# Confirmatory Factor Analysis (CFA)

## Abstract

The capabilities in SmartPLS for covariance-based structural equation modeling (CB-SEM) also support confirmatory factor analysis (CFA). Thus, CB-SEM models in SmartPLS allow for the execution of CFA, which is a statistical method designed to validate the underlying factor structure of a given set of observed variables. It empowers researchers to rigorously assess the hypothesis that a meaningful relationship exists between these observed variables and the unobservable latent variables (i.e., constructs) that underpin them (Hair et al., 2018). SmartPLS supports the creation of graphical CFA models. The model estimation assumes that constructs are represented by common factors and uses the maximum likelihood (ML) approach.

SmartPLS is a clear alternative to IBM SPSS Amos.

The following screenshot shows the CFA results in SmartPLS for Kline's (2023) CFA textbook example on the context of job satisfaction:

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

## CB-SEM Algorithm Settings to run CFA in SmartPLS

### Maximum iterations

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

*default value*is 1,000.### Starting value strategy

#### Apply configured starting values.

Only if you check this option, the use-specified starting values from the model will be used. If this option is not selected, we will always use the default starting values.

#### Default strategy

This strategy mimics the default starting values from lavaan. It uses Fabin style estimates for loadings, 0.0 for path coefficients and covariances, 0.5*indicator variance for residual variances, 0.05 for latent variable variances.

#### One zero strategy

This strategy uses much more simple starting value, 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 are 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 are 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 small enough. 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.

## CFA examples in SmartPLS from renowned textbooks

SmartPLS provides directly computable CFA 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 CFA 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.

# 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