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系統識別號 U0026-2708201716182600
論文名稱(中文) 右設限資料下檢定兩組母體中位數的經驗概似比方法
論文名稱(英文) Empirical Likelihood Ratio Test for Comparing Two Population Medians under Right-Censored Data
校院名稱 成功大學
系所名稱(中) 統計學系
系所名稱(英) Department of Statistics
學年度 105
學期 2
出版年 106
研究生(中文) 邱承德
研究生(英文) Chen-De Chiu
學號 R26031024
學位類別 碩士
語文別 中文
論文頁數 28頁
口試委員 指導教授-嵇允嬋
口試委員-黃錦輝
口試委員-陳嘉民
中文關鍵字 存活中位數檢定  經驗概似比檢定  自助法  Wald檢定 
英文關鍵字 median survival time  empirical likelihood ratio test  bootstrap  Wald test 
學科別分類
中文摘要 在完整資料下,已知有不少檢定兩母體中位數是否相等的無母數統計方法被提出,例如:Mann-Whitney檢定(1947)、Mood’s中位數檢定(1950)。然而上述方法仍有基本假設,即兩母體分布之形狀須相同(如變異數相同)。在完整資料且觀察值不重複時,Yu et al. (2011)提出以經驗概似函數建立的經驗概似比檢定法,來檢定兩母體中位數的相等性。本論文將經驗概似比檢定法,推廣至右設限資料下。此外,在右設限資料下,Tsai et al. (2016)以Wald檢定法搭配其所推估的變異數估計法,對兩母體存活中位數的相等性檢定有良好的表現。因此,本文亦依據其方法進行模擬比較。最後,本論文的模擬結果顯示,本論文所提出的方法在型一誤差率的表現不受樣本數大小影響,而 Wald 檢定法的表現不夠穩定。經驗概似比檢定法與 Wald 檢定法的檢定力大致接近,只有在母體為韋伯分布時,Wald 檢定法在任何樣本數皆大於經驗概似比檢定法,但仍相當接近。而且Wald 檢定法在變異數相當大時的檢定力表現較差。因此,本論文建議不論在何種母體分布及樣本數之下,皆推薦使用經驗概似比檢定法進行檢定。
英文摘要 Several non-parameter tests have been developed to compare the medians of two populations with complete data, such as Mann-Whitney test and Mood’s median test. However, these tests still need the assumption that the shape of the two population distributions has to be the same. As a result, Yu et al. (2011) proposed empirical likelihood ratio tests to get over this limitation with complete data. We extend their procedure for right-censored data. Moreover, the Wald test with variance estimator of sample median developed by Tsai et al. (2016) has well performance for testing the medians of two populations. Accordingly, we compare the Wald test with the extended empirical likelihood ratio test (ELR_RC) by a simulation study. The type I error rates of ELR_RC are reasonable for both small and large sample sizes; on the other hand, the type I error rates of Wald test are not within reasonable range. The power of the two methods are similar, but Wald test has worse performance under large variance of the populations. Therefore, we recommend to use ELR_RC for comparing two population medians with right-censored data.
論文目次 第一章 緒論............................1
第二章 文獻回顧........................3
第一節 符號定義........................3
第二節 經驗概似比檢定...................4
第三節 Wald 檢定.......................6
第四節 自助法..........................8
第三章 經驗概似比檢定...................9
第一節 右設限資料下的經驗概似函數........9
第二節 最大經驗概似估計量之推導..........10
第四章 模擬比較.........................14
第一節 模擬設計.........................14
第二節 模擬結果.........................15
第五章 實例分析.........................19
第六章 結論.............................21
參考文獻.................................22
附錄一 .................................23
附錄二 .................................26
附錄三 .................................28
表4.1.1 樣本數設定........................14
表4.2.1 樣本數相同,各存活時間分布之型一誤差率...16
表4.2.2 樣本數相異,各存活時間分布之型一誤差率...16
表4.2.3 樣本數相同,各存活時間分布之檢定力..18
表4.2.4 樣本數相異,各存活時間分布之檢定力..18
圖5.1 四種腫瘤類型之存活函數...............20
參考文獻 Brookmeyer, R. and Crowley, J. (1982). A confidence interval for the median survival time. Biometrika, 38, 29-41.
Chen, Y. Y. (2015). Comparison of several statistical tests for comparing two population medians. Master Thesis, Department of Statistics, National Cheng-Kung University.
Efron, B. (1981). Censored data and the bootstrap. Journal of the American Statistical Association, 76, 312-319.
Greenwood, M.(1926). The natural duration of cancer. Reports on Public Health and Medical Subjects, 33, 1-26. London:HMSO.
Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53, 457-481
Li, P. S. (2014). A k-sample median test based on the length of confidence interval with the right-censored data. Master Thesis, Department of Statistics, National Cheng-Kung University.
Sander, J. M. (1975). The weak convergence of quantiles of the product-limit estimator. Stanford University Technical Reports, No.5.
Tsai, T. H., Tsai, W. Y., Chi, Y. C. and Chang, S. M.(2014). Estimation of the ratio of two median residual lifetimes with left-truncated and right-censored data. Technical report, Department of Statistics, National Cheng-Kung University.
Yu, J., Vexler, A., Kim, S. E. and Hutson, A. D.(2011). Two-sample empirical likelihood ratio tests for medians in application to biomarker evaluations. The Canadian Journal of Statistics, 39, 671-689.
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