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Centering decisions and equivalence in multilevel regression : A re-examination of the issue

Centering decisions and equivalence in multilevel regression : A re-examination of the issue. Georges Van Landeghem K.U.Leuven Georges.VanLandeghem @ ped.kuleuven.be. References.

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Centering decisions and equivalence in multilevel regression : A re-examination of the issue

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  1. Centeringdecisions and equivalence in multilevelregression: A re-examination of the issue Georges Van Landeghem K.U.Leuven Georges.VanLandeghem@ped.kuleuven.be EARLI SigEE

  2. References Kreft, I. G. G., de Leeuw, J., & Aiken, L. S. (1995). The effect of different forms of centering in hierarchicallinear models. MultivariateBehavioral 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). Multilevelmodeling: 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-sectionalmultilevel models: A new look at anold issue. PsychologicalMethods, 12(2), 121-138. Selection EARLI SigEE

  3. “The issue of centering predictor variables in MLM has been discussed in the methodologicalliterature () butit is stillwidelymisunderstoodbysubstantiveresearchers and methodologistsalike.” Enders and Tofighi(2007), p. 121 EARLI SigEE

  4. Data preparation • Reduction (items to scales, factor analysis, IRT-scores, …) • Transformation (rotation, rescaling, centering, …) Preparatory to linearregression: X1, X2, X12, X22, X1X2, X13, … General EARLI SigEE

  5. Data preparation • Two types of information: (1) Variable scores (response and explanatory) (2) Cluster membershipinformation Preparatory to linearregression: Xij, X.j, … Multilevel data EARLI SigEE

  6. Centeringoptions [1 Xij-X..X.j] “Centeredaround grand mean” representation [1 Xij-X.jX.j] “Centeredwithin context” representation Example EARLI SigEE

  7. Recommendation Dedrick et al. (2009) advise to “explicitly state whethercentering was used , and ifused, provide details onwhich variables werecentered and howtheywerecentered” (p. 96) because “knowledge of centeringdecisionswillaid in the interpretation of regressioncoefficients and varianceestimates” (p. 96) Reporting EARLI SigEE

  8. Centering and (non)equivalence “The conclusion is thatcenteringaround the groupmeanamounts to fitting a different model fromthatobtainedbycenteringaround the grand mean ().” (Kreft, de Leeuw, & Aiken, 1995, p. 1) “Althoughthere are uniquesituations in which CGM and CWC produce equivalent estimates () this is the exceptionratherthan the norm.” (Enders & Tofighi, 2007, p. 121) The prevailing view EARLI SigEE

  9. Substantiveconsiderations “The decision to base centeringdecisionsonempiricalresults is problematicbecauseitshouldbebasedon the substantivequestion of interest. Kreft et al. (1995) underscoredthis point, statingthat ‘There is nostatistically correct choiceamong RAS [the rawmetric], CGM, and CWC’ (p. 17).” Enders and Tofighi, 2007, p. 122 … and centeringoptions EARLI SigEE

  10. Centering and (non)equivalence “Centeredwithin context” view and “grand meancentered” parameterization are equivalent, i.e.: • all the usualoperations of multilevelregressionanalysiscarried out in onerepresentation have their counterpart in the other view; • the result of anoperation in one view canbetranslateddirectlyinto the result of its counterpart in the other view. re-examined EARLI SigEE

  11. List of operations Model specification Estimation FIML estimation REML estimation GLS estimation Covariance of fixedeffectsestimates EmpiricalBayesestimation of cluster-levelresiduals Testing Deviance tests F-tests, t-tests of linearcombinations of fixedeffects Specificationsearches EARLI SigEE

  12. x T [1 Xij-X..X.j] [1 Xij- X.jX.j] x T-1 Operation in CGM view Operation in CWC view Result in CGM view Result in CWC view EARLI SigEE

  13. Range of models Multilevelregression model with Anynumber of covariates; Anynumber of modes of clustering (hierarchicalornot); Anycovariancestructure. EARLI SigEE

  14. A theoreticalillustration In “centered within context” representation: ‘CWC2’ model of Kreft et al. (1995) EARLI SigEE

  15. A theoreticalillustration In “centered around grand mean” representation: ‘CWC2’ model of Kreft et al. (1995) (continued) EARLI SigEE

  16. A theoreticalillustration Linking the “CWC” and “CGM” representations: ‘CWC2’ model of Kreft et al. (1995) (continued) EARLI SigEE

  17. Recommendations • Report centeringoptions • Report to allowfortransformation of results to otherrepresentations • Preferably look at model in multiple representations (especiallyany view withaninterestingsubstantiveinterpretation) EARLI SigEE

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