CENTERING IN HLM

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.

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.

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 .

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

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

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:

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:

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.

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,

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”

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”

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”

OUTPUT

Effect SES(raw score model)

SES(Grand mean centered)

SES (Group mean centered)

00(s.e) 0.187984 0.187984 0.244502

10(s.e) 0.105719 0.105719 0.108655

Var(rij) 37.03440 37.03440 37.01040

Var(oj) 4.76815 4.76815 8.67252

From Ihui’s “Issues with centering”

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.

EXAMPLE – GROUP MEAN CENTERING

Level-1 model: 

Mathachij= βoj+β1j(SESij-SES.j)+rij

 Level-2 model : 

βoj=00+ 01(MEANSESj) +oj

β1j=10

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).

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)

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)

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.