Absolutely, MLM in CFA can be confusing with all these CFI/TLI variations. Standard CFI/TLI are the usual suspects, but they're unreliable for wonky data (not perfectly normal). Scaled CFI/TLI are a bit better, but still not ideal for super messed up data.

In confirmatory factor analysis (CFA), a common practice is to evaluate model fit using various fit indices such as comparative fit index (CFI) and Tucker-Lewis index (TLI). When using the 'MLM' (maximum likelihood estimation with robust standard errors and mean- and variance-adjusted chi-square test statistic) estimator, the terms "standard," "scaled," and "robust" refer to how these fit indices are calculated or adjusted to account for potential issues in the data.

In summary, when using the 'MLM' estimator in CFA, it is recommended to use scaled or robust versions of CFI and TLI to obtain more accurate estimates of model fit, especially when dealing with non-normal data or potential violations of assumptions. These adjusted indices can provide more reliable information about how well the proposed model fits the observed data.

Thank you! Dilshad Altalabani

Viola Lyu - I'm late replying, but I wanted to contribute further to how scaled and robust indices differ. I just looked into this for the last couple hours though I was primarily concerned with scaled vs. robust for WLSMV estimation, not MLM.

First, a related question was asked in the lavaan Google Group which may interest you: https://groups.google.com/g/lavaan/c/DlVaoP41MyI

Second, the "robust" options are alternatives to the scaled versions that, to my knowledge, emerged upon realization that the scaled options were not as appropriate for non-normal estimators as commonly assumed. Scaled options take the test statistic determined through, eg, MLM estimation, and then plug it into the usual CFI, TLI, etc formula without other adjustments. The `lavaan` package was updated with robust fit indices for certain continuous data estimators in Version 0.5-21 (September 2016): https://lavaan.ugent.be/history/dot5.html#version-0.5-21. Robust fit indices were added for other estimators, including those for categorical data, estimator "MLMV", and cases where `missing = "ml"` (i.e., where maximum likelihood imputation is used), in Version 0.6-13 (January 2023): https://lavaan.ugent.be/history/dot6.html#version-0.6-13

Relevant papers cited for the lavaan changes are (among others):

I am unfamiliar with MLM estimation. Scaled or robust may be appropriate; standard most definitely isn't. As to which of standard and robust is better, hopefully the articles above (likely the first two more than the third) will be helpful.

Joseph Sexton Thank you so much, Joseph! Very useful information to me! Good luck with your research.