Model Fit Indices for Random Effects Models: Translating Model Fit Ideas from Latent Growth Curve Models

Abstract

The latent growth curve modeling (LGM) and random effects modeling (REM) frameworks are analytically and empirically equivalent for intrinsically linear models and used interchangeably for intrinsically nonlinear models. However, while LGM provides overall model fit indices, REM does not. Overall model fit indices are useful because they evaluate how well a specified model fits data. This paper proposes to translate model fit concepts from LGM to REM to help researchers compute overall model fit indices, including the model chi-square (${\chi }^2$), comparative fit index (CFI), root mean squared error of approximation (RMSEA), and standardized root mean squared residual (SRMR). Three empirical examples were used as illustrations.

Keywords:

• Latent growth curve model
• linear and nonlinear models
• overall model fit indices
• random effects model

Acknowledgments

We would like to thank the Longitudinal Methods Development lab at the University of Minnesota for providing SAS software to conduct this study.

Notes

1 This definition is used by PROC NLMIXED in SAS.

2 This definition is used by PROC CALIS in SAS and lavaan package in R. Specifically, the number of unique observations is defined as: $\frac{\mathit{p}\left(\mathit{p}+1\right)}{2\mathrm{ }}+\mathit{p}$ where$\mathrm{ }\mathit{p}$ is the number of measurement occasions. The number of free parameters for a fitted model is the sum of the number of intercepts corresponding to latent variables, number of variances corresponding to latent variables, number of covariances among the latent variables, and number of error variances.

3 The appendix can be found at https://www.dropbox.com/sh/dts69we8be8ozx9/AABlTA-qoW3Gwtn2SJ8_bFl8a?dl=0

4 We want to acknowledge that there is an existing article that presents a workaround for estimating banded error structures in PROC NLMIXED (Harring & Blozis, 2014). However, the code provided in the article requires considerable manipulation of the default code/options and this kind of error structure is not often implemented in real data applications (see e.g., Kohli et al., 2015; Sullivan et al., 2017).