GLM and Common Factors Model

[Saul Gabriel]

Greetings, In some programs (e.g., Mplus), the common factors model can be extended to categorical, binary, and even count indicators of an underlying latent variable. However, generalized linear modeling approach can also accommodate such a mixture of items without presuming an underlying latent variable using maximum likelihood. The latter can be implemented in Stata's gsem, for example. And in some cases not evoking an underlying latent variable makes sense. The former directly fits a probability model with the observed/measured variables using a data likelihood function as opposed to the multi-step approach in the common factors model (e.g., computing a correlation matrix S* and then fitting the model to S* using a summary statistic fit function; cf. Curran-Hancock). My question is how the generalized linear modeling approach estimates parameters for a latent variable (e.g., mean, variance) as, again, it does not presume an underlying latent variable or construct. How is it producing these? Thanks, Saul

[William Lisowski]

Have you reviewed the full documentation for the SEM/GSEM commands in Stata Structural Equation Modeling Reference Manual PDF included in your Stata installation and accessible from Stata's Help menu?

As I understand it, I believe your question is addressed in the "Intro 4 – Substantive concepts" section of the documentation, specifically in the subsection entitled "Identification 2: Normalization constraints (anchoring)".

[Saul Gabriel]

William Lisowski. Indeed, and my question was not about model identification or the scaling of the latent factor, but more about where and how latent variables are estimated within this particular approach compared to the common factors approach.
Although it is clear to me that the latent factor(s) are incorporated in the linear predictor, it is still not clear how the GLM approach completely circumvents the notion of an underlying latent variable as in the common factors approach given that it is still included in the linear predictor. Unless the latent variable in the GLM approach means something else... Or because a modified correlation matrix is not used as in the common factors approach. In any case, I wonder how to interpret the latent variable in the GLM approach.

As far as the calculation of latent variable (or assigning values to it), I believe it is estimated or calculated using empirical Bayes or EAP.