A data frame that provides the summary of population misfit and misspecified-parameter values imposed on the real parameters.
The discrepancy value (\(f_0\); Browne & Cudeck, 1992) is calculated by
$$ F_0 = tr\left( \tilde{\Sigma} \Sigma^{-1} \right) - \log{\left| \tilde{\Sigma} \Sigma^{-1} \right|} - p + \left( \tilde{\mu} - \mu \right)^{\prime} \Sigma^{-1} \left( \tilde{\mu} - \mu \right). $$
where \(\mu\) is the model-implied mean from the real parameters, \(\Sigma\) is the model-implied covariance matrix from the real parameters, \(\tilde{\mu}\) is the model-implied mean from the real and misspecified parameters, \(\tilde{\Sigma}\) is the model-implied covariance matrix from the real and misspecified parameter, p is the number of indicators. For the multiple groups, the resulting \(f_0\) value is the sum of this value across groups.
The root mean squared error of approximation (rmsea) is calculated by
$$rmsea = \sqrt{\frac{f_0}{df}}$$
where \(df\) is the degree of freedom in the real model.
The standardized root mean squared residual (srmr) can be calculated by
$$srmr = \sqrt{\frac{2\sum_{g} \sum_{i} \sum_{j \le i} \left( \frac{s_{gij}}{\sqrt{s_{gii}}\sqrt{s_{gjj}}} - \frac{\hat{\sigma}_{gij}}{\sqrt{\hat{\sigma}_{gii}}\sqrt{\hat{\sigma}_{gjj}}} \right)}{g \times p(p + 1)}}$$
where \(s_{gij}\) is the observed covariance between indicators i and j in group g, \(\hat{\sigma}_{ij}\) is the model-implied covariance between indicators i and j in group g, p is the number of indicators.