CB-SEM Model Comparison (LRT)
Abstract
In covariance-based structural equation modeling (CB-SEM), it is often important to compare alternative models to evaluate whether additional restrictions (e.g., 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 (comparison model) fits the data significantly worse than a less restrictive model (focal model). This feature allows researchers to:
- Test theoretical assumptions by adding or removing constraints.
- Assess measurement invariance through nested model comparisons.
- 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 models must be nested, meaning that one is a submodel of the other, with the same or fewer indicators and constructs.
- Both models must share at least one endogenous construct with the same name and identical indicators.
- Comparisons are made for all endogenous constructs that are common across the two models.
Special assumptions
Special assumptions can be applied when comparing models:
- 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:
- 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
References
- Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2018). Multivariate Data Analysis (8 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
