Joop Hox: Mplus can Do More Than Two-level SEM

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Most modern SEM software includes routines for two-level SEM. Having only two levels may seem an important limitation, but one must appreciate that SEM is an inherently multivariate technique, and multilevel regression is univariate. Consequently, for multivariate analysis or including a measurement model, multilevel regression needs an extra ‘variable’ level, and for longitudinal analysis it needs an ‘occasion’ level. Multilevel SEM does not need this.
Nevertheless, having only two levels can put strong limits on the analysis. At the time, only the software Gllamm (Rabe-Hesketh & Skrondal, 2008) can analyze multiple-level SEM. Using direct estimation as described above, more than two levels can be accommodated, but this is restricted to multivariate normal variables, and the example shows that the estimates and
standard errors are not very accurate.

The two-stage approaches are simpler than the general random coefficient model. They are comparable to the multilevel regression model with random variation only for the intercepts. There is no provision for randomly varying slopes (factor loadings and path coefficients). Although it would be possible to include cross-level interactions, introducing interaction
variables of any kind in structural equation models is complicated (cf. Bollen, 1989; Marcoulides & Schumacker, 2001). An interesting approach is allowing different within groups covariance matrices in different subsamples, by combining two-level and multigroup models.

When maximum likelihood estimation is used, multilevel SEM can include varying slopes.  At the time, only Mplus and Gllamm support this. Muthén and Muthén (1998-2007) have extended the standard path diagram by using a black dot on an arrow in the level-1 model to indicate a random slope. This slope appears in the level-2 model as a latent variable. This is consistent with the use of latent variables for the level-2 intercept variances. This highlights an important link between multilevel regression and multilevel SEM: random coefficients are latent variables, and many multilevel regression models can also be specified in the SEM context (Curran, 2003; Mehta & Neale, 2005).


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关键词：two-level Level Mplus Leve More inherently limitation important including software