Hi Drs. Muthen, I am not sure where to post this, but I just finished reading "Generalized Latent Variable Modeling: Multilevel, Longitudinal and Structural Equation Models" (similar to the Psychometrica piece with Pickles) by Skrondal & Rabe-Hesketh. I was wondering where you stood on their approach (i.e., GLLAMM). They critize the multigroup approach to M-level anlayses, but only LISREL seems to do that now. I am ingorant as to much of the math underlying Mplus, but it seems Mplus's approach is quite similar (and I've heard as being much faster). Do you know where the two approaches/programs differ markedly and when one might be more advantageous than the other.
The spirit of general latent variable modeling introduced with the emergence of Mplus in 1998 is also present in the nice book of 2004 by Skrondal-Rabe-Hesketh and in their related computer program GLLAMM, but there are some key differences with respect to interface, models, and algorithms. GLLAMM has a technical-statistical interface where the user needs to specify models in terms of matrices, whereas Mplus has a simple, non-technical interface. The modeling framework of Mplus is more general than that of GLLAMM, for example modeling with a very flexible combination of continuous and categorical latent variables and random slopes with continuous latent variables. The computations of Mplus are considerably faster than those of GLLAMM both because Mplus has a more efficient executable platform and because with full ML estimation Mplus avoids numerical integration wherever possible and Mplus also offers other, quicker estimators. If you point me to the pages where the critique of the multigroup approach is given, I can respond to that aspect.
Thanks Bengt, I have returned the book to the library already, but the multi-group critique exists in the Psychometrika piece as well. Here's an exerpt:
When structural equation modeling is instead taken as a starting point, we note that some limited multilevel structural equation modeling is possible using the traditional approaches where models are fitted to sample covariance matrices and sometimes means. This is achieved by treating the highest level of the multilevel model as “level 1” and the combinations of all lower-level units as a high-dimensional multivariate response (e.g., Muth´en, 1997). For example, 4 variables observed at 3 time points on 2 twins from each of a large number of families could be modeled by letting the families be the “level 1” units and the responses for each family a 24-dimensional multivariate response. Obviously, this method can only be applied if there are not too many units at any of the levels apart from the highest, since the dimensionality of the response vector would otherwise become excessive. In addition, the method requires complete multivariate responses and balanced multilevel designs. Random coefficients can be included via factor models, but this is only possible if the corresponding covariates are balanced, thus requiring all three types of balance mentioned above. Using the traditional approach to structural equation modeling, all three types of imbalance can be handled to some degree by multi-group analysis. However, a more flexible approach is to base parameter estimation on likelihood contributions from individual units (e.g., Arminger & Sobel, 1990) instead of sample covariance matrices and means from listwise samples. However, clusters with large numbers of lower-level units still pose problems due to excessive dimensionality...
...For continuous responses or responses modeled by latent underlying variables, such multilevel structural equation models are typically defined by specifying separate models for the within-cluster and between-cluster covariance matrices (e.g., Longford& Muth´en, 1992; Poon & Lee, 1992; Linda, Lee, & Poon, 1993; Muth´en, 1994; Lee & Shi, 2001) although Goldstein and McDonald (Goldstein & McDonald, 1988; McDonald & Goldstein, 1989) propose a more general framework. There are six limitations to models specified via two separate conventional structural equation models. First, random coefficients cannot be included at “level 2.” Second, random coefficients at the lower levels are not permitted if the corresponding covariates are continuous or highly unbalanced. Third, the models cannot include large clusters at more than one level (“level 2”) since this would render the dimensionality excessive. Fourth, this framework does not allow direct inclusion of regressions of “level 1” latent variables on “level 2” latent variables. Such models must be specified in a roundabout way by using a large number of parameters and imposing a large number of nonlinear parameter constraints.We will return to this point later. Fifth, modeling the “level 1” covariance matrix presupposes that the responses are either continuous or can be viewed as generated by categorizing continuous underlying variables. This accommodates models for dichotomous and ordinal responses, but not, for instance, Poisson and gamma models often used for counts and durations in continuous time. Finally, the typical specification of multivariate normality for underlying variables corresponds to a probit link and therefore rules out several useful links such as the log, logistic, and complementary log-log.
