lisrel: LISREL Structural Equations with Latent Variables.

In jeksterslabds/jeksterslabRds: Jekster's Lab Data Science and Statistics

Description

Model-implied variance-covariance matrix (\boldsymbol{Σ} ≤ft( \boldsymbol{θ} \right)) using the LISREL notation for structural equations with latent variables.

Usage

lisrel(LY, LX, TE, TD, BE, I, GA, PS, PH)

Arguments

LY	\boldsymbol{Λ}_{\mathbf{y}} p \times m matrix of factor loadings (\boldsymbol{λ}). p is the number of observed indicators (\mathbf{y}) and m is the number of latent endogenous variables (\boldsymbol{η}).
LX	\boldsymbol{Λ}_{\mathbf{x}} q \times n matrix of factor loadings (\boldsymbol{λ}). q is the number of observed indicators (\mathbf{x}) and n is the number of latent exogenous variables (\boldsymbol{ξ}).
TE	\boldsymbol{Θ_{\boldsymbol{ε}}} p \times p matrix of residual variances and covariances for \mathbf{y} (\boldsymbol{ε}).
TD	\boldsymbol{Θ_{\boldsymbol{δ}}} q \times q matrix of residual variances and covariances for \mathbf{x} (\boldsymbol{δ}).
BE	\mathbf{B}_{m \times m} coefficient matrix for endogenous variables.
I	\mathbf{I}_{m \times m} identity matrix.
GA	\boldsymbol{Γ}_{m \times n} coefficient matrix for exogenous variables.
PS	\boldsymbol{Ψ}_{m \times m} variance-covariance of \boldsymbol{ζ}. \boldsymbol{ζ} is a matrix of residual variances and covariances in regression equations.
PH	\boldsymbol{Φ}_{n \times n} variance-covariance matrix of \boldsymbol{ξ}.

Details

Combines \boldsymbol{Σ}_{\mathbf{yy}} ≤ft( \boldsymbol{θ} \right) (from lisrel_yy), \boldsymbol{Σ}_{\mathbf{yx}} ≤ft( \boldsymbol{θ} \right) (from lisrel_yx), \boldsymbol{Σ}_{\mathbf{xy}} ≤ft( \boldsymbol{θ} \right) (from lisrel_xy), and \boldsymbol{Σ}_{\mathbf{xx}} ≤ft( \boldsymbol{θ} \right) (from lisrel_xx) to produce \boldsymbol{Σ} ≤ft( \boldsymbol{θ} \right) . The variance-covariance matrices are derived using the following equations

\boldsymbol{Σ}_{\mathbf{yy}} ≤ft( \boldsymbol{θ} \right) = \boldsymbol{Λ}_{\mathbf{y}} ≤ft( \mathbf{I} - \mathbf{B} \right)^{-1} ≤ft( \boldsymbol{Γ} \boldsymbol{Φ} \boldsymbol{Γ}^{T} + \boldsymbol{Ψ} \right) ≤ft[ ≤ft( \mathbf{I} - \mathbf{B} \right)^{-1} \right]^{T} \boldsymbol{Λ}_{\mathbf{y}}^{T} + \boldsymbol{Θ}_{\boldsymbol{ε}}

\boldsymbol{Σ}_{\mathbf{yx}} ≤ft( \boldsymbol{θ} \right) = \boldsymbol{Λ}_{\mathbf{y}} ≤ft( \mathbf{I} - \mathbf{B} \right)^{-1} \boldsymbol{Γ} \boldsymbol{Φ} \boldsymbol{Λ}_{\mathbf{x}}^{T}

\boldsymbol{Σ}_{\mathbf{xy}} ≤ft( \boldsymbol{θ} \right) = \boldsymbol{Λ}_{\mathbf{x}} \boldsymbol{Φ} \boldsymbol{Γ}^{T} ≤ft[ ≤ft( \mathbf{I} - \mathbf{B} \right)^{-1} \right]^{T} \boldsymbol{Λ}_{\mathbf{y}}^{T}

\boldsymbol{Σ}_{\mathbf{xx}} ≤ft( \boldsymbol{θ} \right) = \boldsymbol{Λ}_{\mathbf{x}} \boldsymbol{Φ} \boldsymbol{Λ}_{\mathbf{y}}^{T} + \boldsymbol{Θ}_{\boldsymbol{δ}} .

Value

Returns the model-implied variance-covariance matrix (\boldsymbol{Σ} ≤ft( \boldsymbol{θ} \right)) derived from the \boldsymbol{Λ}_{\mathbf{y}} (LY), \boldsymbol{Λ}_{\mathbf{x}} (LX), \boldsymbol{Θ}_{\boldsymbol{ε}} (TE), \boldsymbol{Θ}_{\boldsymbol{δ}} (TD), \mathbf{B} (BE), \mathbf{I} (I), \boldsymbol{Γ} (GA), \boldsymbol{Ψ} (PS), and \boldsymbol{Φ} (PH) matrices.

Author(s)

Ivan Jacob Agaloos Pesigan

References

Bollen, K. A. (1989). Structural equations with latent variables. New York: Wiley.

Jöreskog, K. G., & Sörbom, D. (1996). Lisrel 8: User's reference guide (2nd ed.). Scientific Software.

See Also

Other SEM notation functions: eqs_mu(), eqs(), lisrel_fa(), lisrel_obs_xy(), lisrel_obs_yx(), lisrel_obs_yy(), lisrel_obs(), lisrel_xx(), lisrel_xy(), lisrel_yx(), lisrel_yy(), ram_mu(), ram_m(), ram_s(), ram(), sem_fa(), sem_lat(), sem_obs()

jeksterslabds/jeksterslabRds documentation built on July 16, 2020, 3:41 p.m.