Generalized SEM description

Question

I've become very interested in SEM recently, and while reviewing the literature on the subject I've run into the problem, that almost none of these descriptions are very general per se, they all handle specific cases. Bollen has the most comprehensive treatment but I've noticed he's made two assumptions:

1. No path between latent exogenous variables (i.e. I don't see an $ξ_i$ to $\xi_j$ matrix anywhere) is included. What if we needed a model with interacting ($\xi$'s)
2. The shortest path possible between the X, and Y side involves at least two latent variables. What if we needed only one? There does not seem to be a way to accomplish that in Bollen's notation
3. From my understanding Bollen's measurement model is reflective, what if we needed a formative one (i.e. $\xi$'s are being influenced by the $x$'s rather that the other way around)?

Bollen presents the generalized description of a SEM: $$ \begin{gather} \mathbf{\eta=B\eta+\Gamma \xi + \zeta}\ \ \text{(Structural Model)}\\ \mathbf{X=\Lambda_x\xi+\delta}\ \ \text{(X-side measurement model)}\\ \mathbf{Y=\Lambda_Y\eta+\epsilon}\ \ \text{(Y-side measurement model)}\\ \end{gather} $$ What would be a generalized version of the above system with [1-3]? How would the ML estimator and $\Sigma$ change?

Answer 1

... almost none of these descriptions are very general per se, they all handle specific cases. Bollen has the most comprehensive treatment ...

Like most other texts, Bollen (1989) was describing the LISREL model, which is the predominant representation because it was the first developed (e.g., Jöreskog, 1970; also the software by the same name). There are other representations, such as the reticular action model (RAM; McArdle & McDonald, 1984), which collapses all directed paths (single-headed arrows) into a single asymmetric matrix $A$, and all undirected paths (double-headed arrows) into a single symmetric matrix $S$, then using a filter matrix to indicate which variables are latent or observed, but basically employing the same algebra that the LISREL model capitalizes on to marginalize out the latent variables for estimation.

Regarding your list of 3:

1. An interaction is not a path between 2+ variables, it is another variable (the product between 2+ variables). The product-indicator approaches make this explicit by requiring the user to create indicators for the latent product term, although it can make the model clunky.
2. The X and Y sides can be linked by a single path (in the Beta matrix, regressing a latent endogenous on a latent exogenous variable). If you mean between indicators, you can just use the "all-Y" part of the LISREL model, which is what lavaan uses, and Mplus extends to include direct effects of X on endogenous observed (kappa) or latent (gamma) variables (Muthen, 2002). Then the Theta matrix directly connects any 2 indicators with undirected paths, although no LISREL matrix directly connects them with directed paths (but the Mplus model can, if X is purely exogenous).
3. Bollen (e.g., 2007, 2017) has written extensively about how to specify formative/causal indicators in a SEM. I also highly recommend Florian Schuberth and Jörg Henseler's work (e.g., 2018, 2021) on confirmatory composite analysis.

What would be a generalized version of the above system with [1-3]? How would the ML estimator and Σ change?

RAM absorbs LISREL and other linear models (e.g., the Bentler-Weeks model), and Boker et al. (preprint) are exploring how to extend RAMs to estimate SEMs with product terms from summary statistics alone. You can also find some ad-hoc representations that generalize to products among latent variables (e.g., Song & Lee, 2006, who used MCMC; also Wall & Amemiya's work).

Some other models generalize SEM even further to include also generalized linear mixed models as special cases (e.g., Muthen, 2002, implemented in Mplus; Skrondal & Rabe-Hesketh, 2004, implemented for Stata in GLLAMM). Frank Buckler has done a lot of work and developed the NEUSREL software to generalize SEM. Making something even more general tends to limit its practicality. For example, GLLAMM uses numeric integration to obtain all ML estimates, which means it typically takes a long time to run. Basically, it is incredibly difficult to come up with a single general representation to rule them all.