On the Computation of the RMSEA and CFI from the Mean-And-Variance Corrected Test Statistic with Nonnormal Data in SEM

ABSTRACT

A new type of nonnormality correction to the RMSEA has recently been developed, which has several advantages over existing corrections. In particular, the new correction adjusts the sample estimate of the RMSEA for the inflation due to nonnormality, while leaving its population value unchanged, so that established cutoff criteria can still be used to judge the degree of approximate fit. A confidence interval (CI) for the new robust RMSEA based on the mean-corrected (“Satorra-Bentler”) test statistic has also been proposed. Follow up work has provided the same type of nonnormality correction for the CFI (Brosseau-Liard & Savalei, 2014). These developments have recently been implemented in lavaan. This note has three goals: a) to show how to compute the new robust RMSEA and CFI from the mean-and-variance corrected test statistic; b) to offer a new CI for the robust RMSEA based on the mean-and-variance corrected test statistic; and c) to caution that the logic of the new nonnormality corrections to RMSEA and CFI is most appropriate for the maximum likelihood (ML) estimator, and cannot easily be generalized to the most commonly used categorical data estimators.

KEYWORDS:

• Structural equation modeling
• nonnormal data
• RMSEA
• fit index

Notes

1 In some software packages, n may instead be equal to the total sample size minus 1.

2 An older version of the mean-and-variance-corrected chi-square that required an adjustment to the degrees of freedom is no longer recommended, as it performs essentially identically to the new version (e.g., Savalei & Rhemtulla, 2013), which does not require this adjustment.

3 EQS 6.3 (Bentler, 2008 is different in that the “MLM” and “MLMV” options (using lavaan and Mplus terminology) are combined into a single run (“METHOD=ML,ROBUST”), which produces a host of test statistics for nonnormal data, including the MLM and MLMV chi-squares, but only one nonnormality-adjusted RMSEA, which is obtained by substituting the MLM chi-square into equation (1)(1) . However, the robust RMSEA and CI advocated here are also available in EQS if one sets RFIT = 1 in /SPECIFICATIONS. The robust CFI advocated here is not available.

4 It is best to check the printed value using this computation anyway. For instance, in EQS 6.3, the “scaling factor” printed is actually 1/c_n rather than c_n. Similarly, Asparouhov and Muthen (2010) define what is a_n in this manuscript as 1/a_n. The notation in this manuscript is consistent with what lavaan prints for scaling corrections.

5 In fact, one can imagine a “zero-order” sample RMSEA estimate computed simply as . This estimate will have the same population value as the ML-based RMSEA, but very poor small sample performance.

6 In software that does not print the second-order scaling correction, one can obtain it from all three test statistics using the equation in Appendix B. However, if this is the case, it is much easier to compute the robust RMSEA from the MLM statistic using one of the expressions given in (6(6) ).