Hu and Bentler's Cut-offs in CFA and SEM: What They Mean, How to Report Them and When They Fail You

Your CFA is done. Four numbers come out that you have seen a thousand times in other papers, you put them in a table, and now comes the sentence: "the model showed acceptable fit". The question you are asking yourself is whether it really is acceptable, and the short answer is that it depends on which figure you put under each index, what you are comparing it against, and which model you estimated. This guide explains where the cut-offs everyone cites come from, what each index measures, how to write the fit line so a reviewer does not send it back, and in which cases, and there are some, the cut-offs do not say what you think.

Where the .95, .06 and .08 come from

Before 1999 the convention was Bentler and Bonett's (1980): an incremental index above .90 indicated good fit. Hu and Bentler (1999) ran a simulation with three-factor, fifteen-indicator factor models, samples between 150 and 5,000 cases, and two deliberate kinds of misspecification (omitted cross-loadings and misplaced factor covariances). They then looked at which value of each index best separated correct models from misspecified ones. That is where the figures cited today as if they were thresholds come from: CFI and TLI close to .95, RMSEA close to .06, SRMR close to .08.

Two qualifications that the article itself spells out and that get lost when it is cited from memory. First: they propose a combinational rule, not four independent cut-offs. A model is retained when SRMR is close to .08 and, in addition, one of the other indices goes with it (CFI or TLI close to .95, or RMSEA close to .06). Second: they write "close to" in every case, because the values depended on sample size and on the kind of misspecification simulated. A CFI of .94 is not a fail; a CFI of .95 is not a guarantee.

Marsh, Hau and Wen (2004) said it in the title of their paper: In search of golden rules. They warned that Hu and Bentler's cut-offs were being generalised to models with far more indicators, ordinal items and structures the simulation never evaluated, and that under those conditions .95 rejects reasonable models. That warning is twenty years old and still has not reached many Results sections.

What each index checks

The four look at different things in the same model, which is why they are reported together: a model can pass one and fail another, and it is the combination that informs.

χ² and its degrees of freedom

The χ² tests the hypothesis that the covariance matrix reproduced by the model equals the population one. With large samples any small discrepancy makes it significant, so in a CFA with 500 cases you get p < .001 almost always and that tells you nothing on its own. It is reported all the same, with its degrees of freedom, because the other indices are computed from it and because the reviewer wants to see the size of the model. The χ²/df ratio that some textbooks recommend below 3 (or 5) is a convention with no clear simulation basis: you may report it, but do not rest the fit on it.

CFI and TLI: what you are being compared against

These are incremental indices: they compare your model with the null model, in which all observed variables are uncorrelated. CFI (Bentler, 1990) runs from 0 to 1 and does not penalise complexity; TLI (Tucker and Lewis, 1973) does, and can exceed 1 or drop below 0. The .95 cut-off applies to both, and TLI usually comes out a little below CFI for the same model. One practical consequence: if your observed variables are weakly correlated with each other, the null model already fits relatively well and your CFI stays low even when the model is correct. That is why Kenny (2015) recommends looking at the RMSEA of the null model: if it is below .158, CFI and TLI are not informative.

RMSEA and its 90% interval

The RMSEA (Steiger, 1990; Browne and Cudeck, 1993) measures discrepancy per degree of freedom, correcting for sample size. Browne and Cudeck proposed .05 as close fit and .08 as reasonable; Hu and Bentler lowered the cut-off to .06. It is reported with its 90% confidence interval, not 95%, because the close-fit test carried out with it is one-sided: you look at whether the lower bound is below .05 and whether the upper bound is below .08 or .10 (MacCallum, Browne and Sugawara, 1996). The p value printed next to it, p(RMSEA ≤ .05), is the probability from that test.

SRMR: the only one that looks at the residuals

The SRMR is the root of the mean squared standardised residual: the average difference between the observed correlations and those the model reproduces. It is the most direct index of all, does not depend on the null model, and is the one Hu and Bentler put at the base of their combinational rule. An SRMR of .08 means that, on average, your reproduced correlations deviate .08 from the observed ones.

The table you can copy, and what it does not say

What the table does not say is under which conditions those figures were computed. Hu and Bentler simulated with maximum-likelihood estimation on continuous variables, fifteen indicators and three factors. If your model departs from that, and most do, the table is a starting point, not a verdict.