These 2 paragraphs deal with multilevel modeling using the "multivariate" approach and the "within-between covariance matrix" approach, respectively. Mplus covers both approaches but is not limited to these, but can do multilevel modeling in a general maximum-likelihood fashion. In a sense, the authors' critique of these 2 methods is a comment on the limits of conventional SEM used for multilevel modeling. Mplus does not limit itself to conventional SEM, but takes a more general latent variable approach. The multivariate approach is as the authors say limited to smaller numbers of members of clusters, but for those cases I find the method more powerful than conventional multilevel modeling in that explicit relationships can be modeled between members of a cluster. For example, older siblings can be modeled to influence younger siblings as in the Khoo-Muthen article on the Mplus web site. The within-between covariance matrix approach, represented as the MUML (as opposed to FIML) estimator in Mplus, is limited in that it cannot handle random slopes only random intercepts and cannot handle MAR missingness, but has the advantage of giving fast computations for the models it does handle with results close to those of FIML. For example, it is convenient with large numbers of variables.
That makes much sense. However (in terms of, for example, MUML), because it seperates within and between components in seperate analyses, will it handle, for example, multilevel mediation in an appropriate way? As far as I've read (and understood), multilevel mediation requires a dynamic interplay between within and between component estimation to examine for changes in multilevel relations (e.g., the Sobel method) to be able to claim that mediation has occured. For example, if an IV is at L2, with mediator and DV at L1, will a method which seperates between and within components allow one to examine for coefficient changes in IV-DV relations after accounting for a mediator? We have emailed in the past about this type of question, but my understanding of the issue is that unless the between and within components of a computation are done simultaneously (i.e., in one equation & not as seperate groups), then multilevel mediation is problematic to address. If I am thinking about this issue incorrectly, please let me know. However, if I am not, and should avoid the MUML estimator in such cases, please also let me know this (I only ask because I am writing a piece for Organizational Research Methods on multilevel SEM and want give Mplus much limelight, but am somewhat unsure if I should redirect my efforts and also discuss GLLAMM's approach).
Thank you for your time.
bmuthen posted on Thursday, June 02, 2005 - 9:06 am
Just to clarify - Mplus has 2 estimators for multilevel modeling: MUML and FIML. So Mplus is not synonymous with MUML. The Mplus FIML estimator is the same as that used in GLLAMM (and other programs). FIML analysis in Mplus offers more generality than FIML in GLLAMM.
Now, regarding your question of what MUML does - yes, it does sound like you are thinking about the MUML between and within covariance matrix analysis incorrectly. MUML allows random intercept modeling, albeit not random slope modeling (FIML handles both). The best way to think of MUML is that for random intercept models (that both MUML and FIML can handle) - MUML gives a good approximation to FIML results. MUML can handle multilevel mediation appropriately because random intercept modeling implies that a level 2 variable can influence a level 1 variable by influencing the random intercept in the level 1 outcome equation. MUML of course analyzes the between and within matrices simultaneously since these two parts of the model share parameters - see the Technical Appendix to Mplus on our web site and see also my 1994 Soc Meth & Res article. Let me know if I can give you further clarification for writing your article.
Prof. Muthen, would you characterize the FIML approach in Mplus as a multivariate approach while GLLAMM's is univariate? Also, is Mplus limited to 2-level models, while GLLAMM is not, because Mplus relies on the computation of within/between matrices while GLLAMM does not?
I think both Mplus and GLLAMM can take univariate and multivariate approaches. Mplus does not rely on computation of within/between matrices when the ML estimator is used - that would only be a correct characterization of the MUML estimator. Mplus is currently limited to 2 levels unless one of the levels is longitudinal in which case 3-level modeling is done. GLLAMM is for a general number of levels, but the GLLAMM generality is more academic than practical - numerical integration is always carried out even when not in principle needed which significantly increases computational time.