1. C ENTERING IN HLM

2. W HY CENTERING ? In OLS regression, we mostly focus on the slope but not intercept. Therefore, raw data (natural X metric) is perfectly fine for the purpose of the study. The slope indicates expected increase in DV for a unit increase in IV. The intercept represents the expected value of DV when all predictors are 0.

3. W HY CENTERING ? In HLM, however, we are interested in not only slope, but intercept. We use level l coefficients (intercept and slopes) as outcome variables at level 2 Thus, we need clearly understand the meaning of these outcome variables.

4. W HY CENTERING ? Intercept in behavior researches sometimes are meaningless. e.g. Y - math achievement. X -IQ. Without centering, the intercept is expected math achievement for a student in school j whose IQ is zero. But we know it does not make sense. Centering is a method to change the meaning of the intercept, especially for.

5. F OUR POSSIBILITIES FOR LOCATION OF X Natural X metric Centering around the grand mean (grand mean centering): Centering around the level-2 mean (group-mean centering) Other specialized choices of location for X

6. M EANINGS OF INTERCEPTS UNDER THE FIRST 3 LOCATIONS OF X (1) Example: Y – math achievement. X - IQ score. Natural X metric: expected math achievement for a student in school j whose IQ is zero. Caution: only used it if x=0 is meaningful, not in this case. When X ij =0, µ y =E(Y ij )= β oj

7. M EANINGS OF INTERCEPTS UNDER THE FIRST 3 LOCATIONS OF X (2) Example: Y - math achievement. X - IQ score. Grand-mean centering ( ): expected math achievement for a student in school j whose IQ is equal to the mean of all students from all schools. The intercept is adjusted mean for group j:

8. M EANINGS OF INTERCEPTS UNDER THE FIRST 3 LOCATIONS OF X (3) Example: Y - math achievement. X - IQ score. Group-mean centering ( ): expected math achievement for a student in school j whose IQ is equal to the mean of school (group) j. The intercept is unadjusted mean for group j:

9. C ONSEQUENCES OF CENTERING In both cases, the intercept is more interpretable than the natural X metric alternative. Grand mean centering and natural X metric produce equivalent models (estimates could be recalculated from one model to another), but grand mean centering has computational advantage. Mostly, group mean centering produces non- equivalent model to either natural X metric or grand mean centering.

10. C HOICE OF CENTERING “there is no statistically correct choice” among the three models. The choice between grand mean (more preferable than natural X metric) and group mean centering “must be determined by theory.” Therefore, if the absolute values of level 1 variable is important, then use grand-mean centering. If the relative position of the person to the group’s mean is important, then use group-centering. 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: 1-21,

11. E XAMPLE – WITHOUT CENTERING Level-1 model: Mathach ij = β oj +β 1j (SES ij )+r ij Level-2 model : β oj =  00 +  oj β 1j =  10 From Ihui’s “Issues with centering”

12. E XAMPLE – GRAND MEAN CENTERING Level-1 model: Mathach ij = β oj +β 1j (SES ij -SES..)+r ij Level-2 model : β oj =  00 +  oj β 1j =  10 From Ihui’s “Issues with centering”

13. E XAMPLE – GROUP MEAN CENTERING Level-1 model: Mathach ij = β oj +β 1j (SES ij -SES. j )+r ij Level-2 model : β oj =  00 +  oj β 1j =  10 From Ihui’s “Issues with centering”

14. OUTPUT EffectSES(raw score model) SES(Grand mean centered) SES (Group mean centered)  00 (s.e) 0.187984 0.244502  10 (s.e) 0.105719 0.108655 Var(r ij )37.03440 37.01040 Var(  oj ) 4.76815 8.67252 From Ihui’s “Issues with centering”

15. REMARKS Under grand-mean centering or no centering, the parameter estimates reflect a combination of person-level effects and compositional effects. But when we use a group-centered predictor, we only estimate the person-level effects. In order not to discard the compositional effects with group-mean centering, level-2 variables should be created to represent the group mean values for each group-mean centered predictor.

16. E XAMPLE – GROUP MEAN CENTERING Level-1 model: Mathach ij = β oj +β 1j (SES ij -SES. j )+r ij Level-2 model : β oj =  00 +  01 (MEANSES j ) +  oj β 1j =  10

17. C ENTERING FOR DUMMY VARIABLES (1) Mathach ij = β oj +β 1j X ij +r ij where dummy variable X ij =1 for female, X ij =0 for male for student i in school j Without centering, the intercept is the expected math achievement for male student in school j (i.e., the predicted value for student with X ij =0).

18. C ENTERING FOR DUMMY VARIABLES (2) Grand mean centering: if a student is female, is equal to the proportion of male students in the sample. If a student is male, is equal to the minus proportion of female students in the sample. For example, we have n 1 male, n 2 female students, the total is n=n 1 +n 2. (X ij =1 female, X ij =0 male). Then, =n 2 /n  For female, =1-n 2 /n=n 1 /n (% of male)  For male, =0-n 2 /n=-n 2 /n (-% of female)

19. C ENTERING FOR DUMMY VARIABLES (3) Group mean centering: if a student is female, is equal to the proportion of male students in school j. If a student is male, is equal to the minus proportion of female students in school j. For example, we have n 1 male, n 2 female students in school j, the group mean = n 2 /(n 1 +n 2 )=n 2 /n  For female, =n 1 /n (% of male in school j)  For male, =-n 2 /n (-% of female in school j )

20. W HAT ABOUT THE INTERCEPTS AFTER C ENTERING FOR DUMMY VARIABLES Grand mean centering: the intercept is now the expected math achievement adjusted for the differences among the units in the percentage of female students. Group mean centering: the intercept is still the average outcome for unit j, µ yj.