Better Confidence Intervals for RMSEA in Growth Models given Nonnormal Data

Abstract

Currently, the best confidence interval (CI) for RMSEA in covariance structure analysis given nonnormal data is proposed by Brosseau-Liard, Savalei, and Li (BSL). A key assumption for the BSL CI often overlooked is that all the nonzero eigenvalues are equal in a matrix related to the model and data nonnormality. This assumption rarely holds in practice, especially for mean and covariance structure analysis, and violating this assumption can entail serious mistakes when the model’s degrees of freedom are small. One important application of moment structure analysis with small degrees of freedom is growth models. In this paper, we propose a new CI method for RMSEA in growth models given nonnormal data, without assuming equal eigenvalues. Although we focus on growth models, our method applies to any other models in moment structure analysis. Simulation results verify the new method is trustworthy and better than all the current methods.

Keywords:

• The root mean square error of approximation
• confidence interval
• nonnormal data
• robust method

Notes

¹ The “lavaan” package in R denotes ${\hat{\epsilon }}_{\mathrm{M}\mathrm{L}\mathrm{M}}$ and MLM CI as “scaled RMSEA,” and denotes ${\hat{\epsilon }}_{\mathrm{B}\mathrm{S}\mathrm{L}}$ and BSL CI as “robust RMSEA.” When the argument “estimator = MLM” is used in lavaan, it will calculate both the MLM and BSL estimates.

² The central limit theorem for independent non-identical variables (e.g., Lehmann, 2004, Section 2.7) can be used to study ${\sum }_{j=1}^d{\pi }_j{\chi }_1^2$. Because ${\chi }_d^2$ is the sum of independent and identical variables, the regular central limit theorem is applicable.

³ See https://bit.ly/2L5GcfB for all the R functions and population moments for the simulation studies.