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研究生:江珮鳳
研究生(外文):Chiang,Pei-Feng  
論文名稱:檢驗變異數分析模型及混合模型之常態假設
論文名稱(外文):Assessing normality assumptions under the ANOVA model and mixed model
指導教授:黃怡婷黃怡婷引用關係汪群超
學位類別:碩士
校院名稱:國立臺北大學
系所名稱:統計學系
學門:數學及統計學門
學類:統計學類
論文種類:學術論文
論文出版年:2007
畢業學年度:95
語文別:中文
論文頁數:63
中文關鍵詞:變異數分析檢驗常態分配混合模型資料轉換
外文關鍵詞:ANOVA modelexamine normality assumptionmixed modeltransforming
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變異數分析模型為統計模型中最常被用來檢驗組別間是否有差異的方法,模型假設資料來自常態分配。欲正確使用ANOVA模型,常態分配之假設需被確認。要正確的判定組間是否有差異,必須檢驗資料是否服從常態分配。在固定效應模型且組內樣本較大時, Bonett 及 Woodward (1990) 透過蒙地卡羅模擬,發現使用偏態及峰態係數檢定殘差可以有效檢驗模型之常態假設。在單一因子ANOVA模型的架構下, Shapiro 及 Wilk (1968), Pettitt (1977) 建議利用單變量常態檢定先檢驗各組的常態假設,再將各組檢驗結果合併以檢驗總樣本是否來自常態。利用條件機率積分轉換法, Quesenberry 及 Giesbrecht (1983) 將原樣本進行轉換,若資料服從常態,則轉換後之資料會服從均勻分配。現有之檢驗方式均在樣本較大時可以有很好的檢定結果,但小樣本的結果則未知。再者,在混合模型的架構下,樣本間之獨立性並不存在,要檢驗模型之常態假設就更困難。本論文在單因子變異數分析模型及小樣本的架構下,檢驗現有之檢測方式是否可以有效地驗證常態的假設,在混合模型中則利用 Hwang 及 Wei (2006) 所提的轉換方式來解決資料相依的問題,再利用模擬來檢驗現有常態檢測方式是否可有效檢測轉換後資料之常態分配。
The ANOVA model is the most commonly used method to examine whether the difference between groups exists. To distinguish the difference is statistically significant, the normal assumption is required. Thus, the normal assumption has to be validate accordingly in order to use the model legitimately. Bonett and Woodward (1990) indicated that using the skewness and kurtosis tests to examine the normal assumption is sufficient under the large sample. Under a one-way ANOVA model, normality assumption for each group by the univariate normality test and combining the test results for each group to make an overall inference about the normal assumption for the entire sample. Quesenberry and Giesbrecht (1983) suggested transforming the original data by the conditional probability integral transformation into a uniform sample and performed the uniformity test to make an overall inference. The ability of the current method to examine the normality assumption is validated only for the large sample. In practice, the small sample is encountered very often.
In addiation, under the mixed model, the normality assumption is also imposed. However, since the sample is not independent, it is more difficulty to examine the normality assumption. The first objective of this thesis is to validate whether the existing method can provide decent performance under the small sample for the ANOVA model.The second objective is to propose a way to examine the normality assumption under the mixed model.
第一章 緒論
第二章 固定效應變異數分析模型之常態檢定
第三章 混合模型之常態檢定
第四章 模擬
第五章 結論
Bonett, D. G. and Woodward, J. A. (1990). Testing residual normality in the ANOVA model. Journal of Applied Statistics, 17, 383-387.
Bowman, K. O. and Shenton, L. R. (1975). Omnibus contours for departures from normality based on sqrt{b_1} and b_2. Biometrika, 62, 243-250.
Lilliefors, H. W. (1967). On the Kolmogorov-Smirnov test for normality with mean and variance unknown. American statistical association journal, 62, 399-402.
Hwang, Y. T. and Wei, P. F. (2006). A novel method for testing normality in a mixed model of a nested classification. Journal of Applied Statistics, 51, 1163-1183.
Hwang, Y. T. and Wei, P. F. (2007). A remark on Zhang omnibus test for normality based on the $Q$ statistic. Computational Statistics and Data Analysis, 34, 177-184.
Weisberg, S and Bingham, C. (1975). An approximate analysis of variance test for non-normality suitable for machine calculation. Technometrics, 17, 133-134.
White, H. and Macdonald, G. M. (1980). Some large-sample tests for normality in the linear regression model. Journal of the American Statistic Association, 75, 16-30.
Wilk, M. B. and Shapiro, S. S. (1965). An analysis of variance test for normality. Biometrika, 52, 591-611.
Wilk, M. B. and Shapiro, S. S. (1968). The joint assessment of normality of normality of several independent samples. Technometrics, 10, 825-839.
O'Reilly, F. J. and Quesenberry, C. P. (1973). The conditional probability integral transformation and applications to obtain composite chi-square goodness-of-fits tests. The Annals of Statistics, 1, 74-83.
Pettitt, A. N. (1977). Testing the normality on several independent samples using the Anderson-Darling statistic. Appl. Statist., 26, 156-161.
Quesenberry, C. P. and Giesbrecht, F. G. (1983). Some methods for studying the validity of normal model assumptions for multiple samples. Biometrics, 39, 735-739.
Royston, J. P. (1982). An extension of Shapiro and Wilk's W test for normality to large samples. Appl. Statist., 31, 115-124.
Zang, P. (1999). Omnibus test of normality using the Q statistic. Journal of Appl. Statist., 26, 519-528.
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