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研究生:孫德宇
研究生(外文):De-Yu Sun
論文名稱:單、雙變數區間刪減資料的廣義秩檢定
論文名稱(外文):Generalized rank tests for univariate and bivariate interval-censored failure time data
指導教授:黎進三黎進三引用關係
指導教授(外文):Chin-San Lee
學位類別:碩士
校院名稱:國立中山大學
系所名稱:應用數學系研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2003
畢業學年度:91
語文別:中文
論文頁數:117
中文關鍵詞:雙變數區間刪減資料無母數檢定模擬Turnbull演算法單變數區間刪減及截斷資料
外文關鍵詞:Turnbull''s algorithmunivariate interval-censored and truncated failunon-parametric testsbivariate interval-censored failure time datasimulation
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在這篇論文的第一部份,對於單變數區間刪減及截斷資
料情形下的存活時間分布,我們改寫Turnbull的演算法以
估計這種數據的分布函數,並提出四種無母數檢定,來判
斷兩個存活時間的母體是否來自相同的分布,再由模擬的
結果去比較所提出的統計量在不同分布情形下的檢定力。
此外,我們用所提出的方法分析一個AIDS的研究。

在第二部份中,關於雙變數區間刪減情形下的存活時間
分布,首先我們想要模擬出這種雙變數區間刪減的數據,
並定義兩個變數之間的相關程度。接著提出幾種無母數秩
檢定,來判斷雙變數區間刪減資料的兩個變數其分布是否
相互獨立並且是否相同,並用模擬的結果來顯示所提出的
統計量的檢定力。我們亦將所提出的方法應用到一個
AIDS的研究(ACTG 181)。
In Part 1 of this paper, we adapt Turnbull’s algorithm to estimate the distribution function of univariate interval-censored and truncated
failure time data. We also propose four non-parametric tests to test whether two groups of the data come from the same distribution. The
powers of proposed test statistics are compared by simulation under different distributions. The proposed tests are then used to analyze an AIDS study.

In Part 2, for bivariate interval-censored data, we propose some models of how to generate the data and several methods to measure the
correlation between the two variates. We also propose several nonparametric tests to determine whether the two variates are mutually independent or whether they have the same distribution. We demonstrate the performance of these tests by simulation and give an application to AIDS study(ACTG 181).
PART 1. Univariate interval-censored and truncated data
1. Introduction……………………………………….………………….1
2. Adapted Turnbull’s algorithm…………………………….…………3
3. Urn model to select data………………………...……………………6
3.1. The selection of interval-censored data…………..…...…………..6
3.2. The selection of interval-censored and truncated data……..…....10
4.The proposed four non-parametric tests…………………..…...….13
4.1. The multinomial test…..…………….…………………………...14
4.2. The modified Sun’s rank test………...……….………………….16
4.3. The generalized weighted rank test……..….…………………....18
4.4. The generalized Mann-Whitney test……………..……...……….20
5. Simulation result.………………………….………………………...24
5.1. Exponential distribution…………………………...…………….26
5.2. Weibull distribution…………………………………..………….31
5.3. Extreme-value distribution…………………...………………….36
5.4. Logistic distribution………………...……………………………41
5.5. Conclusion of the simulation………………………………...…..46
6. Application to an AIDS study……..……………..…..……………..48
7. Reference of Part 1………………………………….………………51

