臺灣博碩士論文加值系統

共變數分析(analysis of covariance)為結合變異數分析(analysis of variance)及迴歸分析(regression)的統計方法，其目的為控制某共變數水準下，檢定處理方法對反應變數是否有差異，此模型假設包含(a)每組處理方法的共變數對反應變數的影響皆為相同即各組迴歸線斜率相同(b) 處理方法間變異數同質性(c)各處理方法之共變數期望值相同(d)誤差項的獨立性與常態性(e)共變數與反應變數為線性關係。
在實務上，經常會遭遇到與模型假設不一致的情形，例如處理方法與共變數間有交互作用，產生各組處理方法的迴歸線並不平行，或處理方法間的變異數並不一致，導致傳統共變數分析模型之$F$檢定表現不佳。
本論文將討論放寬傳統共變數分析的假設(a)各組迴歸線斜率相同，(b)組間變異數同質性(c)各處理方法之共變數期望值相同，針對放寬此三項假設，且參考Neter (1999)之推導方式，導出適當的統計量及其分配，進而檢定兩處理方法反應變數平均值是否有差異。
再者，除了放寬共變數分析的假設，本論文進一步將共變數分析加入隨機效應(random effect)，即將模型由共變數分析推廣至混合模型( mixed model )，此研究的目的與共變數分析相同，考慮混合模型下，推導出適當的統計量及其分配，來檢測處理方法對反應變數是否有差異。所有統計量的表現均利用蒙地卡羅法來衡量。

The analysis of covariance is a statistical methodology that combines features of the analysis of variance and the regression. Its purpose is to test if there is a significant mean difference in the outcome between two treatments. To use ANCOVA, some statistical assumptions are needed. If one of model's assumptions violates, the $F$ test is then not appropriate.
The purpose of the thesis is to release certain assumptions in the ANCOVA model including: (a) constancy of slopes (b) homogeneity (c) equality of covariate means for different treatments. Derivations of new proposed tests are based on the method that Neter(1999) derived tests for simple regression model adding a covariate.
In addition to release model assumptions, a random effect is added into the ANCOVA model, which is a special case in the mixed model .Finally, the Monte Carlo simulations are used to evaluate the performance of the Type I error and power of the proposed tests.

1緒論 ………………………………………………………………………1

 1.1研究背景與動機 ……………………………………………………1
1.2研究目的 ……………………………………………………………5

2研究方法 …………………………………………………………………6

2.1共變數分析 …………………………………………………………6
2.1.1推廣共變數分析一 …………………………………………9
2.1.2推廣共變數分析二 …………………………………………17

2.2混合模型 ……………………………………………………………25

2.2.1反應變數之共變異矩陣中V為已知 ………………………30
2.2.2反應變數之共變異矩陣中V為未知 ………………………36

3蒙地卡羅模擬 ……………………………………………………………38

3.1推廣共變數分析的模擬結果 ………………………………………40
3.2混合模型的模擬結果 ………………………………………………43

4結論與建議 ………………………………………………………………45

5參考文獻 ……………………………………………………………………47

1.王昱昇 (民93). 推廣共變數模型兩獨立樣本T檢定. 國立台北大學統計學系碩士論文.
2.范德鑫 (民81). 共變數分析功能、假設及使用之限制. 師大學報 37, 133-163.
3.Casella G.. & Berger R. L. (2001). Statistical inference 2nd ed. Duxbury advanced series, CA.
4.Diggle, P. J.(1998). Analysis of longitudinal Data. Oxford University Press Inc., New York.
5.Fleiss, J. L. (1999). The design and analysis of clinical experiments. Wiley series in probability and mathematical statistics, New York.
6.Neter, J. (1999). Applied linear statistical models 4th ed. McGraw-Hill series in bussiness statistics, New York.
7.Potthoff, R. F. (1964). On the Johnson-Neyman technique and some extensions thereof. Psychometrika, 15, 241-256.
8.Rencher, A. C. (2000). Linear models in statistics.Wiley series in probability and statistics, New York.
9.Rogosa, D. (1980). Comparing nonparallel regression lines. Psychological Bulletin, 88, 307-321.
10.Wang & Chow (2002) A practical approach for comparing means of two groups without equal variance assumption. Statistics in medicine, 21, 3137-3151.