This FAQ considers how to interpret the coefficients from multilevel models when different kinds of centering are used. Although the examples are illustrated with HLM, these principles apply to multilevel models solved in any statistical package.

The data file for this example is made like this

use https://stats.idre.ucla.edu/stat/paperexamples/singer/hsb12 , clear keep school student ses mathach * make ses uncentered with a mean of 12.5 generate sesu = ses + 12.5 egen meanses = mean(ses), by(school) egen meansesu = mean(sesu), by(school) hlm mkmdm using hsb12hlm, id2( school) l1( mathach ses sesu ) l2( meanses meansesu ) replace

You can download the MDM file here: hsb12hlm

Model 1. SES Group mean centered at level 1, MeanSES Grand mean centered at level 2.

Consider this multilevel model where we are predicting math achievement (MATHACH) based on the student level SES (USES) that has been group mean centered at level 1 and school level average SES (MEANSES) that has been grand mean centered at level 2. This model is shown below.

MATHACH_{ij }= β_{0j }+ β_{1j}
(Group Mean Centered SES) + r_{ij}
β_{0j }= γ_{00 } + γ_{01
}(Grand Mean Centered Mean SES) + u_{0j
}β_{1j }= γ_{10 } + u_{1j}

We can specify this in HLM like this.

When we run this in HLM using Maximum Likelihood Estimation, we get these results.

Final estimation of fixed effects: ---------------------------------------------------------------------------- Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value ---------------------------------------------------------------------------- For INTRCPT1, B0 INTRCPT2, G00 12.647291 0.148414 85.216 158 0.000 MEANSESU, G01 5.865601 0.359373 16.322 158 0.000 For SESU slope, B1 INTRCPT2, G10 2.191172 0.108660 20.165 7182 0.000 ----------------------------------------------------------------------------

In terms of the model, the results look like this.

MATHACH_{ij }= β_{0j }+ β_{1j}
(USES) + r_{ij}
β_{0j }= 12.64 + 5.86 (Grand Mean Centered Mean SES)
+ u_{0j
}β_{1j }= 2.19

Interpretation. This decomposes the relationship between SES and MATCHACH into two components, the level 1 (within school) component and the level 2 (between school) component. At level 1, the regression coefficient is 2.19, meaning that for a one unit increase in a student’s SES, their math achievement would be predicted to increase by 2.19 units. The level 2 coefficient of 5.86 means that when the SES for a school increases by 1 unit, the school level math achievement would be expected to increase by 5.86 units.

Model 2. SES Grand mean centered at level 1, MeanSES Grand mean centered at level 2.

Let’s now try this model where as compared to Model 1 SES was group mean centered it is now Grand Mean Centered. We continue to Grand mean center MEANSES at level 2.

MATHACH_{ij }= β_{0j }+ β_{1j}
(Grand Mean Centered SES) + r_{ij}
β_{0j }= γ_{00 } + γ_{01
}(Grand Mean Centered Mean SES) + u_{0j
}β_{1j }= γ_{10 }

We can specify this in HLM like this.

When we run this in HLM using Maximum Likelihood Estimation, we get these results.

Final estimation of fixed effects: ---------------------------------------------------------------------------- Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value ---------------------------------------------------------------------------- For INTRCPT1, B0 INTRCPT2, G00 12.661166 0.148416 85.309 158 0.000 MEANSESU, G01 3.674429 0.375441 9.787 158 0.000 For SESU slope, B1 INTRCPT2, G10 2.191172 0.108660 20.165 7182 0.000 ----------------------------------------------------------------------------

In terms of the model, the results look like this.

MATHACH_{ij }= β_{0j }+ β_{1j}
(USES) + r_{ij}
β_{0j }= 12.66 + 3.67 (Grand Mean Centered Mean SES)
+ u_{0j
}β_{1j }= 2.19

Interpretation. Like model 2, this decomposes the relationship between SES and MATCHACH into two components, the level 1 (within school) component and the level 2 (between school) component. Also, like model 2 the level 1 regression coefficient remains 2.19 with the same meaning. But, note how the coefficient associated with the grand mean centered SES has changed to 3.67. This value reflects the difference of the Between school regression coefficient for meanses minus the Within school regression coefficient for SES. Referring to the estimates from Model 2, 5.865601 – 2.191172 = 3.674429.

Model 3. Group Centered SES at level 1.

Suppose that we imagine that we are only concerned with the level 1 effect of SES (at the student level) and really do not care about the level 2 effect. We might construct a model like this to examine just the student level effect of SES.

MATHACH_{ij }= β_{0j
}+ β_{1j} (Group Centered SES) + r_{ij}
β_{0j }= γ_{00 } + u_{0j
}β_{1j }= γ_{10 }

We can specify this in HLM like this.

When we run this in HLM, we get these results.

Final estimation of fixed effects: ---------------------------------------------------------------------------- Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value ---------------------------------------------------------------------------- For INTRCPT1, B0 INTRCPT2, G00 12.636226 0.243729 51.845 159 0.000 For SESU slope, B1 INTRCPT2, G10 2.191172 0.108647 20.168 7183 0.000 ----------------------------------------------------------------------------

In terms of the model, the results look like this.

MATHACH_{ij }= β_{0j }+ β_{1j}
(USES) + r_{ij}
β_{0j }= 12.63 + u_{0j
}β_{1j }= 2.19 + u_{1j}

Model 4. Grand Mean Centered SES at level 1.

Suppose that we imagine that we are only concerned with the level 1 effect of SES (at the student level) and really do not care about the level 2 effect. We might construct a model like this to examine just the student level effect of SES.

MATHACH_{ij }= β_{0j
}+ β_{1j} (Uncentered SES) + r_{ij}
β_{0j }= γ_{00 } + u_{0j
}β_{1j }= γ_{10 }

We can specify this in HLM like this.

When we run this in HLM, we get these results.

Final estimation of fixed effects: ---------------------------------------------------------------------------- Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value ---------------------------------------------------------------------------- For INTRCPT1, B0 INTRCPT2, G00 12.657979 0.187263 67.595 159 0.000 For SESU slope, B1 INTRCPT2, G10 2.391617 0.105691 22.628 7183 0.000 ----------------------------------------------------------------------------

In terms of the model, the results look like this.

MATHACH_{ij }= β_{0j }+ β_{1j}
(USES) + r_{ij}
β_{0j }= 12.65 + u_{0j
}β_{1j }= 2.39 + u_{1j}

Model 5. Uncentered SES at level 1.

Suppose that we imagine that we are only concerned with the level 1 effect of SES (at the student level) and really do not care about the level 2 effect. We might construct a model like this to examine just the student level effect of SES.

MATHACH_{ij }= β_{0j
}+ β_{1j} (Uncentered SES) + r_{ij}
β_{0j }= γ_{00 } + u_{0j
}β_{1j }= γ_{10 }

We can specify this in HLM like this.

When we run this in HLM, we get these results.

Final estimation of fixed effects: ---------------------------------------------------------------------------- Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value ---------------------------------------------------------------------------- For INTRCPT1, B0 INTRCPT2, G00 -17.237571 1.333826 -12.923 159 0.000 For SESU slope, B1 INTRCPT2, G10 2.391617 0.105691 22.628 7183 0.000 ----------------------------------------------------------------------------

In terms of the model, the results look like this.

MATHACH_{ij }= β_{0j }+ β_{1j}
(USES) + r_{ij}
β_{0j }= -17.23 + u_{0j
}β_{1j }= 2.39 + u_{1j}