SEM & CFA Fit Index Checker

Which cutoffs it checks

The tool compares each index against the standard thresholds and colour-codes them: CFI and TLI ≥ .95 (good) or ≥ .90 (acceptable); RMSEA ≤ .06 (good) or ≤ .08 (acceptable), always reported with its 90% confidence interval; SRMR ≤ .08 (good) or ≤ .10 (acceptable). The strict good-fit values (CFI/TLI ≥ .95, RMSEA ≤ .06, SRMR ≤ .08) follow Hu & Bentler (1999; https://doi.org/10.1080/10705519909540118); the more lenient acceptable values come from other, mostly older sources: CFI/TLI ≥ .90 from Bentler & Bonett (1980; https://doi.org/10.1037/0033-2909.88.3.588), applied to CFI by convention; RMSEA ≤ .08 from Browne & Cudeck (1993); SRMR ≤ .10 from Schermelleh-Engel, Moosbrugger & Müller (2003). The model χ² is shown for completeness but carries no verdict, because it is sensitive to sample size: with large N it is almost always significant even when the model fits well, so the overall verdict rests on the approximate-fit indices.

APA style drops the leading zero for statistics that cannot exceed 1 (for example .95, not 0.95), and reports p < .001 rather than an exact tiny value. The tool applies both rules for you. The χ², df and the information criteria print normally because they can exceed 1.

RMSEA, its confidence interval and PCLOSE

RMSEA measures how badly the model misses, per degree of freedom and per case. It takes the part of the χ² that exceeds its df — the estimated noncentrality — and scales it by the df and the sample size:

RMSEA equals the square root of the larger of chi-square minus df and zero, divided by the square root of df times the sample size.

Because RMSEA is estimated from a single sample, it carries uncertainty, and the interval is the honest way to show it. The 90% confidence interval is found by asking which population noncentrality values would still make your χ² a plausible result: the lower bound is the noncentrality that puts your χ² at the 95th percentile, the upper bound the one that puts it at the 5th. Both bounds are then put back on the RMSEA scale with the same formula. This needs the noncentral χ² distribution, which is why the tool loads a small extra script the moment you give it a sample size — and why so many people type the interval by hand instead. A wide interval is itself a finding: with a small sample the upper bound can sit far above .08 even when the point estimate looks fine, and reviewers read that bound.

PCLOSE is the p value of the test of close fit. Its null hypothesis is that RMSEA is .05 or lower in the population, so a PCLOSE above .05 means close fit cannot be rejected — the result you want — and a PCLOSE below .05 says the model is worse than close fitting. It is not a test of perfect fit, and it inherits the sample-size sensitivity of every χ²-based test. lavaan prints it as rmsea.pvalue and AMOS prints it as PCLOSE. The idea of judging models by close fit rather than exact fit, and the interval itself, come from MacCallum, Browne and Sugawara (1996; https://doi.org/10.1037/1082-989X.1.2.130).

Comparing two nested models

Two models are nested when they are fitted to the same data, the same cases and the same variables, and one of them is the other plus constraints — equal loadings across groups, a path fixed to zero, equal intercepts. For such a pair the difference in χ² is itself a χ² statistic on the difference in df, so a p value follows directly. In measurement invariance this is the whole workflow: configural, then metric (equal loadings), then scalar (equal intercepts), each step tested against the one before it.

The catch is sample size again: with several hundred cases the Δχ² test rejects differences too small to matter. That is why Cheung and Rensvold (2002; https://doi.org/10.1207/S15328007SEM0902_5) proposed reading ΔCFI instead, with a drop of more than .01 marking a real loss of fit, and why Chen (2007; https://doi.org/10.1080/10705510701301834) added ΔRMSEA ≤ .015 as a companion rule. Report both, and say which one your decision rests on. The Compare tab does the arithmetic and writes the sentence.

χ²/df, AIC, BIC and SABIC

The χ²/df ratio is an old rule of thumb: 2 or lower is usually called good and 3 or lower acceptable (Carmines & McIver, 1981; Schermelleh-Engel, Moosbrugger & Müller, 2003). It shares the weakness of the χ² test itself, so treat it as a rough signal and never as the deciding number. AIC, BIC and SABIC are for comparing models that are not nested, where a Δχ² test does not apply — the model with the lower value is preferred. This page writes them on the χ² scale: AIC = χ² + 2q, BIC = χ² + q·ln(N) and SABIC = χ² + q·ln((N+2)/24), where q is the number of free parameters. Those forms differ from the likelihood-based values lavaan prints by a constant that is identical for every model fitted to the same data, so the difference between two models is the same as lavaan's while the absolute numbers are not. Take the values you report in a paper from your software, and use the ones here to compare models on the same data.