CCA is a type of covariance structure analysis that assesses whether two or more covarying composites. It can also be extended to assess models containing interrelated composites and common factors. This extension is also referred to as confirmatory composite factor analysis.
Jörg Henseler and Theo K. Dijkstra sketched the initial idea of CCA. Subsequently, Florian Schuberth together with Jörg Henseler and Theo K. Dijkstra provided the first full description of CCA and demonstrated its efficacy by means of a Monte Carlo simulation.
CCA consists of the four steps, namely (i) model specification, (ii) model identification, (iii) model (parameter) estimation, and (iv) model assessment.
In its original form, CCA is used to study models containing covarying composites in which disjunct sets of observed variables form composites, i.e., each observed variable is connected to one composite only.
The exact identification rules depend on the employed estimator. In general, the scale of each composite needs to be fixed, e.g., by scaling the weights to ensure a unit variance of each composite. Moreover, no composite is allowed to be isolated in the model, i.e., each composite needs to be connected to another variable next to its components.
In CCA, various estimators can be used. Originally, approaches that emerged outside the realm of structural equation modeling were proposed such as partial least squares path modeling and approaches to generalized canonical correlation analysis. Next to these approaches, a recently introduced specification, i.e., the Henseler-Ogasawara specification, allows to express the composite model, i.e., the model underlying CCA, as a special type of structural equation model. As a consequence, estimators implemented SEM software such as the maximum likelihood estimator and the generalized least squares estimator can be used to obtain the parameter estimates.
In CCA, model assessment comprises of two main steps, namely (i) assessment of the overall model, e.g., by tests for exact overall model and approximate fit measures, and (ii) assessment of the model parameter estimates, i.e., weights, composite loadings and covariances among composites.
Both CCA and CFA are types of covariance structure analysis. Whereas CFA deals with models of covarying common factors, CCA deals with models of covarying composites. In practice, it is possible that models contain composites and common factors. In this case, a confirmatory composite factor analysis is applied.
Measurement models represent auxiliary theories. According to Popperian dictum, theories can only be rejected but never be confirmed. Since the confirmation of theories is impossible, there cannot be any method that confirms reflective and causal-formative measurement models. Moreover, CCA assess composite models and not reflective or causal-formative measurement models.
In its original form, CCA assumes that the observed variables are free from random measurement errors. However, the reliability of observed variables can be taken into account by specifying observed variables as single-indicator latent variables with a fixed random measurement error variance. Moreover, CCA does not assess reflective measurement models.