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Centering decisions and equivalence in multilevel regression: A re-
examination of the issueGeorges Van Landeghem
K.U.Leuven
27 August 2010
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References
Kreft, I. G. G., de Leeuw, J., & Aiken, L. S. (1995). The effect of different forms of centering in hierarchical linear models. Multivariate Behavioral Research, 30(1), 1-21.
Dedrick, R. F., Ferron, J. M., Hess, M. R., Hogarty, K. Y., Kromrey, J. D., Lang, T. R., Niles, J. D., & Lee, R. S. (2009). Multilevel modeling: A review of methodological issues and applications. Review of Educational Research, 79(1), 69-102.
Enders, C. K., & Tofighi, D. (2007). Centering predictor variables in cross-sectional multilevel models: A new look at an old issue. Psychological Methods, 12(2), 121-138.
Selection
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“The issue of centering predictor variables in MLM has been discussed in the methodological literature () but it is still widely misunderstood by substantive researchers and methodologists alike.”
Enders and Tofighi (2007), p. 121
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Data preparation
• Reduction(items to scales, factor analysis, IRT-scores, …)
• Transformation(rotation, rescaling, centering, …)
Preparatory to linear regression:X1, X2, X1
2, X22, X1X2, X1
3, …
General
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Data preparation
• Two types of information:(1) Variable scores (response and explanatory)(2) Cluster membership information
Preparatory to linear regression:Xij, X.j, …
Multilevel data
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Centering options
[1 Xij-X.. X.j]“Centered around grand
mean” representation
[1 Xij-X.j X.j]“Centered within context”
representation
Example
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Recommendation
Dedrick et al. (2009) advise to “explicitly state whether centering was used , and if used, provide details on which variables were centered and how they were centered” (p. 96)
because“knowledge of centering decisions will aid in the interpretation of regression coefficients and variance estimates” (p. 96)
Reporting
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Centering and (non)equivalence
“The conclusion is that centering around the group mean amounts to fitting a different model from that obtained by centering around the grand mean ().” (Kreft, de Leeuw, & Aiken, 1995, p. 1)
“Although there are unique situations in which CGM and CWC produce equivalent estimates () this is the exception rather than the norm.” (Enders & Tofighi, 2007, p. 121)
The prevailing view
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Substantive considerations
“The decision to base centering decisions on empirical results is problematic because it should be based on the substantive question of interest. Kreft et al. (1995) underscored this point, stating that ‘There is no statistically correct choice among RAS [the raw metric], CGM, and CWC’ (p. 17).”
Enders and Tofighi, 2007, p. 122
… and centering options
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Centering and (non)equivalence
“Centered within context” view and “grand mean centered” parameterization are equivalent, i.e.:
(1) all the usual operations of multilevel regression analysis carried out in one representation have their counterpart in the other view;
(2) the result of an operation in one view can be translated directly into the result of its counterpart in the other view.
re-examined
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List of operations
Model specificationEstimation
FIML estimationREML estimationGLS estimationCovariance of fixed effects estimatesEmpirical Bayes estimation of cluster-level residuals
TestingDeviance testsF-tests, t-tests of linear combinations of fixed effects
Specification searches27 August 2010
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[1 Xij-X.. X.j] [1 Xij- X.j X.j]x T
x T-1
Operation in CGM view
Operation in CWC view
Result in CGM view
Result in CWC view
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Range of models
Multilevel regression model with
Any number of covariates;Any number of modes of
clustering (hierarchical or not);
Any covariance structure.
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A theoretical illustration‘CWC2’ model of Kreft et al. (1995)
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))(2(
22))(2(
12
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0
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MMUUY
In “centered within context” representation:
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A theoretical illustration‘CWC2’ model of Kreft et al. (1995)(continued)
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1))(2(
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In “centered around grand mean” representation:
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A theoretical illustration‘CWC2’ model of Kreft et al. (1995)(continued)
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Linking the “CWC” and “CGM” representations:
)(2
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ccg
cg
ccg M
))(2(22
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M
MM
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Recommendations
• Report centering options• Report to allow for
transformation of results to other representations
• Preferably look at model in multiple representations (especially any view with an interesting substantive interpretation)
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