HLM: What happens if I enter level2 variables at level 1?

The goal of this exercise is to find out what happens if I intentionally use a level2 variable at level 1 in HLM.   I found that the coefficients and standard errors remain about the same.  The parameter that differed was just the degree of freedom, which was consistent with my expectation.

Using my old NELS dataset, I ran two different HLM models using Bryk and Raudenbush’s software (See model 1 and model 2 equations in the table below).

  • The URBAN as level 1 covariate model (I entered the level2 variable URBAN wrongly at level 1)
  • The URBAN as level 2 covariate model (I entered the level2 variable URBAN correctly at level 2)

The outcome variable is the achievement composite (POSTTEST), students are level 1 and schools are level 2.  When expressed as mixed models, the two models are identical, which is why I expected most parameters to come out the same.

POSTTESTij = γ00
γ10*URBANij  + u0jrij

The first model (MODEL 1; see below) included URBAN (students are in urban school) as a level 1 predictor.  Of course this is a wrong specification because urban is a school characteristic.  In the second model (MODEL 2), I used it at the expected level, which is at level 2 (school level).

These models look different, but AGAIN when expressed as mixed models, they are identical.  As the third model (MODEL 3), I replicated the same HLM model using SAS PROC GLIMMIX.  SAS requires that the equation be expressed as a mixed model.

Results showed that coefficients and standard errors are more or less the same across three models.  The only one thing that was different was degree of freedom.

Conclusion: As long as variables enter the model as fixed effects as done here, there is nothing magical about the HLM model.  HLM software or SAS PROC GLIMMIX (option ddfm=kr) adjust degree of freedom values, accounting for the fact that URBAN is a school-level variable and thus should not be awarded a value that is too large.  Notice that under the correct specification (MODEL 2 and MODEL 3), the degree of freedom for URBAN is close to the number of schools, not to the number of students.

Thanks for any comments you may have.

MODEL 1 MODEL 2 MODEL 3
URBAN as LEVEL 1 covariate URBAN as level 2 covariate SAS PROC GLIMMIX
Level-1 Model

POSTTESTij = β0j + β1j*(URBANij) + rij

Level-2 Model

β0j = γ00 + u0j
β1j = γ10

Mixed Model

POSTTESTij = γ00
γ10*URBANij  + u0jrij

Level-1 Model

POSTTESTij = β0j + rij

Level-2 Model

β0j = γ00 + γ01*(URBAN_LEj) + u0j

Mixed Model

POSTTESTij = γ00 + γ01*URBAN_LEj  + u0jrij

proc glimmix data=kaz.level1;

class schoolID;

model

posttest =

urban

/solution ddfm=kr dist=normal link=identity s ;

random schoolID;

run;

 

Final estimation of fixed effects
(with robust standard errors)

Fixed Effect  Coefficient  Standard
error
 t-ratio  Approx.
d.f.
 p-value
For INTRCPT1, β0
    INTRCPT2, γ00 52.643432 0.526139 100.056 125 <0.001
For URBAN slope, β1
    INTRCPT2, γ10 -0.450022 1.157924 -0.389 692 0.698

 

Final estimation of variance components

Random Effect Standard
Deviation
Variance
Component
  d.f. χ2 p-value
INTRCPT1, u0 3.76951 14.20923 125 292.48369 <0.001
level-1, r 8.39004 70.39271

Statistics for current covariance components model

 

Final estimation of fixed effects
(with robust standard errors)

Fixed Effect  Coefficient  Standard
error
 t-ratio  Approx.
d.f.
 p-value
For INTRCPT1, β0
    INTRCPT2, γ00 52.643459 0.526140 100.056 124 <0.001
    URBAN_LE, γ01 -0.449983 1.157920 -0.389 124 0.698

 

Final estimation of variance components

Random Effect Standard
Deviation
Variance
Component
  d.f. χ2 p-value
INTRCPT1, u0 3.76919 14.20678 124 292.48068 <0.001
level-1, r 8.39007 70.39334

Statistics for current covariance components model

 

Solutions for Fixed Effects
Effect Estimate Standard Error DF t Value Pr > |t|
Intercept 52.6434 0.5460 95.01 96.41 <.0001
urban -0.4501 1.1095 132.4 -0.41 0.6856
 

 

Covariance Parameter Estimates
Cov Parm Estimate Standard Error
schoolID 14.2150 3.4610
Residual 70.3913 3.7455

 

Datasets:

www.nippondream.com/file/datafiles_HLM.zip

Results from Model 1

Results from Model 2

Results from PROC GLIMMIX

 

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