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Maximum likelihood with robust standard errors (MLR) is a commonly used estimation method for structural equation models when observed data are continuous. MLR is an estimation method under normal theory maximum likelihood where the observed data are assumed to follow a multivariate normal distribution. The robust nature of MLR is in order to more accurately estimate standard errors. In this study, the standard errors are not directly of interest, so we will focus on the estimation of the fit function under MLR where the resulting \(F_{ML}\) is dervied. A discussion of the standard errors is left a paper forthcoming on parameter estimation and recover in ML-CFA.

For ML-CFA under MLR estimation, the general idea is to find parameters (\(\theta\)) that maximize the likelihood function of the observed data given a distributional assumption (nromal theory typically in social science). In ML-CFA, the model is composed of two major pieces: 1) a model for the pooled within-group covariance matrix (\(\Sigma_W\)), and 2) a model for the between group covariance matrix (\(\Sigma_B\)).

As shown in Ryu (2014), Bentler & Liang (2003), and Liang & Benlter (2004), the sample estimators for the two population covariance matrices for each group (\(j\)) are \[S_{W_j} = {(n_j -1)}^{-1} \sum_i^{n_j} (\mathbf{y}_{ij} - \bar{\mathbf{y}}_{j}){(\mathbf{y}_{ij} - \bar{\mathbf{y}}_{j})}^{\prime}\] \[S_{B_j} = n_j (\bar{\mathbf{y}}_{j} - \bar{\mathbf{y}}){(\bar{\mathbf{y}}_{j} - \bar{\mathbf{y}})}^{\prime}\] where,

• \(n_j\) is the sample size of group \(j\) (\(N = \sum_{\forall j} n_j\));
• \(\mathbf{y}_{ij}\) is the observed vector of responses for individual \(i\) in group \(j\);
• \(\bar{\mathbf{y}}_{j}\) is the vector of average of observed responses in group \(j\);
• \(\bar{\mathbf{y}}\) is the vector of average responses across all groups;
• \(S_{Wj}\) is the within-group covariance matrix of group \(j\); and
• \(S_{Bj}\) is the between-group matrix of random effects for covariance matrix for group \(j\).

The model-implied covariance matrices \(\Sigma_W(\theta), \Sigma_B(\theta),\) and \(\Sigma_{gj}(\theta)\) are needed for the MLR fit function.
We need the \(\Sigma_{gj}(\theta)\) model-implied covariance matrix to help identify the average difference between the observed and model-implied covariances, which is defined as \[\Sigma_{gj}(\theta) = \Sigma_B(\theta) + n_j^{-1}\Sigma_W(\theta)\] which can be interpreted as the group size weighted deviation of group \(j\) from the average group covariance matrix.

The maximum likelihood fit function has been shown to be:

\[F_{ML}=\sum_{j=1}^{J}(n_j-1) \left\lbrace \mathrm{log}\mid \Sigma_{W}(\theta) \mid + tr\left(\Sigma_W^{-1}(\theta)S_{w_j}\right)\right\rbrace + \sum_{j=1}^{J} \left\lbrace\mathrm{log}\mid \Sigma_{gj}(\theta) \mid + tr\left(\Sigma_{gj}^{-1}(\theta)S_{Bj}\right)\right\rbrace\]

where the interested reader is refered to Muthen (1994), Bentler & Liang (2003), Liang & Bentler (2004), and Ryu (2014) for more information.