When the cut-offs become too demanding

With many indicators. For the same misspecification, CFI and TLI go down and RMSEA goes up when the model has more observed variables (Kenny and McCoach, 2003; Shi, Lee and Maydeu-Olivares, 2019). A CFA of a 40-item scale with a CFI of .91 can be as well specified as a 12-item one with .96. With long instruments, many authors also report fit by subscale or use parcels, and in any case it is worth saying explicitly in the Discussion that instrument length penalises the indices.

With ordinal items and robust estimators. With Likert items of few categories and estimators such as WLSMV or DWLS, CFI and TLI come out systematically higher and RMSEA lower than with ML on the same data, regardless of whether the model is correct (Xia and Yang, 2019). Hu and Bentler's cut-offs were not derived for that case. Report which estimator you used, and if fit is comfortable with WLSMV do not sell it as strong evidence; if it is borderline with ML on four-category items, that is not strong evidence against either.

With small samples. Below 200 cases the RMSEA is biased upwards and its interval widens; the TLI also becomes unstable. There the RMSEA interval matters more than the point value, and a lower bound below .05 with an upper bound above .10 means what it says: there is not enough precision to decide.

In SEM: two fits, not one

A structural equation model has a measurement part (how the items load on the factors) and a structural part (how the factors relate to each other). Global indices blend the two, and that is the problem: an SEM with an excellent measurement model and a misspecified structural part can give a CFI of .96, because the measurement part, which has many more degrees of freedom, dominates the index (Anderson and Gerbing, 1988; Kline, 2023).

The practice that solves this is the two-step procedure: first the measurement model is estimated as a CFA in which all factors correlate freely, its fit is reported, and only then is the structure imposed. The comparison between the two models (the χ² difference with its difference in degrees of freedom, and the drop in CFI) tells you how much fit the structural part costs. If the χ² rises a lot when moving from the measurement model to the structural one, the relations between factors you fixed to zero or constrained are what the model fails to reproduce, and you do not see that in a global CFI of .95.

When reporting an SEM, then, three things go in: the fit of the measurement model, the fit of the structural model and the comparison between them. And in the structural model, besides the indices, the standardised coefficients with their intervals and the explained variance of each endogenous variable, which is what actually answers the hypothesis.

Path analysis: why the RMSEA comes out bad even when the model is fine

A path analysis is an SEM with observed variables instead of factors: each construct enters as a total score. Those models have few variables and, above all, few degrees of freedom: a simple mediation model with three variables has one degree of freedom if you fix the direct effect to zero, and zero if you leave it free. And there the RMSEA fails, not because of your model but because of its formula.

The RMSEA is computed as the square root of (χ² − df) divided by df × (N − 1), truncated at zero. With df in the denominator, when the degrees of freedom are one or two any small excess of the χ² over its degrees of freedom is divided by a tiny number and the index blows up. Kenny, Kaniskan and McCoach (2015) simulated this with correctly specified models: with small df and moderate samples, the RMSEA of a true model frequently falls above .10, and its 90% interval is so wide that it runs from 0 to values any reviewer would reject. Their recommendation is not to compute the RMSEA, or not to interpret it, when the degrees of freedom are very few, and not to use the close-fit test in that case.

With zero degrees of freedom the situation is different and simpler: the model is saturated, it reproduces the covariance matrix exactly, χ² is zero and every index gives perfect fit by construction. There is no fit to report there, and writing "the model showed excellent fit (CFI = 1.00, RMSEA = .00)" is reporting a tautology. In a saturated model what gets reported are the coefficients, their intervals and the indirect effects with bootstrap.

What you can do in a path analysis with few degrees of freedom: report the χ² with its df and its p, which with few df is interpretable because the sample no longer forces it to be significant; report SRMR and CFI, which do not divide by df; and if you report the RMSEA, give it with its interval and a sentence saying that with df = 1 the index is not informative, citing Kenny et al. (2015). A reviewer who knows the problem appreciates it; one who does not has the citation to check it.

How the fit line is written

The fit line goes in the text, not only in the table, and carries every index in the same format. In APA 7, statistics are italicised when they are Latin letters (N, p, df), χ² is not, decimals drop the leading zero when the value cannot exceed 1, and the RMSEA carries its interval in brackets.

For a CFA:

The three-factor correlated model showed adequate fit to the data, χ²(87) = 214.36, p < .001, CFI = .953, TLI = .943, RMSEA = .062, 90% CI [.052, .072], SRMR = .046. Standardised loadings ranged from .48 to .84 and all were significant (p < .001).

