Confirmatory composite analysis

Statistical model

A composite is typically a linear combination of observable random variables. However, also so-called second-order composites as linear combinations of latent variables and composites, respectively, are conceivable.

For a random column vector ${\displaystyle \mathbf{x}}$ of observable variables that is partitioned into sub-vectors ${\displaystyle {\mathbf{x}}_i}$, composites can be defined as weighted linear combinations. So the i-th composite ${\displaystyle c_i}$ equals:

${\displaystyle c_i={\mathbf{w}}_i^{\prime }{\mathbf{x}}_i}$,

where the weights of each composite are appropriately normalized (see Confirmatory composite analysis#Model identification). In the following, it is assumed that the weights are scaled in such a way that each composite has a variance of one, i.e., ${\displaystyle {\mathbf{w}}_i^{\prime }{\mathbf{\Sigma }}_{ii}{\mathbf{w}}_i}$. Moreover, it is assumed that the observable random variables are standardized having a mean of zero and a unit variance. Generally, the variance-covariance matrices ${\displaystyle {\mathbf{\Sigma }}_{ii}}$ of the sub-vectors are not constrained beyond being positive definite. Similar to the latent variables of a factor model, the composites explain the covariances between the sub-vectors leading to the following inter-block covariance matrix:

${\displaystyle {\mathbf{\Sigma }}_{ij}={\rho }_{ij}{\mathbf{\Sigma }}_{ii}{\mathbf{w}}_i({\mathbf{\Sigma }}_{jj}{\mathbf{w}}_j{)}^{\prime }}$,

where ${\displaystyle {\rho }_{ij}}$ is the correlation between the composites ${\displaystyle c_j}$ and ${\displaystyle c_i}$. The composite model imposes rank one constraints on the inter-block covariance matrices ${\displaystyle {\mathbf{\Sigma }}_{ij}}$, i.e., ${\displaystyle \text{rank}({\mathbf{\Sigma }}_{ij})=1}$. Generally, the variance-covariance matrix of ${\displaystyle \mathbf{x}}$ is positive definite iff the correlation matrix of the composites ${\displaystyle \mathbf{R}:=({\rho }_{ij})}$ and the variance-covariance matrices ${\displaystyle {\mathbf{\Sigma }}_{jj}}$'s are both positive definite.

In addition, the composites can be related via a structural model which constrains the correlation matrix ${\displaystyle \mathbf{R}}$ indirectly via a set of simultaneous equations:

${\displaystyle \mathbf{B}{\mathbf{c}}_{\text{endogenous}}=\mathbf{C}{\mathbf{c}}_{\text{exogenous}}+\mathbf{z}}$,

where the vector ${\displaystyle \mathbf{c}}$ is partitioned in an exogenous and an endogenous part, and the matrices ${\displaystyle \mathbf{B}}$ and ${\displaystyle \mathbf{C}}$ contain the so-called path (and feedback) coefficients. Moreover, the vector ${\displaystyle \mathbf{z}}$ contains the structural error terms having a zero mean and being uncorrelated with ${\displaystyle {\mathbf{c}}_{\text{exogenous}}}$. As the model needs not to be recursive, the matrix ${\displaystyle \mathbf{B}}$ is not necessarily triangular and the elements of ${\displaystyle \mathbf{z}}$ may be correlated.

Model identification

To ensure identification of the composite model, each composite must be correlated with at least one variable not forming the composite. Additionally to this non-isolation condition, each composite needs to be normalized, e.g., by fixing one weight per composite, the length of each weight vector, or the composite’s variance to a certain value. If the composites are embedded in a structural model, also the structural model needs to be identified. Finally, since the weight signs are still undetermined, it is recommended to select a dominant indicator per block of indicators that dictates the orientation of the composite.

The degrees of freedom of the basic composite model, i.e., with no constraints imposed on the composites' correlation matrix ${\displaystyle \mathbf{R}}$, are calculated as follows:

Model estimation

To estimate the parameters of a composite model, various methods that create composites can be used such as approaches to generalized canonical correlation, principal component analysis, and linear discriminant analysis. Moreover, a maximum-likelihood estimator and composite-based methods for SEM such as partial least squares path modeling and generalized structured component analysis can be employed to estimate weights and the correlations among the composites.

Evaluating model fit

In CCA, the model fit, i.e., the discrepancy between the estimated model-implied variance-covariance matrix ${\displaystyle \hat{\mathbf{\Sigma }}}$ and its sample counterpart ${\displaystyle \mathbf{S}}$, can be assessed in two non-exclusive ways. On the one hand, measures of fit can be employed; on the other hand, a test for overall model fit can be used. While the former relies on heuristic rules, the latter is based on statistical inferences.

Fit measures for composite models comprises statistics such as the standardized root mean square residual (SRMR), and the root mean squared error of outer residuals (RMS${\displaystyle {}_{\theta }}$) In contrast to fit measures for common factor models, fit measures for composite models are relatively unexplored and reliable thresholds still need to be determined. To assess the overall model fit by means of statistical testing, the bootstrap test for overall model fit, also known as Bollen-Stine bootstrap test, can be used to investigate whether a composite model fits to the data.

Alternative views on CCA

Besides the originally proposed CCA, the evaluation steps known from partial least squares structural equation modeling (PLS-SEM) are dubbed CCA. It is emphasized that PLS-SEM's evaluation steps, in the following called PLS-CCA, differ from CCA in many regards:. (i) While PLS-CCA aims at conforming reflective and formative measurement models, CCA aims at assessing composite models; (ii) PLS-CCA omits overall model fit assessment, which is a crucial step in CCA as well as SEM; (iii) PLS-CCA is strongly linked to PLS-PM, while for CCA PLS-PM can be employed as one estimator, but this is in no way mandatory. Hence, researchers who employ need to be aware to which technique they are referring to.