The main families of index
The chi-square test asks whether the model reproduces the data exactly. With large samples it flags trivial discrepancies, which is why researchers supplement it with descriptive indices. Each index answers a somewhat different question, so reports usually give several.
- Absolute fit, such as SRMR, summarizes the average gap between observed and implied correlations
- Incremental fit, such as CFI and TLI, compares the model with a baseline in which variables are unrelated
- Parsimony-adjusted fit, such as RMSEA, penalizes unnecessary complexity
- Information criteria, such as AIC and BIC, compare competing models
The cutoff debate
Hu and Bentler's 1999 simulations are widely cited for suggesting cutoffs near .95 for CFI and TLI, .06 for RMSEA, and .08 for SRMR. Marsh and colleagues warned that these were never meant as universal golden rules. Performance of an index depends on sample size, number of indicators, factor loadings, estimator, and the type of misspecification.
Reading fit sensibly
Good global fit does not show that a model is correct. Equivalent models can fit identically while implying different causal stories, and a poorly specified part can be hidden by a well-fitting whole. Good practice examines residuals and parameter estimates, compares theoretically plausible alternatives, and reports modifications openly, because changes driven by the data capitalize on chance.
Further reading
- Bentler, P. M. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107(2).
- Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6(1).
- Marsh, H. W., Hau, K.-T., & Wen, Z. (2004). In search of golden rules. Structural Equation Modeling, 11(3).