Confirmatory factor analysis (CFA) is a structural equation modeling technique used to test whether a hypothesized factor structure is consistent with the observed data. Unlike exploratory factor analysis, which discovers factor structure from the data, CFA begins with a theoretically specified model — specifying which items load on which factors, which factor loadings are zero, and whether factors are correlated — and evaluates how well this model reproduces the observed covariance matrix.
Model Specification
Σ(θ) = ΛΦΛ′ + Θ
where x = vector of observed indicators
Λ = matrix of factor loadings
ξ = vector of latent factors, Φ = factor covariance matrix
δ = measurement errors, Θ = error covariance matrix
The model implies a specific structure for the population covariance matrix. The model-implied covariance matrix Σ(θ) is a function of the free parameters θ = {Λ, Φ, Θ}. Model estimation finds the parameter values that minimize the discrepancy between the observed covariance matrix S and the model-implied covariance matrix Σ(θ). Maximum likelihood estimation minimizes F_ML = ln|Σ(θ)| + tr(SΣ(θ)⁻¹) − ln|S| − p, where p is the number of observed variables.
Model Identification and Estimation
A model is identified when its parameters can be uniquely determined from the observed covariance matrix. Necessary conditions include having at least three indicators per factor (for single-factor models) or two indicators per factor (for multi-factor models with correlated factors). Each factor's scale must be set either by fixing one loading to 1.0 or by standardizing the factor variance. Underidentified models have infinite parameter solutions; overidentified models have testable constraints.
CFA models are evaluated using multiple fit indices. The chi-square test (χ² = (N − 1) × F_ML) tests exact fit but is sensitive to sample size. Approximate fit indices include RMSEA (values ≤ 0.06 suggest close fit), CFI and TLI (values ≥ 0.95 suggest good fit), and SRMR (values ≤ 0.08 suggest adequate fit). No single index is definitive; researchers evaluate the pattern across indices. Modification indices suggest parameter additions that would improve fit, but post-hoc modifications should be guided by theory, not solely by statistical criteria.
Applications in Psychometrics
CFA is the standard method for evaluating the factorial validity of psychological instruments. It tests whether a questionnaire's items cluster into the theoretical dimensions claimed by its developers. Multi-group CFA is used to test measurement invariance across populations. Higher-order CFA models test whether first-order factors are themselves indicators of a broader construct. Bifactor models test whether items reflect both a general factor and specific group factors.
CFA also serves as the measurement component of structural equation models. The quality of the measurement model — factor loadings, error variances, factor correlations — directly affects the interpretability of structural relationships among latent variables. Researchers are advised to evaluate the measurement model thoroughly before proceeding to structural modeling, following Anderson and Gerbing's (1988) two-step approach. CFA's ability to separate measurement error from construct variance makes it an indispensable tool in modern psychometrics.