Why model fit matters
In covariance-based SEM, the estimated model implies a covariance matrix for your observed variables. Fit indices summarise how far that implied matrix is from the one you observed. Good fit does not prove a model is true, since other models may fit equally well, but poor fit means the model, as specified, is not consistent with the data.
Cut-offs at a glance
| Index | Type | Good fit | Acceptable |
|---|---|---|---|
| χ² (p) | Exact fit test | p > .05 | Often significant in large samples |
| χ²/df | Normed chi-square | ≤ 3 | ≤ 5 |
| CFI | Incremental | ≥ 0.95 | ≥ 0.90 |
| TLI | Incremental | ≥ 0.95 | ≥ 0.90 |
| RMSEA [90% CI] | Absolute, parsimony-adjusted | ≤ 0.06 | ≤ 0.08 |
| SRMR | Absolute | ≤ 0.08 | ≤ 0.10 |
| GFI | Absolute | ≥ 0.95 | ≥ 0.90 |
Most of these cut-offs come from simulations by Hu and Bentler (1999). They are guidelines, not laws: they depend on sample size, the number of indicators and loading size, so report several indices together.
Chi-square
The model chi-square tests the null hypothesis that the model reproduces the population covariance matrix exactly. It equals the sample size times the minimised maximum likelihood discrepancy. Because power grows with N, even small, unimportant misfit becomes significant with a few hundred cases. The normed chi-square (χ² divided by its degrees of freedom) is a rough correction.
CFI and TLI
The comparative fit index (CFI) and the Tucker–Lewis index (TLI) compare your model with a baseline model in which all variables are uncorrelated. CFI ranges from 0 to 1. TLI penalises complexity and can exceed 1. Both are less sensitive to sample size than chi-square.
RMSEA
The root mean square error of approximation estimates misfit per degree of freedom in the population, rewarding parsimony. Report its 90% confidence interval and PCLOSE, the test of close fit (RMSEA ≤ 0.05). RMSEA can be inflated in models with very few degrees of freedom and small samples.
SRMR
The standardized root mean square residual is the average difference between the observed and model-implied correlations. It is easy to interpret: an SRMR of 0.05 means correlations are reproduced to within about 0.05 on average. Large individual residual correlations (above 0.10) show where the model fails.
GFI, AIC and BIC
The goodness-of-fit index (GFI) is the share of observed covariance explained by the model; it is reported less often today because it depends on sample size. AIC and BIC do not judge absolute fit; they compare models estimated on the same data, with lower values being better. BIC penalises extra parameters more strongly.
What to do when fit is poor
- Check the data: coding errors, reverse-coded items, outliers and strong non-normality.
- Check the measurement model first: drop items with low loadings or large cross-loadings if content validity allows.
- Inspect modification indices and residual correlations. Free a parameter only with a theoretical justification, one change at a time, and report it.
- Consider whether a construct is multidimensional or two constructs should be merged.
How to report fit
“The measurement model fit the data well, χ²(48) = 82.15, p = .002, χ²/df = 1.71, CFI = .974, TLI = .964, RMSEA = .049, 90% CI [.030, .067], SRMR = .041.”
Frequently asked questions
What is a good CFI value?
CFI of 0.95 or higher is commonly read as good fit and 0.90 or higher as acceptable (Hu and Bentler, 1999).
What is a good RMSEA value?
RMSEA of 0.06 or lower indicates close fit, up to 0.08 reasonable fit, and above 0.10 poor fit. Also report its 90% confidence interval.
What is a good SRMR value?
SRMR of 0.08 or lower is generally considered good fit.
Why is my chi-square significant even though the other indices are good?
The chi-square test grows with sample size, so with a few hundred cases even trivial misfit becomes significant. Report it, but judge fit mainly with CFI, TLI, RMSEA and SRMR.
References
- Browne, M. W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 136–162). Sage.
- Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis. Structural Equation Modeling, 6(1), 1–55.
- Kline, R. B. (2023). Principles and practice of structural equation modeling (5th ed.). Guilford.
- Marsh, H. W., Hau, K.-T., & Wen, Z. (2004). In search of golden rules. Structural Equation Modeling, 11(3), 320–341.