Last updated: 2019-09-29
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Knit directory: mcfa-fit/
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The standardized root mean square residual (SRMR) is an aggregate measure of the deviation of the observed correlation matrix to the model implied correlation matrix Joreskog1981. Ideally, the average difference between the observed and expected correlations is minimal, and smaller values represent better fitting models. In MCFA, two difference SRMR are reported based on each level’s covariance matrix. The level-1 SRMR is known as SRMR-Within (SRMRW). The level-2 SRMR is known as SRMR-Between (SRMRB). The computation of each of these measures is roughly equivalent conditional on which covariance matrix is under consideration. Generally, SRMR is computed as the standardized difference between the observed correlations and the model implied correlations about variables as shown below. \[ SRMR = \sqrt{\frac{2 \sum_{j=1}^{p} \sum_{k=j}^{i} {\left( \frac{s_{jk} - \hat{\sigma}_{jk} }{ \sqrt{ s_{jj}s_{kk}} } \right)}^2 }{p(p+1)}} \] where \(p\) is the total number of variables in the model, \(s_{jk}\) and \(\sigma_{jk}\) are the sample and model implied, respectively, covariance between the \(j^{th}\) and \(k^{th}\) variables. For the SRMR, generally acceptable values less than .08 are used. However, Hu1999 suggested values less .06 alone or .08 in combination of other within recommended ranges are indicative of good fit.
In Mplus, SRMR is estimated slightly differently than shown in Equationeq:srmr. The computation extends the definition above by accounting for the meanstructure, multilevel structure, and categorical nature if applicable. The technical details of the computation of SRMRW and SRMRB in Mplus v8.2 is out of scope for this article, and the interested reader is referred to MuthenSRMR for more details.