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題名：透過廣義動差法估計時間相依協變量的修正羅吉斯-韋伯模型，和修正混合多類別模型
作者：潘宥亦
作者(外文)：Pan, Yu-Yi
校院名稱：國立成功大學
系所名稱：統計學系
指導教授：馬瀰嘉
學位類別：博士
出版日期：2023
主題關鍵詞：時間相依協變量；廣義動差估計法；多類別狀態；支援向量機；隨機森林；機率校正；Time-dependent covariates；Generalized method of moments；Multiple states；Support vector machine；Random forest；Probability calibration
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在分析患有慢性疾病或是一些癌症的案例中，有一定比例的患者在實驗期間可能永遠不會經歷研究者感興趣的事件。這種現象會導致生存曲線在觀察期結束時保持平穩的狀態。為了解決這個問題，使用混合治癒模型來分析具有重尾特性的存活時間資料。一般常見的混合治癒模型包含針對患者的治癒狀態和針對未治癒患者的生存函數部分。在本篇研究中，使用修正混合治癒模型，其中治癒患者的生存函數不固定為一，而是賦予一個特定的函數。此修改並不會對於參數估計有影響；取而代之的是，它會調整個別患者的生存機率，使其達到零。接著，本篇論文探討了兩個在使用修正混合治療模型會出現的問題。第一個問題是在考慮廣義估計方程式處理存活時間受到時間依賴協變量影響時，如果使用了不正確的工作相關矩陣，則參數估計量可能會有不一致性。在這項研究中，我提出了一種新穎的演算法，該演算法使用了三種修正受限已知基底矩陣的廣義矩方法來估計具有時間相依協變量的修正混合治癒模型的參數。基於二次推論函數概念，利用修正受限已知基底矩陣重建工作相關矩陣的逆矩陣，來確保參數估計量的一致性。而為了估計這些估計量的變異數，本篇論文推導了路易斯方法於三明治方差估計量上。模擬部分比較了兩種使用不同修正受限已知基底矩陣的演算法。第一種演算法是使用平均全一修正的受限已知基底矩陣和原始全一受限已知基底矩陣來處理型一時間相依協便量。而第二種演算法是使用平均下三角修正的受限已知基底矩陣和原始下三角的受限已知基底矩陣來處理型二時間相依協便量。而在實際資料分析，使用原發性膽汁性肝硬化(PBC)資料集說明了所提出方法的實際效用。第二個問題是，當潛在變數具有多種狀態時，需要對修正混合治癒模型的治癒狀態部分進行修改。解決多類別分類問題的一種方法是使用多項羅吉斯回歸模型(MLRM)，它是羅吉斯回歸模型的擴展。此外，支援向量機(SVM) 和隨機森林(RF) 等方法也適用於處理兩類別和多類別反應變數。在本研究中，提出了修正混合多類別模型，其中治癒患者的生存分佈來自於韋伯分布。而為了適應多類別狀態的判定，本篇論文提出了三種模型：修正多項羅吉斯-韋伯模型、修正支援向量機-韋伯模型和修正隨機森林-韋伯模型。提出貝氏平行模型(BPM) 校準估計類別機率時，增強RF的投票結果。模擬部分，為了評估三種修正混合多類別模型在多項式線性Logit函數、多項式二次Logit函數和非線性形式下的類別機率預測效能。實例部分，透過使用結直腸癌(CRC) 資料集證明了所提出的修正混合多類別模型的實際用途。
A certain proportion of patients may never experience an event of interest during the duration of an experiment in cases of chronic diseases or cancers, and it results in the survival curve exhibiting a plateau at the end of the observed period. The use of a mixture cure model is appropriate for fitting survival time data with a heavy tail. The common mixture cure model comprises the incidence part and the latency part. A modified mixture cure model is utilized in which the survival function of a cured patient is not fixed at 1 but is given a specific function. This modification has no bearing on parameter estimation; instead, it adjusts a patient's survival probability, allowing it to reach 0. This dissertation explores two issues. One concern with using the modified mixture cure model arises when survival times are influenced by time-dependent covariates. In such cases, parameter estimators may become inconsistent when using the generalized estimating equations method with an incorrect working correlation matrix. I introduce a novel algorithm that uses three modified restricted known basis matrices to estimate the parameters of the modified mixture cure model with time-dependent covariates via the generalized method of moments. The inverse of working correlation matrix is reconstructed by using modified restricted known basis matrices to ensure the consistency of parameter estimators based on the concept of the quadratic inference function. For estimating the variances of these estimators, I derive sandwich variance estimators with Louis' method. The simulation involves comparing algorithms with the original and modified restricted known basis matrices for Type I and Type II time-dependent covariates. The practical utility of the proposed methodology is illustrated with a primary biliary cirrhosis (PBC) data set. Another issue is the modifications for the incidence part of the modified mixture cure model when the latent variable has multiple states. One way to address multiclass problems is by employing the multinomial logistic regression model (MLRM). Additionally, methods like support vector machine (SVM) and random forest (RF) can be suitable for handling both binary and multiclass response variables. The modified mixture multiclass models are proposed where the survival distribution of cured patients follows a Weibull distribution. To accommodate the determination of multiple states, I introduce the modified multinomial logistic-Weibull model, the modified SVM-Weibull model, and the modified RF-Weibull model. Furthermore, I propose Bayesian Parallel Model calibration to enhance the smoothness of voting results for RF when estimating class probabilities. The simulations involving comparing the predictive performance of class probabilities for the three modified mixture multiclass models under setting 1: multinomial linear logit, setting 2: multinomial quadratic logit, and setting 3: nonlinear form. The practical use of the proposed modified mixture multiclass models is demonstrated by using a colorectal cancer (CRC) data set.
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