Covariance-based Structural Equation Modeling (CB-SEM)
Covariance-based structural equation modeling (CB-SEM) is a statistical approach for estimating structural equation models (SEM) that relies on the covariance matrix of the observed data to test whether a hypothesized model fits that data (Hair et al., 2018; Kline, 2023). CB-SEM treats constructs as common factors and typically uses maximum likelihood (ML) estimation, which positions it as a confirmatory, structure-testing procedure (Rigdon et al., 2017; Sarstedt et al., 2016). This approach belongs to the broader family of SEM methods that researchers across the social and behavioral sciences, epidemiology, economics, and other fields use to model causal relationships between unobservable latent variables (constructs such as attitudes) and observed variables (e.g., survey responses) through equations and path diagrams (Byrne, 2016; Schumacker & Lomax, 2010). As demonstrated in the tutorial article on CB-SEM in SmartPLS by Hair et al. (2025), SmartPLS supports graphical model building and CB-SEM estimation with ML, offering a modern alternative to other SEM software such as SPSS Amos.
The following screenshot shows CB-SEM results in SmartPLS for Kline's (2023) SEM textbook example on job satisfaction:
CB-SEM Algorithm Settings in SmartPLS
Maximum Iterations
Specifies the maximum number of iterations the optimizer will perform. This should be set high enough to ensure convergence to a good solution.
- Default value: 1,000 (can be increased for complex models).
Starting Value Strategy
Researchers can let SmartPLS apply user-specified starting values defined in the theoretical model (apply configured starting values), or choose between two default strategies if this option is unchecked:
| Parameter | Default strategy (mimics Lavaan) | One-zero strategy |
|---|---|---|
| Loading estimates | 1.0 | 1.0 |
| Path coefficients & covariances | 0.0 | 0.0 |
| Residual variances | 0.5 × indicator variance | 1.0 |
| Latent variable variances | 0.05 | 1.0 |
Stop Criteria
- Gradient criterion
The optimizer stops when:
||g|| < stop criterion × max(1, ||x||)- Default value: 10^-6
- Function value criterion
The optimizer stops when the improvement in the ML objective function is negligible:
(f' – f) / f < stop criterion- Default value: 10^-9
Special Assumptions
- Imply latent variable correlations Estimates correlations between all exogenous latent variables, even if no correlation arrow is drawn.
- Imply causal indicator correlations per construct Estimates correlations between all causal indicators of a latent variable, even without correlation arrows.
- Imply a variance of 1.0 for causal indicators Constrains all causal indicator variances to 1.0, mimicking Lavaan's defaults (overrides user-specified values).
Mean Structure
Most SEM analyses focus on modeling the covariance structure of observed variables. In special cases (e.g., latent growth curve modeling, multigroup analysis), including a mean structure may be necessary. A mean structure involves means and intercepts of latent and observed variables and requires constraints for identification (since only p observed means are available, with p = number of observed variables).
| Option | Description |
|---|---|
| No mean structure (default) | Ignores means and estimates only covariances (sufficient for standard SEM). |
| Estimate mean structure, fix factor means to zero | Includes mean structure. Factor means are constrained to zero, observed variable intercepts are estimated freely. Additional constraints can be added by the user. |
| Estimate mean structure with only user-specified constraints | Includes mean structure without predefined constraints. The user must specify all necessary identification constraints. |
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 is CB-SEM used for?
CB-SEM tests whether a hypothesized model's relationships are consistent with the observed data. It relies on the covariance matrix of the data and treats constructs as common factors, which makes it a confirmatory, structure-testing procedure rather than a prediction-oriented method.
How does SmartPLS estimate CB-SEM models?
SmartPLS builds CB-SEM models graphically and estimates them using maximum likelihood (ML). Researchers can control the optimizer's maximum iterations, starting value strategy, and stop criteria to ensure convergence to a good solution.
What starting value strategies does SmartPLS offer for CB-SEM?
SmartPLS lets researchers apply their own configured starting values, or choose between a default strategy that mimics Lavaan's defaults and a simpler one-zero strategy. Both differ in how they set initial loading estimates, path coefficients, covariances, and variances before optimization begins.
When do I need a mean structure in CB-SEM?
A mean structure is not needed for standard SEM, since most analyses only model the covariance structure. It becomes necessary in special cases such as latent growth curve modeling or multigroup analysis, and requires identification constraints because only as many observed means are available as there are observed variables.
Are there ready-made CB-SEM examples I can try in SmartPLS?
Yes. SmartPLS provides directly computable CB-SEM examples from leading SEM textbooks (Byrne, 2016; Hair et al., 2018; Kline, 2023; Schumacker & Lomax, 2010), and the results in SmartPLS replicate these textbook examples exactly.
How is CB-SEM different from PLS-SEM in SmartPLS?
CB-SEM treats constructs as common factors and estimates them with maximum likelihood to test global model fit, whereas PLS-SEM estimates constructs as composites. SmartPLS supports both approaches, so researchers can choose the estimation method that fits their research objective, or use both for a multimethod SEM strategy.
Related SmartPLS Methods
- Confirmatory factor analysis (CFA)
- Principal component analysis (PCA)
- PLS-SEM algorithm
- Consistent PLS-SEM (PLSc)
References
- Byrne, B. M. (2016). Structural equation modeling with AMOS: Basic concepts, applications, and programming (Multivariate Applications) (3rd ed.). Routledge.
- Hair, J. F., Babin, B. J., Ringle, C. M., Sarstedt, M., & Becker, J.-M. (2025). Covariance-based structural equation modeling (CB-SEM): A SmartPLS 4 software tutorial. Journal of Marketing Analytics, 13, 709–724.
- Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2018). Multivariate data analysis (8th ed.). Cengage Learning.
- Kline, R. B. (2023). Principles and practice of structural equation modeling (5th ed.). Guilford Press.
- Schumacker, R. E., & Lomax, R. G. (2010). A beginner's guide to structural equation modeling (3rd ed.). Routledge.
- Rigdon, E. E., Sarstedt, M., & Ringle, C. M. (2017). On comparing results from CB-SEM and PLS-SEM. Five perspectives and five recommendations. Marketing ZFP, 39(3), 4–16.
- Sarstedt, M., Hair, J. F., Ringle, C. M., Thiele, K. O., & Gudergan, S. P. (2016). Estimation issues with PLS and CBSEM: Where the bias lies! Journal of Business Research, 69(10), 3998–4010.
- 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
