Confirmatory Factor Analysis
Confirmatory factor analysis (CFA) tests how well a hypothesised factor structure reproduces an observed covariance matrix. Unlike exploratory factor analysis, which lets the data suggest a structure, CFA specifies in advance which observed variables load on which latent factors and constrains all other loadings to zero. It is the measurement-model component of structural equation modelling and a central tool in modern construct-validation research.
Specification
The researcher draws a model linking observed indicators to latent factors, specifies inter-factor correlations, and identifies the model by fixing one loading per factor or fixing factor variance to one. The model is then fitted, typically by maximum likelihood, and the implied covariance matrix is compared with the observed one. Parameters of interest include factor loadings, factor correlations, and indicator residuals.
Fit indices
Because the chi-square test of exact fit is sensitive to sample size and trivial misspecification, CFA reporting relies on a battery of approximate fit indices. The most commonly cited are the Comparative Fit Index (CFI) and Tucker–Lewis Index (TLI), where values above .95 indicate close fit; the Root Mean Square Error of Approximation (RMSEA), where values below .06 indicate close fit and below .08 acceptable fit; and the Standardised Root Mean Square Residual (SRMR), where values below .08 are conventional. Hu and Bentler's combinational rules are widely cited but not universally accepted, and authors increasingly report multiple indices alongside the chi-square.
Use in language assessment
CFA is the standard procedure for testing whether a multi-skill battery is best represented as separate skill factors, a single proficiency factor, or a hierarchical structure with subskills loading on a higher-order factor. Applications to TOEFL iBT, IELTS, and the Cambridge English suite have informed scoring and reporting decisions. CFA also underpins modern operationalisations of convergent and discriminant validity through average variance extracted and the heterotrait-monotrait ratio.
References
- AERA, APA, & NCME (2014). Standards for Educational and Psychological Testing. American Educational Research Association.
- Brown, T. A. (2015). Confirmatory Factor Analysis for Applied Research (2nd ed.). Guilford Press.
- Kline, R. B. (2016). Principles and Practice of Structural Equation Modeling (4th ed.). Guilford Press.