Reticular Action Model - Model-Implied Mean Vector \(\boldsymbol{\mu} \left( \boldsymbol{\theta} \right)\)

Derives the model-implied mean vector \(\boldsymbol{\mu} \left( \boldsymbol{\theta} \right)\) using the Reticular Action Model (RAM) notation.

rammutheta(M, A, filter)

Arguments

M	m x 1 numeric vector \(\mathbf{M}_{m \times 1}\). Mean structure. Vector of means and intercepts.
A	m x m numeric matrix \(\mathbf{A}_{m \times m}\). Asymmetric paths (single-headed arrows), such as regression coefficients and factor loadings.
filter	k x m numeric matrix \(\mathbf{F}_{k \times m}\). Filter matrix used to select variables.

Value

Returns the model-implied mean vector \( \boldsymbol{\mu} \left( \boldsymbol{\theta} \right)\) derived from the \(\mathbf{A}\), \(\mathbf{F}\), \(\mathbf{I}\), matrices and \(\mathbf{M}\) vector.

Details

The model-implied mean vector \(\boldsymbol{\mu} \left( \boldsymbol{\theta} \right)\) as a function of Reticular Action Model (RAM) matrices is given by

$$ \boldsymbol{\mu} \left( \boldsymbol{\theta} \right) = \mathbf{F} \left( \mathbf{I} - \mathbf{A} \right)^{-1} \mathbf{M} $$

where

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

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

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

• \(\mathbf{M}_{m \times 1}\) represents the mean structure, that is, a vector of means and intercepts,

• \(k\) 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(), ramSigmatheta(), ramsigma2()