隨著生物臨床醫學研究的發展, 右設限資料分析方法在文獻中已廣為發展與應用, 區間設限資料為醫療研究收集過程中易遇到的資料, 例如病患的定期一段時間回診觀察, 未能確切知道發生事件的準確時間, 只觀察到某兩次回診時間區間中發生。
此外, 在資料結構上亦會收集到不易感受性的資料, 有些被觀察者在研究期間內不會發生我們所感興趣的事件, 通常被歸類為右設限資料, 但是這些資料為確切不發生之事件資料。
本篇考慮使用線性轉換模型(transformationmodel) 分析不易感受性區間設限資料, 使用EM 演算法(EM algorithm) 和牛頓迭代法(Newton-Raphsoniterationmethod) 估計參數, 並透過模擬驗證之。

There are numerous statistical methods reported for the analysis of right-censored failure time data in the past 30 years. In a medical follow-up study, additional problems arise in the analysis of interval censoring. For example, patients are observed periodically, we don't know the exact onset time of the disease, thus the observed failure time falls into a time period.
In addition, we consider a data set with two populations. Some subjects (non-susceptibility) do not become events we are interested in and some subjects (susceptibility) become events we are interested in. The non-susceptible rate (cured rate) represents a combination of cure data and survival data.
This thesis considers transformation model to analysis the interval censoring data with cured proportion. The EM algorithmis developed for the estimation and simulation studies are conducted.

目錄
第一章序論.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1
1.1 失效時間.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3
1.1.1 精神疾病病人失效時間資料.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3
1.2 區間設限失效時間.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4
1.2.1 水凝膠人工晶體鈣化風險研究(型一區間設限) .. .. .. .. .. .. .. 5
1.2.2 乳癌研究(型二區間設限) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6
1.3 使用迴歸模型分析型二區間設限資料.. .. .. .. .. .. .. .. .. .. .. 8
1.3.1 比例風險模型.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10
1.3.2 比例勝算模型.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 11
1.3.3 加速失效時間模型.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 12
1.3.4 線性轉換模型.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 14
1.4 不易感受性區間設限失效時間資料.. .. .. .. .. .. .. .. .. .. .. .. .. 15
1.4.1 不易感受性區間設限資料結構.. .. .. .. .. .. .. .. .. .. .. .. .. .. 15
1.4.2 文獻探討.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 18
1.5 論文架構.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 19
第二章模型建立與假設.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 20
2.1 線性轉換模型.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 21
2.2 EM演算法(Expectation-maximization Algorithm) .. .. .. 23
第三章參數估計.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 27
3.1 基底風險函數.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 27
3.2 E步驟.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 29
3.3 M步驟.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 31
3.4 Fisher InformationMatrix .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 34
3.5 拔靴法.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 36
第四章模擬分析.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 38
4.1 生成區間設限資料.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 38
4.2 模擬結果.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 39
4.3 風險函數(Hazard function) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 42
4.4 AIC .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 45
第五章結論與未來發展.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 47
附錄一.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 49
附錄二.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 54
參考文獻.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 57
表目錄
1.1 精神疾病病人失效時間資料. . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 人工晶體鈣化資料. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 乳房外觀攣縮時間. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 血友病患者感染HIV-1病毒風險資料. . . . . . . . . . . . . . . . . . . 16
4.1 模擬一(n = 200) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 模擬一(n = 400) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 AIC (模擬一,n = 200) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 AIC (模擬一,n = 400) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
A.1 模擬二(n = 200) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A.2 模擬二(n = 400) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
A.3 模擬三(n = 200) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
A.4 模擬三(n = 400) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
A.5 AIC (模擬二,n = 200) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
A.6 AIC (模擬二,n = 400) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
A.7 AIC (模擬三,n = 200) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
A.8 AIC (模擬三,n = 400) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
圖目錄
1.1 存活曲線. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 風險函數曲線. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 風險函數曲線. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Banerjee, S. and Carlin, B. P. (2004). Parametric spatial cure rate
models for interval-censored time-to-relapse data.Biometrics,
60, 268-275.
Chen, K., Jin, Z. and Ying, Z. (2002). Semiparametric analysis
of transformation models with censored data. Biometrika, 89,
659-668.
Efron, B. (1979). BootstrapMethods: Another Look at the Jackknife.
The Annals of Statistics, 7, pp.1-26.
Farewell VT. (1982). The use of mixture models for the analysis of
survival data with long-termsurvivors. Biometrics, 38, 1041-1046.
Finkelstein, D. M. and Wolfe, R. A. (1985). A semiparametricmodel
for regression analysis of interval-censored failure time data.
Biometrics, 41, 933-945.
Finkelstein, D.M. (1986). A proportional hazardmodel for interval-
censored failure time data. Biometrics, 42, 845-854.
Goedert, J., Kessler, C. Adedort, L. and et al. (1989). A prospective-
study of human immunodeficiency virus type-1 infection and the
development of AIDS in subjects with hemophilia. New England
Journal ofMedicine, 321, 1141-1148.
Huang, J. and Wellner, J. A. (1997). Interval censored survival
data: A review of recent progress. Proceedings of the First Seattle
Symposium in Biostatistics: Survival Analysis, eds. Lin, D, and
Fleming, T. Springer-Verlag, New York, 123-169.
Huang, J. and Rossini, A. J. (1997). Sieve estimation for the propor-
tional odds failure-time regressionmodelwith interval censoring.
Journal of the American Statistical Association, 92, 960-967.
Li, L. and Pu. Z. (1999). Regressionmodels with arbitrarily interval-
censored observations. Communications in Statistics. Theory
and Methods, 28, 1547-1563.
Lu, W. and Ying, Z. (2004). On semiparametric transformation cure
models. Biometrika, 91, 331-343.
Lam, K. F. and Xue, H. (2005). A semiparametric regression cure
model with current status data. Biometrika, 92, 573-586.
Ma, S. (2009). Cure model with current status data. Statist. Sinica
19, 233-249.
Ma, S. (2010). Mixed case interval censored data with a cured
subgroup, Statist. Sinica 20, 1165-1181.
Rabinowitz, D., Tsiatis, A. A. and Aragon, J. (1995). Regression with
interval-censored data. Biometrika, 82, 501-513.
Shen, X. (1998). Proportional odds regression and sieve maximum
likelihood estimation. Biometrika, 85, 165-177.
Sun, J. (2006). The statistical analysis of interval-censored failure
time data. Springer, New York.
Taylor, J. M. G. (1995). Semiparametric estimation in failure time
mixturemodels. Biometrics, 51, 899-907.
Woolson, R. F. (1981) Rank tests and one-sample log rank test for
comparing observed survival data to a standard population.
Biometrics, 37, 687-696.
Xue, H., Lam, K. F. and Li, G. (2004). Sevie maximum likelihood
estimation for semiparametric regression models with current
status data. Journal of the American Statistical Association, 99,
346-356.
Yu, A. K. F., Kwan, K. Y. W., Chan, D. H. Y. and Fong, D. Y. T. (2001).
Clinical features of 46 eyes with calcified hydrogel intraocular
lenses. Journal of Cataract and Refractive Surgery, 27, 1596-1606.
Zhang, Z., Sun, L., Zhao, X. and Sun, J. (2005). Regression analysis
of interval-censored failure time data with linear transformation
models. The Canadian Journal of Statistics, 33, 61-70.
林建甫(2008) 。存活分析。雙葉書廊, 台北。
許靖涵博士。生醫影像專題系列:EM 演算法。