# Confirmatory Tetrad Analysis in PLS (CTA-PLS)

## Abstract

Confirmatory tetrad analysis in PLS-SEM (CTA-PLS; Gudergan et al., 2008; Hair et al., 2024) allows distinguishing between formative and reflective measurement models. In principal, the analysis follows Bollen and Ting’s (2000) confirmatory approach of testing model-implied vanishing tetrads in the PLS-SEM context.

## Description

The use of confirmatory tetrad analysis in PLS-SEM (CTA-PLS; Gudergan et al., 2008, Hair et al., 2024) allows distinguishing between formative and reflective measurement models. In principal, the analysis follows Bollen and Ting’s (2000) confirmatory approach of testing model-implied vanishing tetrads in the PLS-SEM context with the difference that a bootstrapping procedure is applied to test the significance of the model-implied tetrads.

Gudergan et al. (2008) and Hair et al. (2024) describe the CTA-PLS procedure in detail.

The implemented procedure needs at least 4 manifest variables per construct and can handle a maximum of 25 manifest variables per construct because of the exponentially increasing number of tests if a tetrad is redundant or not.

## CTA-PLS Settings in SmartPLS

### Subsamples

In bootstrapping, subsamples are created with observations randomly drawn from the original set of data (with replacement). To ensure stability of results, the number of subsamples should be large.

For an initial assessment, one may wish to choose a smaller number of bootstrap subsamples (e.g., 500) to be randomly drawn and estimated with the PLS-SEM algorithm, since that requires less time. For the final results preparation, however, one should use a large number of bootstrap subsamples (e.g., 19,000).

**Note:**Larger numbers of bootstrap subsamples increase the computation time.

## Do Parallel Processing

This option runs the bootstrapping routine on multiple processors (if your computer device offers more than one core). Using parallel computing will reduce computation time.

**Important:**The number of processes should not be higher than the number of processors in your computer.

### Test Type

Specifies if a one-sided or two-sided significance test is conducted.

### Significance Level

Specifies the significance level of the test statistic.

## References

- Bollen, K. A., and Ting, K.-f. (2000).
**A Tetrad Test for Causal Indicators**,*Psychological Methods*, 5(1): 3-22. - Gudergan, S. P., Ringle, C. M., Wende, S., and Will, A. (2008).
**Confirmatory Tetrad Analysis in PLS Path Modeling**,*Journal of Business Research*, 61(12): 1238-1249. - Hair, J. F., Sarstedt, M., Ringle, C. M., & Gudergan, S. P. (2024).
**Advanced Issues in Partial Least Squares Structural Equation Modeling (PLS-SEM)**, 2nd Ed., Thousand Oaks, CA: Sage. - More literature ...