PART 2. Bivariate interval-censored data
1. Introduction………………………..…………….…………...……..52
2. Generalized Turnbull’s algorithm……………...…………………..55
3. Selection of bivariate interval-censored data……..……...………..57
3.1. A 3×3 bivariate urn model……..…………...………..…………..57
3.2. A 4×3 bivariate urn model…...…..………………………..…..…67
3.3. A modified bivariate urn model……….…..……………80
3.4. A selection model from conditional probability density function…
…………………………....………..………………....…………90
4. Tests of independence for the two components of the bivariate
interval-censored data……….….…………….……….....…...…...95
4.1. Generalized Pearson’s correlation coefficient.…………….....….95
4.2. An extension of Spearman’s rho………..……...…………...……97
4.3. An extension of Kendall’s tau……………...….…..…..……….100
4.4. Generalized Wilcoxon signed rank test…………...……………103
4.5. Simulation result for independence……..……..………...……..106
5. Tests of whether the two components have the same distribution
in a bivariate interval-censored data……...…….……………….108
5.1. Tests under independence or not……….…..…………….…….108
5.2. Simulation result for identical distributions…….…….………..111
6. Application to an AIDS study(ACTG 181)……...……..……..113
7. Reference of Part 2………………………………..……….………117
Reference of Part 1
[1] De Gruttola, V. and Lagakos, S. (1989):Analysis of doubly-censored
survival data, with application to AIDS. Biometrics. Vol. 45, 1-11.
[2] Turnbull, B.W. (1974):Nonparametric estimation of a surviorship
function with doubly censored data. J. Amer. Statist. Ass. Vol. 69,
169-173.
[3] Turnbull, B.W. (1976):The empirical distribution function with
arbitrarily grouped, cnesored, and truncated data. J. of the Royal
ststistical Society, Series B. Vol. 38, 290-295.
[4] Sun, J. (1995):Empirical estimation of a distribution function with
truncated and doubly interval-censored data and its application to
AIDS studies. Biometrics. Vol. 51, 1096-1104.
[5] Sun, J. (1996):A non-parametric test for interval-censored failure
time data with application to AIDS studies. Statistics in Medicine.
Vol. 15, 1387-1395.
[6] Hung-Yen Hsu (1999):The distribution of a non-parametric test for
interval failure time data. Master of science.
[7] Horng-Huey Luh (1999):The distribution of a non-parametric test
for interval-censored and truncated failure time data. Master of
science.
[8] Yu-Yu Kuo (2000):A generalization of rank tests based on interval-
censored failure time data and its application to AIDS studies.
Master of science.
[9] Ching-Fu Sen. (2001):On the consistency of a simulation procedure
and the construction of a non-parametric test for interval-censored
data. Master of science.
[10]Chinsan, Lee. (1999):An urn model in the simulation of interval
censored failure time data. Statistics & Probability Letters. Vol. 45,
131-139.

Reference of Part 2
[11]W. J. Conover (1999):Practical nonparametric statistics. Wiley.
[12]Lin, D. Y. and Ying, Z. (1993):A simple nonparametric estimator
of the bivariate survival function under univariate censoring.
Biometrika. Vol. 80, 573-581.
[13]Dabrowska, D. M. (1986):Rank tests for independence for bivariate
censored data. Annals of Statistics. Vol. 14, 250-264.
[14]Oakes, D. (1982):A concordance test for independence in the
presence of censoring. Biometrics. Vol. 38, 451-455.
[15]Michael G. Akritas and Clifford C. Clogg (1991):Tests of indepen-
dence for bivariate data with random censoring: a contingency-table
approach. Biometrics. Vol. 47, 1339-1354.
[16]Betensky, R. A. and Finkelstein, D. M. (1999):A non-parametric
maximum likelihood estimator for bivariate interval censored data.
Statistics in Medicine. Vol. 18, 3089-3100.
[17]Betensky, R. A. and Finkelstein, D. M. (1999):An extension of
Kendall’s coefficient of concordance to bivariate interval censored
data. Statistics in Medicine. Vol. 18, 3101-3109.
[18]Shih, J. H. and Louis, T. A. (1996):Tests of independence for
bivariate survival data. Biometrics. Vol. 52, 1440-1449.
[19]Hsu, L. and Prentice, R. L. (1996):A generalization of the Mantel-
Haenszel test to bivariate failure time data. Biometrika. Vol. 83, 905-
911.
[20]Kevin J. Hastings (1997):Probability and statistics. Addison-Wesley.
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