Confirmatory tetrad analysis (CTA) is a useful tool to empirically substantiate the mode of a latent variable’s measurement model (i.e., formative or reflective). The application of CTA enables researchers to avoid incorrect measurement model specification. 

Measurement model misspecification is a threat to the validity of SEM results (Jarvis et al., 2003). For example, modeling latent variables reflectively when the conceptualization of the measurement model should be a formative specification can result in biased results. In addition, formative indicators produce lower outer loadings when represented in a reflective measurement model. 

Any attempt to purify formative indicators based on correlation patterns among the indicators can have adverse consequences for the content validity of the construct. Empirically, the results in the measurement model and structural model can significantly change before and after eliminating indicators. For these reasons, researchers must avoid measurement model misspecification to ensure the validity of their results conceptually, theoretically, and empirically. 

The CTA-PLS (Gudergan et al., 2008) facilitates empirical evaluation of cause-effect relationships for latent variables and their specification of indicators in measurement models. When applied, the statistical test offers empirical information that facilitates determining a latent variable’s mode of measurement model (i.e., reflective or formative) and provides support for avoiding incorrect measurement model specification. 

CTA-PLS builds on the concept of tetrads (τ), which describe the relationship between pairs of covariances. Covarince shows how the two variables differ. For example, consider a reflectively measured latent variable with four indicators. For this construct, we obtain six covariances (s) between all possible pairs of the four indicators, as shown below:

A tetrad is the difference of the product of one pair of covariances and the product of another pair of covariances. The six covariances of four indicator variables result in six unique pairings that form three tetrads: 

τ1234 = σ12 * σ34 – σ13 * σ24

τ1342 = σ13 * σ42 – σ14 * σ32

τ1423 = σ14 * σ23 – σ12 * σ43

Important Note: 

The theoretical reasoning backed by literature review of construct being formative or reflective is on top most priority urrespective of the results of CTA.

• Each construct must have at leat 4 items (as the name is tetrad)

• Maximum of 25 items should be there. 

• Path/relationship does not matter. 

• CTA only concerns about covariance. 

1. Construct the model

2. Check the p-value (preferably confidence interval) of all the tetrads of each construct.

3. If 80% of tetrads have a p-value more than 0.05, then it is a reflective construct.

4. If 80% of the confidence Interval of the estimated tetrad contains zero, then it is a reflective construct.

   • If the signs of the upper and lower CI values are different (positive and negative / +/-), then the construct is reflective.  

   • If the signs of both upper and lower CI values are positive or negative  (+/+ OR -/-), then the construct is formative. 

• Install the European Customer Satisfaction Index (ECSI) in your SmartPLS 4.

• Assess the CTA.

Gamst-Klaussen, T., Gudex, C., & Olsen, J. A. (2018). Exploring the causal and effect nature of EQ-5D dimensions: an application of confirmatory tetrad analysis and confirmatory factor analysis. Health and quality of life outcomes, 16(1), 1-10.

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.

Kono, S., Ito, E., & Loucks-Atkinson, A. (2021). Are leisure constraints models reflective or formative?: Evidence from confirmatory tetrad analyses. Leisure Sciences, 44(1), 55-76.

Valaei, N. (2017). Organizational structure, sense making activities and SMEs’ competitiveness: An application of confirmatory tetrad analysis-partial least squares (CTA-PLS). VINE Journal of Information and Knowledge Management Systems, 47(1), 16-41.