuals share the same change-point; however, this assumption ignores heterogeneity among individuals and appears inadequate for real-world applications under the trend of personalized medicine. Therefore, under the Cox proportional hazards model with random change-points, our study extends normally-distributed random change-points to accommodate more flexible distributions of change-points, including exponential, log normal, and gamma distributions. We adopt the sieve maximum likelihood estimation approach and use B-spline functions to approximate the baseline hazard function, which overcomes the difficult task of estimating the unknown baseline hazard function. Our study derives that an individual log-likelihood function involving the integration of change-point components can be reduced to a sum of two parts via the value of covariate with change-point. It is noted that each part of an individual log-likelihood function involves only the cumulative distribution functions of change-points rather than complex integration with respect to their distribution function. We conduct simulation studies including different sample sizes, censoring rates, and B-spline orders to evaluate the finite-sample statistical properties of the proposed method. The simulation results show that under correct change-point distribution, the parameter estimation performs well. When the change-point distribution is mis-specified, the covariate and change-point effect estimators maintain good performance, but the variance estimation of random change points performs poorly. The proposed method is also applied to a real data from German Breast Cancer Study for illustration.