Notice that the TLI falls below .95 and the sentence says "adequate", not "excellent". That is exactly what to do: describe what comes out with the word it deserves and let the set support the decision. If one index falls clearly outside, say so, and in the Discussion explain why (long instrument, ordinal items, sample) without pretending it did not happen.

For an SEM, with the two-step procedure:

The measurement model fitted well, χ²(146) = 301.12, p < .001, CFI = .961, TLI = .954, RMSEA = .051, 90% CI [.043, .059], SRMR = .041. The structural model retained the fit, χ²(149) = 318.40, p < .001, CFI = .958, TLI = .952, RMSEA = .052, 90% CI [.044, .060], SRMR = .047; the difference from the measurement model was Δχ²(3) = 17.28, p < .001, ΔCFI = .003. The model explained 34% of the variance in the dependent variable.

For a path analysis with few degrees of freedom:

The mediation model, with the direct effect fixed to zero, had one degree of freedom, χ²(1) = 3.91, p = .048, CFI = .987, SRMR = .021. The RMSEA was .071, 90% CI [.004, .162]; with one degree of freedom this index and its interval are not informative (Kenny et al., 2015), so the evaluation of fit rests on the χ², the CFI and the SRMR. The indirect effect was ab = 0.14, 95% bootstrap CI [0.07, 0.23].

The figures in the three examples are illustrative: what you should copy is the structure, the order and the format, not the numbers.

What a reviewer looks at before the cut-offs

Before checking whether your CFI passes .95, a reviewer with experience in models looks at four things a global fit does not show. Whether the estimator is the right one for the type of data (robust ML with non-normal continuous variables, WLSMV with ordinal items of few categories). Whether the model has been compared with any plausible alternative, because a good fit of one model does not rule out another fitting just as well. Whether the modifications you made (correlated errors, cross-loadings) were planned before seeing the data or came out of the modification indices, and in that case whether you say so. And whether the coefficients make sense: a factor with loadings of .30 and a CFI of .97 is a factor that does not measure much, fit aside.

Hu and Bentler's cut-offs are a useful convention: they give you a common language with the reviewer and a reasonable threshold for the typical case. They stop being useful the moment the figure replaces judgement, and judgement, in a CFA or an SEM, consists of knowing which model you estimated, with which data, and what each index is telling you about it.

Frequently asked questions

Is CFI = .94 a bad fit?

Not on its own. Hu and Bentler write "close to .95", and Marsh, Hau and Wen (2004) show that with many indicators a .94 is compatible with a well-specified model. Look at the other indices, the size of the model and the estimator, and describe the fit with the word it deserves.

Do I report the RMSEA with a 90% or a 95% interval?

90%. It is the one that matches the one-sided close-fit test (MacCallum et al., 1996) and the one that lavaan, Mplus and AMOS return by default. Reporting the 95% one is not an error, but it is what nobody expects.

My path analysis has RMSEA = .12 and the other indices are fine. What do I do?

Look at the degrees of freedom. If they are one or two, the RMSEA is not informative (Kenny et al., 2015): report it with its interval, say why you do not interpret it, and rest the fit on the χ², the CFI and the SRMR. If the degrees of freedom are plentiful, then the .12 does say something and you have a specification problem.

Can I use Hu and Bentler's cut-offs with WLSMV?

With care. With estimators for ordinal data the indices come out more favourable than with ML on the same data (Xia and Yang, 2019), so a CFI of .97 with WLSMV is not equivalent to a .97 with ML. Report the estimator and do not present the fit as strong evidence.

Do I need to report all four indices?

At minimum the χ² with its df, CFI, RMSEA with interval and SRMR. TLI is usually added. Reporting only the ones that came out well is the clearest signal that the others came out badly.

References

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Bentler, P. M. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107(2), 238–246. https://doi.org/10.1037/0033-2909.107.2.238

Bentler, P. M., & Bonett, D. G. (1980). Significance tests and goodness of fit in the analysis of covariance structures. Psychological Bulletin, 88(3), 588–606. https://doi.org/10.1037/0033-2909.88.3.588

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: Conventional criteria versus new alternatives. Structural Equation Modeling, 6(1), 1–55. https://doi.org/10.1080/10705519909540118

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Kline, R. B. (2023). Principles and practice of structural equation modeling (5th ed.). Guilford Press.

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Shi, D., Lee, T., & Maydeu-Olivares, A. (2019). Understanding the model size effect on SEM fit indices. Educational and Psychological Measurement, 79(2), 310–334. https://doi.org/10.1177/0013164418783530

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