Centering in hlm

1. Centering in hlm

2. Why 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. Why 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. Why 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. Four 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. Meanings 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 Xij=0, µy=E(Yij)= βoj

7. Meanings 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. Meanings 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. Consequences 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. Choice 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. Example – without centering Level-1 model: Mathachij= βoj+β1j(SESij)+rij Level-2 model : βoj=00+oj β1j=10 From Ihui’s “Issues with centering”

12. Example – grand mean centering Level-1 model: Mathachij= βoj+β1j(SESij-SES..)+rij Level-2 model : βoj=00+oj β1j=10 From Ihui’s “Issues with centering”

13. Example – group mean centering Level-1 model: Mathachij= βoj+β1j(SESij-SES.j)+rij Level-2 model : βoj=00+oj β1j=10 From Ihui’s “Issues with centering”

14. output 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. Example – group mean centering Level-1 model: Mathachij= βoj+β1j(SESij-SES.j)+rij Level-2 model : βoj=00+ 01(MEANSESj) +oj β1j=10

17. Centering for dummy variables (1) • Mathachij= βoj+β1jXij+rij where dummy variable Xij=1 for female, Xij=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 Xij=0).

18. Centering 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 n1 male, n2 female students, the total is n=n1+n2. (Xij=1 female, Xij=0 male). Then, =n2/n • For female, =1-n2/n=n1/n (% of male) • For male, =0-n2/n=-n2/n (-% of female)

19. Centering 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 n1 male, n2 female students in school j, the group mean = n2/(n1+n2)=n2/n • For female, =n1/n (% of male in school j) • For male, =-n2/n (-% of female in school j)

20. What about the intercepts after Centering 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, µyj.