變項中心化與多元共線性的玄機

Variable Mean-Centering and Multicollinearity

李茂能(Mao-Neng Li)

《測驗統計年刊》 20期_上 (2012/06) Pp. 25-52

https://doi.org/10.6773/JRMS.201206.0025

摘要

變項中心化常可令截距的解釋具有實質意義與減輕多元共線性的威脅，但對於交互作用項中心化的議題則尚有爭論。過去研究發現，有時交互作用項的中心化並無法有效改善多元共線性現象。爲探究此矛盾現象，本文乃透過理論的推導與實徵資料的分析，發現迴歸係數變異量涉及三大要素：估計誤差變異量、預測變項之變異量、變項間之相關性。交互作用項會出現多元共線性現象，乃是這三大主因交織而成，並非單一因素所致。過去對於多元共線性的研究，常侷限於變項間之相關性上，因而導致研究結論不一致的現象。多元共線性的診斷不能單靠VIF/Tolerance 指標，其它指標如CI/Variance proportion/Eigenvalue指標亦須同時納入診斷之，才能正確診斷出來。文中除公式推導之外，並以模擬實例闡釋了VIF/Tolerance指標旨在反映預測變項間之相關程度，而CI/Variance proportion/Eigenvalue指標則在反映預測變項的離散量，這兩類指標反映多元共線性的不同面向的資料結構。文中並提出預測變項「變異比差異」理論說明，為何變項中心化有時可以改善多元共線性的威脅，但有時卻不能改善的矛盾現象。

關鍵字

變項中心化 ； 多元共線性 ； 交互作用項 ； 變異比

並列摘要

To reduce multicollinearity and to make the interpretation of the intercept meaningful, it is frequently recommended that an interaction variable be based on mean-centered variables. Previous studies however have found that this approach may be futile. The variance of a regression coefficient is affected by three factors: error variance, variance of the predicting variables, and the correlation between predictors.Multicollinearity for the interaction term can only be detected correctly by analyzing simultaneously each of these three inter-related factors. Previous researchers have only focused on the correlation between predictors by relying on the VIF/Tolerance index to detect multicollinearity. Consequently, inconsistent findings regarding the merits of mean-centering have been reported. Using an illustrated example, we show that the Condition/Variance proportion Index which reflects the variance of the predictors in addition to VIF be examined to effectively detect multicollinearity.Interestingly, mean-centering the predicting variables for reducing the threat of multicollinearity can be valid only under the condition that there is not a significant increased difference of variance ratio (SD/Mean) especially between the variance ratio of the pre-centering interaction variable and the variance ratio of the post-centering interaction term.

並列關鍵字

Mean-Centering ； Multicollinearity ； Interaction Term ； Variance Ratio

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