Reticular Action Model - Model-Implied Variance-Covariance Matrix \(\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)\)

Derives the model-implied variance-covariance matrix \(\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)\) using the Reticular Action Model (RAM) notation.

ramSigmatheta(A, S, filter)

Arguments

A	m x m numeric matrix \(\mathbf{A}_{m \times m}\). Asymmetric paths (single-headed arrows), such as regression coefficients and factor loadings.
S	m x m numeric matrix \(\mathbf{S}_{m \times m}\). Symmetric paths (double-headed arrows), such as variances and covariances.
filter	k x m numeric matrix \(\mathbf{F}_{k \times m}\). Filter matrix used to select variables.

Value

Returns the model-implied variance-covariance matrix \(\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)\) derived from the A, S, and filter matrices.

Details

The model-implied variance-covariance matrix \(\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)\) as a function of Reticular Action Model (RAM) matrices is given by

$$ \boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right) = \mathbf{F} \left( \mathbf{I} - \mathbf{A} \right)^{-1} \mathbf{S} \left[ \left( \mathbf{I} - \mathbf{A} \right)^{-1} \right]^{T} \mathbf{F}^{T} $$

where

• \(\mathbf{A}_{m \times m}\) represents asymmetric paths (single-headed arrows), such as regression coefficients and factor loadings,

• \(\mathbf{S}_{m \times m}\) represents symmetric paths (double-headed arrows), such as variances and covariances,

• \(\mathbf{F}_{k \times m}\) represents the filter matrix used to select the observed variables,

• \(\mathbf{I}_{m \times m}\) represents an identity matrix,

• \(k\) number of observed variables,

• \(q\) number of latent variables, and

• \(m\) number of observed and latent variables, that is \(k + q\) .

References

McArdle, J. J. (2013). The development of the RAM rules for latent variable structural equation modeling. In A. Maydeu-Olivares & J. J. McArdle (Eds.), Contemporary Psychometrics: A festschrift for Roderick P. McDonald (pp. 225--273). Lawrence Erlbaum Associates.

McArdle, J. J., & McDonald, R. P. (1984). Some algebraic properties of the Reticular Action Model for moment structures. British Journal of Mathematical and Statistical Psychology, 37 (2), 234--251.

See also

Other SEM notation functions: ramM(), rammutheta(), ramsigma2()