7.12 Standardized Root Mean Squared Residual (SRMSR)

Standardized Root Mean Squared Residual (SRMSR) is frequently used in Structural equation modleing to compare the absolute fit of the data. The SRMSR is calculated as the square root of the sum of the squared differences between the observed correlations and the model-implied correlations for all pairs of items(Ravand & Robitzsch, 2018).

Let us assume that the standardized residual variances and covariances be(Pavlov et al., 2021),

$$\begin{equation} \hat{\varepsilon}_{\mathrm{ij}}=\frac{s_{\mathrm{ij}}-\hat{\sigma}_{\mathrm{ij}}}{\sqrt{s_{\mathrm{ii}} s_{\mathrm{jj}}}} \end{equation},$$

where $S_{ij}$ is the sample covariances between item pairs $i$ and $j$, with the model implied counterpart $\hat{\sigma}_{\mathrm{ij}}$. If $i = j$, then $S_{ii}$ and $\hat{\sigma}_{\mathrm{ii}}$ are the variances.

The sample SRMSR can be obtained as,

$$\begin{equation} \widehat{\mathrm{RMSR}}=\sqrt{\frac{1}{t} \sum_{i \leq j} \hat{\varepsilon}_{\mathrm{ij}}^2}=\sqrt{\frac{1}{t} \hat{\varepsilon}' \hat{\varepsilon}}, \end{equation}$$

Where $t= p(p+1)/2$ denotes the number variances and covariances, and $\hat{\varepsilon}$ is the vector of $t$ standardized residula covariances.

Maydeu-Olivares (2013) suggested to consider models with SRMSR values below 0.05.

References

Maydeu-Olivares, A. (2013). Goodness-of-fit assessment of item response theory models. Measurement: Interdisciplinary Research and Perspectives, 11(3), 71–101.

Pavlov, G., Maydeu-Olivares, A., & Shi, D. (2021). Using the standardized root mean squared residual (SRMR) to assess exact fit in structural equation models. Educational and Psychological Measurement, 81(1), 110–130.

Ravand, H., & Robitzsch, A. (2018). Cognitive diagnostic model of best choice: A study of reading comprehension. Educational Psychology, 38(10), 1255–1277.