Mplus Users Guide v6.. - Muthén & Muthén

CHAPTER 9Following is the second part of the example where the covariate x isdecomposed into two latent variable parts.TITLE: this is an example of a two-levelregression analysis for a continuousdependent variable with a random interceptand a latent covariateDATA: FILE = ex9.1b.dat;VARIABLE: NAMES = y x w clus;BETWEEN = w;CLUSTER = clus;CENTERING = GRANDMEAN (x);ANALYSIS: TYPE = TWOLEVEL;MODEL:%WITHIN%y ON x (gamma10);%BETWEEN%y ON wx (gamma01);MODEL CONSTRAINT:NEW(betac);betac = gamma01 - gamma10;The difference between this part of the example and the first part is thatthe covariate x is decomposed into two latent variable parts instead ofbeing treated as an observed variable as in conventional multilevelregression modeling. The decomposition occurs when the covariate x isnot mentioned on the WITHIN statement and is therefore modeled onboth the within and between levels. When a covariate is not mentionedon the WITHIN statement, it is decomposed into two uncorrelated latentvariables,x ij = x wij + x bj ,where i represents individual, j represents cluster, x wij is the latentvariable covariate used on the within level, and x bj is the latent variablecovariate used on the between level. This model is described in Muthén(1989, 1990, 1994). The latent variable covariate x b is not used inconventional multilevel analysis. Using a latent covariate may, however,be advantageous when the observed cluster-mean covariate xm does nothave sufficient reliability resulting in biased estimation of the betweenlevelslope (Asparouhov & Muthén, 2006b; Ludtke et al., 2007).242