本論文考慮了比例連續比模型下離散時間區間設限數據的最大概似估計。我們採用R軟體中“optim”和“ hessian”函數來計算最大概似估計和可觀測的費雪信息矩陣。R套件“ discSurv”和哥本哈根中風研究的右設限資料驗證了計算在右設限情形下的正確性。模擬試驗評量了在一般區間設限情形下估計的數值表現，而所提計算方法與乳腺癌研究的區間設限資料則例說了實際應用。

In this thesis, we consider the maximum likelihood estimation of discrete time interval censored data under the proportional continuous ratio model. We employ the ‘optim’ and ‘hessian’ routines in the R environment to compute the estimator and the observed information matrix. The correctness of the computation under right censoring setting is validated with R package ‘discSurv’ and a right censored Copenhagen Stroke example. The numerical performance of the estimator under general interval censoring setting is examined by simulations and the real application is illustrated with our proposal and an interval censored breast cancer example.

一、前言-----------------------------1
二、符號與模型介紹-------------------4
三、在右設限下與 discSurv 的比較-----8
四、模擬----------------------------12
五、實例分析------------------------18
六、討論----------------------------21
七、參考文獻------------------------22
八、附錄----------------------------24

[1]Byrd, R. H.,& Lu, P.,& Nocedal, J. & Zhu, C. (1995). A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing, 16, 1190--1208. 10.1137/0916069. 
[2]Efron, B. (1988). Logistic regression, survival analysis, and the Kaplan-Meier-curve. Journal of the American Statistical Association, 83, 414–425.
[3]Finkelstein, D. M. & Wolfe, R.A (1985).A proportional hazards model for interval-censored failure time data. Biometrics, 42, 845–854.
[4]Hammer, J.C, &  Fisher, J.D, & Fitzgerald, P, et al.(1996) When two heads aren’t better than one: AIDS risk behavior in clllege-age couples. J Appl Soc Psychol 26:275-397.
[5]Jorgensen, H.S, & Nakayama, H., & Raaschou, HO, Olsen TS.(1996) Stroke in patients with diabetes: the Copenhagen Stroke Study. Stroke.; 25:1977-1984.
[6]Mantel, N., & Hankey, B. F. (1978). A logistic regression analysis of response timedata where the hazard function is time dependent. Communications in Statistics – Theory and Methods, A7, 333–347.
[7]Schmid, M. & Tutz, G. &  Welchowski, T. (2018) Discrimination measures for discrete time-to-event predictions. Econom. Stat. 7, 153–164. 62N01 (62G05 62M20)
[8]Scheike, T., & Keiding, N. (2006). Design and analysis of time-to-pregnancy. Statistical Methods in Medical Research, 15, 127–140.
[9]Schmid, M., &  Küchenhoff, H., & Hoerauf, A., & Tutz, G. (2016). A survival tree method for the analysis of discrete event times in clinical and epidemiological studies. Statistics in Medicine, 35, 734–751.
[10]Tutz, G., & Schmid, M.(2016) Modeling discrete time-to-event data. Springer Series in Statistics. Springer, [Cham], 
[11]Thompson, W.A.(1977). Onthetreatment of grouped observations inlifestudies. Biometrics, 33, 463–470.