過去已經有許多學者，在考慮每一個併發症之間彼此獨立下，對於回歸分析應用在多變量區間設限型的資料上，提出相關的邊際模型進行討論。
然而這些模型有著無法探討併發症之間相關性的問題，因此近來學者

For regression analysis of multivariate interval-censored data, several authors proposed some marginal model approaches, which modeled each time of interest individually. For the problem, those models does not allow for inference about the relationship or association between correlated failure times. The frailty model approach has been commonly used in the analysis of multivariate failure time data and it provides a flexible approach for directly modeling the relationship between correlated failure times.

In the thesis, we present a full likelihood approach based on the proportional hazard frailty model and estimate of regression parameters by Expectation Maximization (EM) algorithm. The method is applied to a set of bivariate interval-censored data arising from an AIDS clinical trial.

Contents
Chapter 1 Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1
1.1 Survival Analysis .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2
1.2 Censored Data.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4
1.3 Terminology and Notation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6
1.4 Concepts and Some Regression Models .. .. .. .. .. .. .. .. 9
Chapter 2 Model and Assumptions .. .. .. .. .. .. .. .. .. .. .. .. 11
2.1 Types of Interval-censored Analysis .. .. .. .. .. .. .. .. .. .. 12
2.2 The Proportinal Hazards Model .. .. .. .. .. .. .. .. .. .. .. .. 16
2.3 The Proportional Hazards Frailty Model.. .. .. .. .. .. .. 20
Chapter 3 Parameter Estimation .... .. .. .. .. .. .. .. .. .. .. .. .. 23
3.1 Expectation-maximization Algorithm.. .. .. .. .. .. .. .. .. 24
3.2 E-step .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 28
3.3 M-step .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 32
3.4 Estimative Procedure .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 35
Chapter 4 Application .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .... .. .. .. .. 40
4.1 Source of Data .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 41
4.2 Fundamental Analysis .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 45
4.3 Turnbull’s Algorithm.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 47
4.4 Result of Analysis .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 58
Chpater 5 Conclusion.... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 62
5.1 Baseline Hazard Function .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 63
5.2 Future Work .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 68
Appendix I Fisher Information Matrix .. .. .. .. .. .. .. .. .. .. 71
Appendix II ACTG181.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .... .. .. .. 74
Reference .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .... .. .. .. .. .. .. .. 75
List of Tables
2.1 Summary of Case I and Case II Interval-censoring . 15
4.1 Number of Censored Patients and Censored Status 43
4.2 Numerical Check Difference Between Turnbull and KM for Blood . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Numerical Check Difference Between Turnbull and KM for Blood . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Numerical Check Difference Between Turnbull and KM for Urine . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5 Numerical Check Difference Between Turnbull and KM for Urine . . . . . . . . . . . . . . . . . . . . . . . . 53
4.6 Observe Time and Estimate Survival Function under Turnbull Method . . . . . . . . . . . . . . . . . . . 57
4.7 Proportional Hazards Model . . . . . . . . . . . . . . 59
4.8 Hazard Ratio . . . . . . . . . . . . . . . . . . . . . . . . 59
4.9 Wald test: H0 :   = 0 v.s H1 :   6= 0 . . . . . . . . . . . . 60
4.10Marginal Proportional Hazards Model . . . . . . . . 61
List of Figures
3.1 The idea of algorithm . . . . . . . . . . . . . . . . . . . 35
4.1 Number of Patients in Each Type of Censoring . . . 45
4.2 (a)Pie Chart of Blood Sample. (b)Pie Chart of
Urine Sample . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Estimates of Baseline Survival Functions for Blood
under Turnbull Method . . . . . . . . . . . . . . . . . 49
4.4 Estimates of Baseline Survival Functions for Urine
under Turnbull Method . . . . . . . . . . . . . . . . . 50
4.5 Estimates of Baseline Survival Functions for Blood
Comparing with Turnbull and KM . . . . . . . . . . 51
4.6 Estimates of Baseline Survival Functions for Urine
Comparing with Turnbull and KM . . . . . . . . . . 51
4.7 Estimates of Baseline Survival Functions for Blood
and Urine under Turnbull Method . . . . . . . . . . . 55
5.1 The Relationship of Failure Time . . . . . . . . . . . 64

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