SRMR for Models with Covariates

Abstract

The standardized root mean squared residual (SRMR) is commonly reported to evaluate approximate fit of latent variable models. As traditionally defined, SRMR summarizes the discrepancy between observed covariance elements and implied covariance elements. However, current applications of latent variable models often include additional features like overidentified mean structures and covariates, to which the traditional SRMR definition is not applicable. To date, SRMR extensions for models with covariates have received limited attention.Nonetheless, mainstreamsoftware provide SRMRformodelswith covariates, but values differ based on model specification and differ across programs. The goal of this paper is to formalize SRMR definitions for models with covariates. We develop possible SRMR definitions corresponding to different model specifications with covariates, discussing the advantages and disadvantages of each. Importantly, some SRMR definitions are susceptible to confoundingmisfit andmodel size such that SRMR values systematically decrease and suggest better fit when covariates are present, even if covariates have null effects. The primary conclusion is that there may not be a single unifying SRMR definition for covariates, but practically, researchers reporting SRMR with covariates should be aware (a) which definition is being used and (b) which information is and is not included in the particular definition.