# Covariance-based Structural Equation Modeling (CB-SEM)

## Abstract

Structural equation modeling (SEM) is a versatile methodology used in a variety of scientific fields (Byrne, 2016; Schumacker & Lomax, 2010). Widely used in the social and behavioral sciences, epidemiology, and economics, SEM constructs models to describe causal relationships between unobservable latent variables (or constructs, e.g., attitudes) and observed variables (e.g., survey responses). Equations and diagrams allow the creation and presentation of such models.

Covariance-based structural equation modeling (CB-SEM) is a statistical method for estimating structural equation models (Hair et al., 2018; Kline, 2023). CB-SEM uses a statistical model to estimate and test correlations between dependent and independent variables and the hidden structures in between. Important note: CB-SEM assumes that the constructs are common factors and estimates the model accordingly (e.g., Rigdon et al., 2017; Sarstedt et al., 2016).

SmartPLS supports graphical model building and CB-SEM model estimation using the maximum likelihood (ML) approach. The results allow testing whether the hypotheses assumed for the model are consistent with the given variables. This aspect positions CB-SEM as a structure-testing multivariate procedure with confirmatory character.

SmartPLS as a clear alternative to SPSS Amos.

The following screenshot shows the CB-SEM results in SmartPLS for Kline's (2023) SEM textbook example on job satisfaction:

This algorithm is in beta stage. Changes and additions are likely and feedback is welcome.

## CB-SEM Algorithm Settings in SmartPLS

## Maximum iterations

The maximum number of iterations the optimizer will perform. This parameter should be high enough to ensure that a good model solution can be found. The

*default value*is 1,000 but could be higher for more complex models.### Starting value strategy

#### Apply configured starting values.

Do so by checking this option and inserting the configured use-specified starting values of your theoretical model. If this option is not selected, the software will always use the default starting values.

#### Default strategy

This strategy mimics Lavaanâ€™s default starting values. It uses Fabin-style estimates for its loadings, 0.0 for path coefficients and covariances, a 0.5*indicator variance for its residual variances, and 0.05 for its latent variable variances.

#### One zero strategy

This strategy applies more simple starting values, with 1.0 for loadings and variances, and 0.0 for path coefficients and covariances.

### Stop criterion (gradient)

The optimizer stops when one of the two stop criteria is fulfilled and convergence to the optimum is assumed. In this case, the optimizer terminates when ||g|| <

*stop criterion** max(1, ||x||), where ||.|| denotes the Euclidean (L2) norm. The default value is 10^-6.### Stop criterion (function value)

The optimizer stops when one of the two stop criteria is fulfilled and convergence to the optimum is assumed. In this case, the optimizer terminates when the decrease in the objective function (maximum likelihood value) is smaller than the recommended minimum. The condition is met if (f' - f) / f <

*stop criterion*, where f' is the objective value of the past iteration and f is the objective value of the current iteration. The default value is 10^-9.### Special assumptions

#### Imply latent variable correlations.

Select this option if you want to estimate correlations between all exogenous latent variables. Usually, if no correlation arrow is drawn in the model, the correlation between exogenous latent variables is constrained to zero. With this option correlations are also estimated freely when no arrow is drawn.

#### Imply causal indicator correlations per construct.

Select this option if you want to estimate correlations between all causal indicators of a latent variable. Usually, if no correlation arrow is drawn in the model, the correlation between causal indicators is constrained to zero. With this option correlations are also estimated freely when no arrow is drawn.

#### Imply a variance of 1.0 for causal indicators.

If you choose this option, all variances of causal indicators are constrained to 1.0. This also overwrites use-specified values. This option should help to mimic the default Lavaan results.

## CB-SEM examples in SmartPLS from renowned textbooks

SmartPLS provides directly computable CB-SEM examples from reputable textbooks (Byrne, 2016; Hair et al., 2018; Kline, 2023; Schumacker & Lomax, 2010) and SmartPLS yields the same results as those textbooks provide. Take a look try out these CB-SEM examples in SmartPLS!

## References

- Byrne, B. M. (2016). Structural Equation Modeling with AMOS: Basic Concepts, Applications, and Programming (Multivariate Applications) (3 ed.). Routledge.
- Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2018). Multivariate Data Analysis (8 ed.). Cengage Learning.
- Kline, R. B. (2023). Principles and Practice of Structural Equation Modeling (5 ed.). Guilford Press.
- Schumacker, R. E., & Lomax, R. G. (2010). A Beginner's Guide to Structural Equation Modeling (3 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 (2016), Issue 10, pp. 3998-4010.
- More literature ...

# Cite correctly

## Please always cite the use of SmartPLS!

Ringle, Christian M., Wende, Sven, & Becker, Jan-Michael. (2024). SmartPLS 4. BÃ¶nningstedt: SmartPLS. Retrieved from https://www.smartpls.com