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Confirmatory factor analysis tests whether data fit a pre-specified measurement model, directly evaluating whether items measure intended constructs. Fit indices (CFI, RMSEA, SRMR) and factor loadings assess model adequacy, making CFA essential for validating test structure and detecting unintended multidimensionality.
Specify competing measurement models based on theory and compare fit statistics. Examine modification indices to understand when respecification is theory-driven versus exploratory.
All fit indices should exceed arbitrary cutoffs. Each index has different sensitivity; using multiple indices addresses different aspects of fit. Good fit alone doesn't guarantee validity; it's necessary but insufficient.
Confirmatory factor analysis is the tool you use when you already have a theory about how a set of items should cluster. Whereas exploratory factor analysis lets the data reveal structure, CFA works in reverse: you specify a measurement model first — based on theory or prior EFA results — and then ask whether that model is consistent with the observed pattern of correlations among items. If you believe six questionnaire items all measure a single construct called "depression," CFA lets you test that claim directly against data.
The core operation in CFA is comparing two covariance matrices: the one actually observed in your sample, and the one your model implies should exist if the factor structure is correct. The difference between these matrices is your residual. Fit indices summarize how large that residual is. The CFI (Comparative Fit Index) compares your model to a null model where variables are uncorrelated — values above 0.95 suggest good fit. The RMSEA (Root Mean Square Error of Approximation) estimates the error per degree of freedom — values below 0.06 are conventionally acceptable. The SRMR (Standardized Root Mean Square Residual) reflects average discrepancy between observed and model-implied correlations — below 0.08 is typical. No single index tells the whole story; you look at all three together, and you look at modification indices to understand which specific constraints the model is straining against.
The most important misconception to guard against is equating good fit with validity. A CFA model can fit beautifully and still measure the wrong thing. Fit only tells you that the factor structure is internally consistent — not that the factor corresponds to a real and meaningful construct. To make validity claims, you need convergent evidence (does the scale correlate with other measures it should correlate with?) and discriminant evidence (does it fail to correlate with measures it shouldn't?). CFA is the foundation of measurement validation, not the whole edifice.
CFA also enables powerful model comparison. You can test a one-factor model against a two-factor model and use likelihood ratio tests or AIC/BIC comparisons to evaluate which structure fits better. This is how researchers test whether, for example, anxiety and depression are best represented as one undifferentiated factor or two related but distinct constructs. The ability to pit competing theories against each other — rather than letting data suggest structure post hoc — is what makes CFA a cornerstone of modern psychometrics.