Confirmatory Factor Analysis

Confirmatory factor analysis (CFA) is a specific type of factor analysis that allows one to determine the extent of the hypothesized relationship between observed indicators and factors (underlying latent variables). CFA, unlike path analysis, allows the distinction between latent variables (referred to as factors) and the indicators (variables) used to measure these latent variables. With CFA models, the factors are assumed to cause the variation and covariation between the observed indicators which are fit to a correlation matrix. This assumption is the primary distinction between CFA and exploratory factor analysis models in which no hypothesis about the number of factors and the relationship between those factors and the indicators is proposed. Thus, CFA can be used for psychometric evaluation, construct validation, and testing measurement invariance. This entry begins with a discussion of the model specification followed by model identification, estimation, evaluation of model fit, and advanced applications of CFA.

CFA models have three primary characteristics:

• 1.
  Indicators are continuous variables with two components: (1) one underlying factor that is measured by the indicator and (2) and everything else which is referred to as error.
• 2.
  Measurement errors must be independent of each other and the factors.
• 3.
  The associations between the factors are not analyzed.

The basic steps in SEM are represented in Figure 1.

Figure 1 Steps in SEM implication framework

[Page 365]Within a structural equation modeling framework, CFA models serve two purposes: (1) to obtain parameter estimates for both the factor (i.e. the factor loadings, the variances, and covariances) and the indicator variables (i.e. residual error variances) and (2) to evaluate the extent to which the model fits the data.

Model Specification

Specification is defined as the representation of a hypothesis in the form of a structural equation model. Specification can take place either before or after data are collected. A CFA model can be defined using various models such as the linear structural relations model (LISREL), the covariance structure analysis model, Bentler-Weeks model, and/or reticular action model (RAM). Provided here are the best known models for continuous observable variables, LISREL and RAM. Figure 2 provides a visual representation of examples of a CFA model with two factors and six indicators.

In Figure 2, squares (or rectangles) and circles (or ellipses) represent observed variables and latent variables, respectively. Also, lines with a single arrowhead and a curved line with two arrowheads reflect hypothesized causal directions and covariances, respectively. This model has seven linear regression equations underlying it, a single structural equation, and six measurement equations. We write the LISREL with matrix notation as:

$$\begin{array}{c}y={\Lambda }_y\eta +\epsilon \\ x={\Lambda }_x\xi +\delta \\ \eta =B\eta +\Gamma \xi +\zeta \text{,}\end{array}$$

[Page 366]where x, y: exogenous and endogenous variable vectors;

Λx Λy: factor loading matrices;

ε, δ: uniqueness vectors;

η, ξ: endogenous and exogenous latent variable vectors;

B: regression coefficient matrix relating the latent endogenous variables to each other;

Γ: regression coefficients matrix relating endogenous variables to exogenous variables; and

ζ: structural disturbance vector.

Figure 2 Graphical presentation of an example of a CFA model using LISREL symbols. CFA = confirmatory factor analysis; LISREL = linear structural relations model.

A general covariance matrix for y and x can be written as:

$$\begin{array}{l}{\displaystyle \sum \left[\begin{array}{cc}{\Sigma }_{yy}& {\Sigma }_{yx}\\ {\Sigma }_{xy}& {\Sigma }_{xx}\end{array}\right]}\\ =\left[\begin{array}{cc}{\Lambda }_y{(I-B)}^{-1}(\Gamma \Phi {\Gamma }^{\text{'}}+\Psi ){(I-B)}^{-1}{\Lambda }_y^{\text{'}}+{\Theta }_s& {\Lambda }_y{(I-B)}^{-1}\Gamma \Phi {\Lambda }_x^{\text{'}}\\ {\Lambda }_x\Phi {\Gamma }^{\text{'}}{(I-B)}^{-1^{\text{'}}}& {\Lambda }_x\Phi {\Lambda }_x^{\text{'}}+{\Theta }_{\varsigma }\end{array}\right],\end{array}$$

where Φ: the variance–covariance matrix of the exogenous latent

...