Sigmatheta: Model-Implied Variance-Covariance Matrix \boldsymbol{Sigma}...
In jeksterslab/ram: Reticular Action Model (RAM) Notation

Description Usage Arguments Details Value Author(s) References See Also

View source: R/Sigmatheta.R

Derives the model-implied variance-covariance matrix \boldsymbol{Σ} ≤ft( \boldsymbol{θ} \right) using the Reticular Action Model (RAM) notation.

1	Sigmatheta(A, Omega, filter = NULL)

`A`	`t x t` numeric matrix \mathbf{A}_{t \times t}. Asymmetric paths (single-headed arrows), such as regression coefficients and factor loadings.
`Omega`	`t x t` numeric matrix \boldsymbol{Ω}_{t \times t}. Symmetric paths (double-headed arrows), such as variances and covariances.
`filter`	`j x t` numeric matrix \mathbf{F}_{j \times t}. Filter matrix used to select variables. If `filter = NULL`, the filter matrix \mathbf{F} is assumed to be a t \times t identity matrix.

The model-implied variance-covariance matrix \boldsymbol{Σ} ≤ft( \boldsymbol{θ} \right) as a function of Reticular Action Model (RAM) matrices is given by

\boldsymbol{Σ} ≤ft( \boldsymbol{θ} \right) = \mathbf{F} ≤ft( \mathbf{I} - \mathbf{A} \right)^{-1} \boldsymbol{Ω} ≤ft[ ≤ft( \mathbf{I} - \mathbf{A} \right)^{-1} \right]^{\mathsf{T}} \mathbf{F}^{\mathsf{T}}

where

\mathbf{A}_{t \times t} represents asymmetric paths (single-headed arrows), such as regression coefficients and factor loadings,
\boldsymbol{Ω}_{t \times t} represents symmetric paths (double-headed arrows), such as variances and covariances,
\mathbf{F}_{j \times t} represents the filter matrix used to select the observed variables,
\mathbf{I}_{t \times t} represents an identity matrix,
j number of observed variables,
k number of latent variables, and
t number of observed and latent variables, that is j + k .

Returns the model-implied variance-covariance matrix \boldsymbol{Σ} ≤ft( \boldsymbol{θ} \right) derived from the A, Omega, and filter matrices.

Ivan Jacob Agaloos Pesigan

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.

Other SEM notation functions: Omega(), mutheta(), m()

jeksterslab/ram documentation built on Jan. 8, 2021, 12:45 a.m.