CB-SEM Model Comparison (LRT)
In covariance-based structural equation modeling (CB-SEM), researchers often need to compare alternative models to evaluate whether additional restrictions, such as equality constraints or omitted parameters, significantly worsen model fit (Hair et al., 2018). SmartPLS provides a Likelihood Ratio Test (LRT) to compare nested CB-SEM models (Satorra & Bentler, 2010). The LRT is based on a chi-square difference test and evaluates whether a more restrictive model (the comparison model) fits the data significantly worse than a less restrictive model (the focal model). This feature allows researchers to test theoretical assumptions by adding or removing constraints, assess measurement invariance through nested model comparisons, and compare alternative structural models in a confirmatory framework.
Model Comparison Settings in SmartPLS
Comparison model file
You need to choose an alternative model that will be compared to the focal model. The Likelihood Ratio Test (LRT) for comparing two CB-SEM models requires:
- Same dataset and indicators – Both models must use the identical dataset and the same set of indicator variables (i.e., the same covariance matrix as input).
- Same estimation method – Both models must be estimated with the same estimation method and optimizer settings.
- Identified and admissible – Both models must be identified and converge to admissible solutions.
- Nested model structure – One model must be nested within the other (see details below).
SmartPLS automatically ensures that conditions 2 and 3 are always fulfilled. You must ensure that your models meet conditions 1 and 4.
What is a nested model?
A nested model uses the same indicator variables (and thus the same covariance matrix as input) but specifies at least one additional parameter to be freely estimated:
- The less restricted model has fewer degrees of freedom (more free parameters).
- The more restricted model has more degrees of freedom (fewer free parameters).
- The less restricted model is nested within the more restricted one.
Special assumptions
Special assumptions can be applied when comparing models:
| Special assumption | Effect |
|---|---|
| Imply construct correlations | Estimates correlations between all exogenous latent variables, even if no correlation arrow is drawn. Normally, such correlations are constrained to zero if not explicitly specified. |
| Imply causal indicator correlations per construct | Estimates correlations between all causal indicators of a latent variable, even without correlation arrows. Normally, these are constrained to zero. |
| Imply a variance of 1.0 for causal indicators | Constrains all causal indicator variances to 1.0 (overwriting user-specified values). This helps mimic default results from other SEM software such as Lavaan. |
Mean structure
By default, SEM focuses on modeling the covariance structure of observed variables. In some cases (e.g., latent growth curve modeling), it may be necessary to also include a mean structure in the model. A mean structure includes means and intercepts of latent and observed variables and requires constraints for identification (since only p observed means are available for p observed variables).
Available options:
| Option | Description |
|---|---|
| No mean structure (default) | Ignores means and estimates only covariances. Suitable for most SEM analyses. |
| Estimate mean structure, fix factor means to zero | Considers both covariance and mean structures. Factor means are constrained to zero, while observed variable intercepts are freely estimated. Additional constraints may be specified by the user. |
| Estimate mean structure with only user-specified constraints | Considers both covariance and mean structures, but without predefined constraints. The user must define all necessary constraints to achieve identification. |
Likelihood Ratio Test (Chi-Square Difference Test)
The likelihood ratio test evaluates whether the more restrictive model fits significantly worse:
- Null hypothesis (H0): Both models fit equally well.
- Alternative hypothesis (H1): The less restrictive model fits significantly better.
The chi-square difference is computed as:
[
\Delta \chi^2 = \chi^2*{\text{comparison}} - \chi^2*{\text{focal}}
]
with degrees of freedom equal to the difference in estimated parameters.
If the difference is significant, the restrictions in the comparison model are not supported.
Applications
CB-SEM model comparison in SmartPLS is particularly useful for:
- Testing measurement invariance models (configural vs. weak vs. strong).
- Confirming or rejecting equality constraints (e.g., equal loadings, equal paths).
- Comparing alternative theory-driven models (nested path structures).
CB-SEM Examples in SmartPLS
SmartPLS provides directly computable CB-SEM examples from leading textbooks (Byrne, 2016; Hair et al., 2018; Kline, 2023; Schumacker & Lomax, 2010). The results in SmartPLS replicate the textbook examples exactly. Try out the CB-SEM example projects in SmartPLS
Frequently Asked Questions
What does the CB-SEM Likelihood Ratio Test (LRT) tell me?
The LRT compares two nested CB-SEM models using a chi-square difference test. It evaluates whether a more restrictive model (the comparison model) fits the data significantly worse than a less restrictive model (the focal model). A significant difference means the restrictions in the comparison model are not supported by the data.
What counts as a nested model in CB-SEM?
A nested model uses the same indicator variables and covariance matrix as another model but specifies at least one additional parameter to be freely estimated. The less restricted model (more free parameters, fewer degrees of freedom) is nested within the more restricted model (fewer free parameters, more degrees of freedom).
What do I need to make sure of before running the LRT in SmartPLS?
Both models must use the identical data set and the same set of indicators, and one model must be nested within the other. SmartPLS automatically ensures that both models are estimated with the same estimation method and optimizer settings and that both converge to admissible solutions.
When do I need to estimate a mean structure instead of only a covariance structure?
A mean structure is needed when the analysis requires means and intercepts of latent and observed variables to be modeled explicitly, for example in latent growth curve modeling. By default, SmartPLS ignores means and estimates only covariances, which is suitable for most SEM analyses.
What is CB-SEM model comparison used for in practice?
Typical applications include testing measurement invariance models (configural vs. weak vs. strong), confirming or rejecting equality constraints such as equal loadings or paths, and comparing alternative theory-driven, nested path structures.
Related SmartPLS Methods
- CB-SEM Measurement Invariance Assessment
- CB-SEM Multigroup Analysis (MGA)
- CB-SEM Bootstrapping
- CB-SEM
- Confirmatory Factor Analysis (CFA)
References
- Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2018). Multivariate data analysis (8th ed.). Cengage Learning.
- Satorra, A., & Bentler, P. M. (2010). Ensuring positiveness of the scaled difference chi-square test statistic. Psychometrika, 75(2), 243–248.
- More literature ...
Cite correctly
Please always cite the use of SmartPLS!
Ringle, Christian M., Wende, Sven, & Becker, Jan-Michael. (2024). SmartPLS 4. Bönningstedt: SmartPLS. Retrieved from https://www.smartpls.com
