The Effect of Different Forms of Centering in Hierarchical Linear Models

Multilevel models are becoming increasingly used in applied educational social and economic research for the analysis of hierarchically nested data. In these random coefficient regression models the parameters are allowed to differ over the groups in which the observations are nested. For computational ease in deriving parameter estimates, predictors are often centered around the mean. In nested or grouped data, the option of centering around the grand mean is extended with an option to center within groups or contexts. Both are statistically sound ways to improve parameter estimation. In this article we study the effects of these two different ways of centering, in comparison to the use of raw scores, on the parameter estimates in random coefficient models. The conclusion is that centering around the group mean amounts to fitting a different model from that obtained by centering around the grand mean or by using raw scores. The choice between the two options for centering can only be made on a theoretical basis. Based on this study, we conclude that centering rules valid for simple models, such as the fixed coefficients regression model, are no longer applicable to more complicated models, such as the random coefficient model. We think researchers should be made aware of the consequences of the choice of particular centering options.