Confirmatory Tetrad Analysis in PLS (CTA-PLS)
Confirmatory tetrad analysis in PLS-SEM (CTA-PLS) is a statistical procedure that helps researchers decide whether a construct is more consistent with a formative or a reflective measurement model. Rather than relying only on conceptual reasoning, CTA-PLS tests the model-implied vanishing tetrads of a construct against the null hypothesis of a reflective measurement model, following the confirmatory approach of Bollen and Ting (1993, 2000). In the PLS-SEM context, a bootstrapping procedure is applied to test the significance of the model-implied tetrads (Cefis et al., 2025; Gudergan et al., 2008; Hair et al., 2024).
How CTA-PLS Works
Gudergan et al. (2008) and Hair et al. (2024) describe the CTA-PLS procedure in detail. The implemented procedure requires at least 4 manifest variables per construct and can handle a maximum of 25 manifest variables per construct, because the number of tests needed to check whether a tetrad is redundant increases exponentially with the number of indicators.
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, you may want 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 computation time. For the final results, however, use a large number of bootstrap subsamples (e.g., 10,000).
Note: Larger numbers of bootstrap subsamples increase the computation time.
Parallel Processing
This option runs the bootstrapping routine on multiple processors, if your computer offers more than one core. Using parallel computing reduces computation time.
Important: The number of processes should not be higher than the number of processors in your computer.
Test Type
Specifies whether a one-sided or two-sided significance test is conducted.
Significance Level
Specifies the significance level of the test statistic.
Frequently Asked Questions
What does CTA-PLS test?
CTA-PLS tests whether a construct's measurement model is more consistent with a formative or a reflective specification. It tests the model-implied vanishing tetrads against the null hypothesis of a reflective measurement model, using a bootstrapping procedure to assess statistical significance.
How many indicators does a construct need for CTA-PLS?
The implemented procedure requires at least 4 manifest variables per construct. It can handle a maximum of 25 manifest variables per construct, since the number of required tetrad tests grows exponentially with the number of indicators.
How many bootstrap subsamples should I use for CTA-PLS?
For an initial assessment, a smaller number of subsamples (e.g., 500) is sufficient and faster to compute. For final results intended for reporting, use a larger number of subsamples (e.g., 10,000) to ensure stable results.
Should I use a one-sided or two-sided test in CTA-PLS?
SmartPLS lets you specify the test type (one-sided or two-sided) and the significance level for the CTA-PLS significance test. The appropriate choice depends on your research hypotheses and should be justified accordingly.
Related SmartPLS Methods
- Confirmatory composite analysis (CCA)
- Discriminant validity assessment (HTMT)
- Bootstrapping
- PLS-SEM algorithm
References
- Bollen, K. A., & Ting, K.-f. (1993). Confirmatory tetrad analysis. In P. V. Marsden (Ed.), Sociological methodology (pp. 147–175). American Sociological Association.
- Bollen, K. A., & Ting, K.-f. (2000). A tetrad test for causal indicators. Psychological Methods, 5(1), 3–22.
- Cefis, M., Angelelli, M., Carpita, M., & Ciavolino, E. (2025). Detecting causal relations among indicators with the CTA test: Simulations and applications. Social Indicators Research, 178, 393–417.
- Gudergan, S. P., Ringle, C. M., Wende, S., & 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.). Sage.
- 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
