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Kelley et al. argue that group-mean-centering covariates in multilevel models is dangerous, since—they claim—it generates results that are biased and misleading. We argue instead that what is dangerous is Kelley et al.’s unjustified assault on a simple statistical procedure that is enormously helpful, if not vital, in analyses of multilevel data. Kelley et al.’s arguments appear to be based on a faulty algebraic operation, and on a simplistic argument that parameter estimates from models with mean-centered covariates must be wrong merely because they are different than those from models with uncentered covariates. They also fail to explain why researchers should dispense with mean-centering when it is central to the estimation of fixed effects models—a common alternative approach to the analysis of clustered data, albeit one increasingly incorporated within a random effects framework. Group-mean-centering is, in short, no more dangerous than any other statistical procedure, and should remain a normal part of multilevel data analyses where it can be judiciously employed to good effect.

Group-mean centering of independent variables in multi-level models is widely practiced and widely recommended. For example, in cross-national studies of educational performance, family background is scored as a deviation from the country mean for student’s family background. We argue that this is usually a serious mis-specification, introducing bias and random measurement error with all their attendant vices. We examine five diverse examples of “real world” analyses using large, high quality datasets on topics of broad interest in the social sciences. In all of them, consistent with much (but not all) of the technical literature, group-mean centering substantially distorts results. Moreover the distortions are large, substantively important differences pointing towards seriously incorrect interpretations of important social processes. We therefore recommend that group-mean centering be abandoned.

利用多層次模式或是階層線性模式進行重複觀測資料的分析，如果個體層次解釋變項包含隨時間變動解釋變項時，在個體層次方程式對它不平減或是總平減所獲得的迴歸係數是一個偏誤的結果，因為這個隨時間變動的解釋變項具有追蹤與橫斷面的資料特性，對個體層次結果變項的影響可以拆解為互斥的組間迴歸係數與組內迴歸係數，因此，必須利用組平減並將組平均數置回截距項方程式方能獲得正確的估計結果。但在不平減、總平減與組平減三種方法下都加上組平均數置回截距項方程式，在隨機截距模型下則會獲得等價的估計結果。本研究整理出這些平減方法之間的統計關係，並利用實徵資料示範分析各種模式，說明之間的差異與等價關係，最後提出研究的結論與建議。 When analyzing repeated measures by using multilevel modeling (MLM) or hierarchical linear modeling (HLM), if the individual-level independent variables include a time-varying variable and it is modeled as uncentered or grand-mean centered in a level-one equation, then this regression coefficient is a biased estimate. Because repeated measures data comprise longitudinal and cross-sectional parts, the total effect of the time-varying independent variable on the individual outcomes can be decomposed into within- and between-subject regression coefficients. Therefore, the optimal approach is to use group-mean centered in a level-one equation and group means replaced in the intercept equation. In some cases (e.g., the random intercepts model), the three methods, namely uncentered, grand-mean centered, and group-mean centered time-varying variable approaches with group means replacement, are equivalent in MLM and HLM. We adopted a statistical model and empirical data analysis to determine the equivalent relationships and differences among the three centered methods and present a conclusion and recommendations.

利用多層次模式或是階層線性模式進行重複觀測資料的分析，如果個體層次解釋變項包含隨時間變動解釋變項時，在個體層次方程式對它不平減或是總平減所獲得的迴歸係數是一個偏誤的結果，因為這個隨時間變動的解釋變項具有追蹤與橫斷面的資料特性，對個體層次結果變項的影響可以拆解為互斥的組間迴歸係數與組內迴歸係數，因此，必須利用組平減並將組平均數置回截距項方程式方能獲得正確的估計結果。但在不平減、總平減與組平減三種方法下都加上組平均數置回截距項方程式，在隨機截距模型下則會獲得等價的估計結果。本研究整理出這些平減方法之間的統計關係，並利用實徵資料示範分析各種模式，說明之間的差異與等價關係，最後提出研究的結論與建議

When analyzing repeated measures by using multilevel modeling (MLM) or hierarchical linear modeling (HLM), if the individual-level independent variables include a time-varying variable and it is modeled as uncentered or grand-mean centered in a level-one equation, then this regression coefficient is a biased estimate. Because repeated measures data comprise longitudinal and cross-sectional parts, the total effect of the time-varying independent variable on the individual outcomes can be decomposed into within- and between-subject regression coefficients. Therefore, the optimal approach is to use group-mean centered in a level-one equation and group means replaced in the intercept equation. In some cases (e.g., the random intercepts model), the three methods, namely uncentered, grand-mean centered, and group-mean centered time-varying variable approaches with group means replacement, are equivalent in MLM and HLM. We adopted a statistical model and empirical data analysis to determine the equivalent relationsh