臺灣博碩士論文加值系統

在釵h長期追蹤的研究中，探討具誤差傾向的長期追蹤變數與檢測存活時間的關係是一個重要的議題。而傳統的統計方法，忽略此測量的誤差，因此對於兩者的關係容易獲得不可靠的結論。本文中我們將提出新的統計方法來解決在存活時間與長期追蹤資料中具有測量誤差的問題。假設長期追蹤的資料是一個具有隨機效果的模式(random effects model)，且存活時間服從Cox的風險比例模式(Cox proportional hazards model)時，在聯合建模(joint modeling)的架構下提出的新的估計方法。

首先，我們假設真實的長期追蹤變數的行為服從一個線性的隨機效果模式，提出一個近似的無母數修正分數(approximate nonparametric corrected score)。在隨機效果或誤差項沒有任何分配的假設下，提出新的估計方法外與其大樣本的性質。

在本文的第二部份，我們在長期追蹤變數資料的線性隨機效果模式下，應用無母數的最大概似法來提供聯合建模之估計。另外，我們也提出了一個修正的無母數最大概似法，此方法在計算上相當容易，且能避免傳統上對於隨機效果的須有常態性假設的缺點。

最後，在本文中我們對於長期追蹤資料提出一個具特定對象的改變點聯合模式(a joint model with a subject-specific changepoints model)，且發展兩個半參數的方法來進行估計，分別為半參數化的修正分數方法及半參數化的條件分數方法。這些方法的估計量在文中也證明其是等價的，此外也推導出這些估計量的漸近性質。本文所提出的估計方法，也將經由模擬研究探討其在有限樣本下的表現，同時我們也進一步使用AIDS的臨床資料來說明所有方法的表現。

In many longitudinal studies, an important objective is the investigation of the association between some error-prone longitudinal covariate and censored survival time processes.
Standard statistical methods that ignore measurement error may lead to unreliable conclusion regarding the relationship between the two processes. In this thesis, we propose new methods that address the problem of measurement error in the analysis of survival time and longitudinal data. The methods are built under the joint modeling framework assuming a random effects model for the longitudinal covariate data and the Cox proportional hazards model for the survival data.

Firstly, we postulate a linear random effects model for the true longitudinal covariate process and develop an approximate nonparametric corrected score method for the joint modeling of the survival and longitudinal data. This approach requires no distributional assumption on the random effects or the error. The large sample properties of the estimator that is based on this method are also established.

Secondly, we present a nonparametric maximum likelihood method for a joint model that also includes a linear random effects model for the longitudinal covariate data. In addition, we provide a corrected nonparametric maximum likelihood approach that is computationally attractive and also has the advantage of being able to avoid the traditional normality assumption about the distribution of the random effects.

Finally, we suggest a joint model with a subject-specific changepoint model for the longitudinal data and develop two semiparametric methods, namely corrected score and conditional score approaches. The estimators based on these methods are shown to be equivalent and their asymptotic theories are also derived. The finite sample size performances of all the proposed estimators are investiagted through simulation studies. Furthermore, we illustrate all the methods using data from an AIDS clinical study.

1 Introduction 1
1.1 Longitudinal studies . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Joint modeling of longitudinal and survival data . . . . . . . . . . 3
1.3 Organization of the dissertation . . . . . . . . . . . . . . . . . . . 4
2 Approximate nonparametric corrected score method 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Formulation of the joint model . . . . . . . . . . . . . . . . . . . 10
2.2.1 Submodel for the longitudinal data . . . . . . . . . . . . . 10
2.2.2 Submodel for the survival times . . . . . . . . . . . . . . . 11
2.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 The ideal and naive methods . . . . . . . . . . . . . . . . . 12
2.3.2 Parametric corrected-score approach . . . . . . . . . . . . 13
2.3.3 Conditional score method . . . . . . . . . . . . . . . . . . 15
2.3.4 Approximate nonparametric corrected-score method . . . . 15
2.4 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 The ACTG-175 data example . . . . . . . . . . . . . . . . . . . . 29
2.5.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.2 Analysis models and results . . . . . . . . . . . . . . . . . 29
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Maximum likelihood-based methods for joint models 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Approaches to estimation . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.1 A nonparametric maximum likelihood method . . . . . . . 37
3.2.2 Corrected-nonparametric maximum Likelihood Approach . 40
3.3 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 The ACTG-175 data example . . . . . . . . . . . . . . . . . . . . 46
3.4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4.2 Analysis models and results . . . . . . . . . . . . . . . . . 46
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4 Joint model with subject-specific changepoints in the covariate 49
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.1 Longitudinal data submodel . . . . . . . . . . . . . . . . . 52
4.2.2 Submodel for the survival data . . . . . . . . . . . . . . . 53
4.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.1 Corrected score method . . . . . . . . . . . . . . . . . . . 57
4.3.2 Conditional score approach . . . . . . . . . . . . . . . . . 59
4.4 Simulation studies . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 ACTG 175 data analysis . . . . . . . . . . . . . . . . . . . . . . . 65
4.5.1 Analysis models . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5.3 Choice of the smoothing parameter . . . . . . . . . . . . . 70
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 Concluding Discussion 78
References 80
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