違反迴歸斜率同質性之假定對共變數分析強?度之影響

The Effects of Violating the Homogeneous Regression Slopes Assumption on the Robustness of the Analysis of Covariance

共變數分析（ANCOVA）是一種經常使用於教育和心理學研究上的統計方法。在某些情況下，這種經由統計法則來控制實驗者不能直接控制之變異的方法要比使用變異數分析（ANCOVA）更為恰當。然而使用共變數分析時，除了必須滿足變異數分析所應具備的假定外，仍須附帶其他假定。其中最重要的一項是同質迴歸斜率之假定，即依變項Y在共變項X上之直線迴歸情形不受組別影響。換言之，各組中的迴歸係數都應該一樣。毫無疑問，在實際情況中這種假定無法全然成立。因此便產生名義（nominal ）與實際（actual ）間差異的問題，亦即一般所謂強靭度（robustness）的問題。研究者在使用ANCOVA時必須特別注意在某些情況下雖然違反了同質迴歸的假設仍不會嚴重地影響使用該統計法之強靭度。只要實際與名義相差未達顯著水準，雖然違背某些基本假定的現象發生，仍可採用該統計方法。 本研究針對ANCOVA強靭度採文獻探討方式進行研究，結果發現： 1.若各組迴歸斜率差異並非很大，則ANCOVA具備強靭度。 2.若各組迴歸斜率不同而且各組樣本大小、誤差變異不等的現象同時發生，則ANCOVA無法具備強靭度。 3.當兩個標準化迴歸係數的差異大於0.4且樣本大小一致，則採用個別斜率來計算考驗統計數的方法要比利用傳統方法只採用一個組內迴歸係數更具強靭度。 4.當兩個歸係數的差異超過0.4又發生樣本大小不一，誤差變異不等之現象，則不管用同一組內迴歸係數或以個別迴歸係數來處理，其結果都使ANCOVA無法達到其應有之強靭度。 5.當兩個標準化迴歸斜率差異小於0.4，不論使用那一種模式處理迴歸係數均使ANCOVA具強靭度。

The analysis of covariance (ANCOVA) is a frequently used statistical approach in educational and psychological research. It provides a method of achieving increased precision over the corresponding analysis of variance (ANCOVA) by employing statistical control of one or more of the sources of variation not directly controlled by the experimenters. However， the use of ANCOVA requires additional assumptions beyond analysis of variance. One important additional assumption is that the linear regression of dependent variable Y on covariate X is the same for all treatment groups; i.e.， homogeneous regression slopes. Exact satisfaction of this assumption is undoubtedly impossible in reality. Thus， researchers who use ANCOVA should be aware to what extent the assumption of homogeneous regression may be violated without seriously affection the interpreting of the results. This study reviewed the literature on the robustness of the ANCOVA and found: 1.ANCOVA is robust if the inequality of regression slopes is not extreme. 2.ANCOVA is not robust if the inequality of regression slopes is associated with unequal sample sizes and unequal error variance. 3.When the difference of two standardized regression slopes exceeds 0.4 and the sample sizes are equal， the model which incorporates estimates of individual regression slopes is more robust than the conventional model which uses a pooled within regression coefficient. 4.When the difference between slopes exceeds 0.4 and unequal sample sizes are associated with unequal error variance， neither of the models are robust. 5.When the two standardized regression slopes differ less than 0.4， both models are robust.