Extending the SRMR for SEMs With Mean Structures and Covariates

• In Event: Past, Present, and Future of Model Fit: Modern Solutions to Historical Challenges

Abstract

The standardized root mean squared residual (SRMR) serves as an indicator of model-data fit by calculating the discrepancy between the sample and model-implied covariance matrices. The SRMR has been characterized as a standardized effect size for the likelihood ratio test of the null hypothesis that states the model-implied covariance matrix is equal to the covariance matrix from the observed data (Maydeu-Olivares, 2017; Maydeu-Olivares, Shi, & Rosseel, 2018). The original SRMR definition is valid for covariance structure models like factor analyses because the mean structure is typically saturated or absent. It is not, however, suitable for structural models of means and covariances such as the latent curve model (Leite & Stapleton, 2011; Wu & West, 2010). When applied to latent curve models (or any model with an overidentified mean structure), information about the model fit for the means (e.g., the average growth trajectory) is absent despite its central importance to the modeling exercise.

In addition to the mean structure being left out of the SRMR, there has been little study of the SRMR in the context of structural models with manifest covariates. Specifically, how a covariate is specified impacts the likelihood function of the model. If a covariate is specified as exogenous, the likelihood is expressed conditionally with respect to that covariate. Alternatively, an endogenous covariate specification brings the covariate into the likelihood resulting in a likelihood for the joint distribution of the outcome variables and that covariate. The model-implied moments produced by the model will differ depending on which covariate specification is adopted, as will the resulting SRMR. Furthermore, it is not clear if models employing exogenous covariates should compute the SRMR based on the residual (conditional) moments or the marginals. Unfortunately, SEM software will provide SRMR for these types of models despite the lack of rigorous investigation in the methodological literature about the best way to compute SRMR in such contexts. Our intent is therefore to clarify these issues so that both researchers and software can adopt best practice.

This paper illustrates the appropriate implementation of SRMR for structural equation models with mean structures and covariates with a particular focus on the latent curve model with covariates. We start by reviewing and extending residual fit indices (the broader family of fit indices to which SRMR belongs) for mean structures. We then review the SEM with covariates and show how exogenous and endogenous covariate specifications impact the model structure, which impact the model-implied moments and the SRMR. We then discuss how to properly calculate the SRMR so that it accurately reflects the fit of a model when covariates are present. An empirical example is provided to show how SRMR for latent curve models can vary widely depending on the SRMR definition used. A simulation then demonstrates how some definitions of SRMR are susceptible to be artificially optimistic when covariates are present whereas the implementation we propose is not.

Authors

• Tyler Matta, NWEA

• Daniel McNeish, Arizona State University