# Formula for standardized root mean square residual (SRMR) in longitudinal latent variable models (SEM, CFA)

I was wondering what the adaption to the formula should be in case of multiple groups or longitudinal data (i.e. several time points).  As provided in the related / linked question, source: Hu, L.; Bentler, Peter (1999). "Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives". Structural Equation Modeling. 6 (1): 1–55. https://dx.doi.org/10.1080%2F10705519909540118

## 1 Answer

For the multi group SEM, the SRMR is calculated by using a weighted average under the square root where each sample covariance matrix is compared to the model predicted covariance matrix. I did not locate a reference, but I did run a quick multi-group SEM, and then calculated the single group SEMs and confirmed this formula to be correct.

Requested Addition
Demonstration for calculating SRMR for two groups: $$SRMR = \sqrt{\frac{n_1·SRMR_1^2 + n_2·SRMR_2^2}{n_1+n_2}}$$ where $n_i$ and $SRMR_i$ are the sample size and $SRMR$ of group $i$, respectively. (Worked example in comments below.)

• Thanks for replying. Although I understand the idea of a 'weighted average', I am not really good at translating this into a formula. My background involves few mathematical courses. Would you mind illustrating the formula? – Amonet Apr 22 '18 at 13:27
• If you have two groups, $n_1 = 200$, $SRMR_1 = 0.04$, $n_2 = 500$, $SRMR_2 = 0.02$, then $SRMR = \sqrt{\frac{n_1SRMR_1^2 + n_2SRMR_2^2}{n_1+n_2}}=\sqrt{\frac{200·(0.04)^2 + 500·(0.02)^2}{200+400}} = \sqrt{\frac{0.52}{700}} = .0273$. – Gregg H Apr 22 '18 at 14:03
• Much appreciated, 100% clear now. Perhaps you could copy / paste this into your answer. Then I'll accept it as an answer, so other ppl will benefit from this as well :) – Amonet Apr 22 '18 at 14:16
• Happy to do so. – Gregg H Apr 22 '18 at 14:18
• Hi Gregg, I recently returned to this topic and compared the results of 4 cross-sectional CFA models (1 time point each), to a longitudinal CFA model with the 4 time points jointly. I did this in Lavaan and computed the 4 SRMRs separately and then used the formula you illustrated above (the sample sizes slightly differ due to complete case analysis where the number of missing data points differs per time point). Then I compared this number to the SRMR calculated by Lavaan based on the longitudinal model. However, the numbers are rather different: 0.06921215 vs 0.085. Any suggestions why? – Amonet Jun 30 '18 at 19:44