R: What is the difference between the "baseline model" and "unrestricted model" in lavaan model summaries?

Question

Usingsummary(fit.measures = TRUE) I am able to access extensive information about the fit of models stored in lavaan model objects. In this output (exemplified in the accompanying image), several lines compare the user's specified model to two alternative models:

1. "Baseline Model"
2. "Unrestricted Model"

I am looking for a somewhat precise explanation of the models implied by each of these terms, since they can mean different things within the structural equation modeling community. Ideally, I would be able to extract the model itself that implied by this term after e.g. using lavaan::cfa().

Currently, the tutorial does not provide any explanation, while the package documentation states the baseline model is "the independence model, assuming all variables are uncorrelated." However, it is not clear what is meant by "all variables" and the example it provides of an independence model on p.79 assumes exogenous various are correlated due to the default settings in lavaan.

Similarly, p.34 of the documentation does not explain what is meant by a "variable" when it notes:

"...the model is defined as the unrestricted model. The following free parameters are included: all covariances/correlations among the variables, variances for continuous variables, means for continuous variables, thresholds for ordered variables, and if exogenous variables are included (ov.names.x is not empty) while some variables are ordered, also the regression slopes enter the model"

Answer 1

Not sure this is an appropriate post because it is not about programming. The answers can be found in introductory SEM textbooks.

the independence model, assuming all variables are uncorrelated." However, it is not clear what is meant by "all variables"

All endogenous (or modeled, explained) variables are uncorrelated in the independence model. Exogenous variables are not explained by the model, but instead are taken as given (like in OLS regression). The independence model is used to calculate incremental fit indices (e.g., CFI and TLI; see Bentler & Bonett, 1980, regarding their basic structure). The default baseline.model is independence, but you can fit whatever custom model you want to use (Widamin & Thompson, 2003), which should be nested within your target model(s), and pass it to fitMeasures().

The unrestricted model is on the opposite end of the continuum. Your target model(s) are nested within it because it is saturated, which reproduces your observed means and covariance matrix better than any restricted model could. Thus, it serves as the reference model for the likelihood ratio test of exact model fit (the chi-squared test statistic under Model Test User Model:).

References

Bentler, P. M., & Bonett, D. G. (1980). Significance tests and goodness of fit in the analysis of covariance structures. Psychological Bulletin, 88(3), 588–606. https://doi.org/10.1037/0033-2909.88.3.588

Widaman, K. F., & Thompson, J. S. (2003). On specifying the null model for incremental fit indices in structural equation modeling. Psychological Methods, 8(1), 16–37. https://doi.org/10.1037/1082-989X.8.1.16