R/lisrel.R
In jeksterslabds/jeksterslabRds: Jekster's Lab Data Science and Statistics
Defines functions
lisrel_residual
lisrel
lisrel_xx
lisrel_xy
lisrel_yx
lisrel_yy
lisrel_fa
lisrel_obs
lisrel_obs_xy
lisrel_obs_yx
lisrel_obs_yy
lisrel_mat
Documented in
lisrel
lisrel_fa
lisrel_mat
lisrel_obs
lisrel_obs_xy
lisrel_obs_yx
lisrel_obs_yy
lisrel_residual
lisrel_xx
lisrel_xy
lisrel_yx
lisrel_yy
# LISREL Model Related Functions
# Ivan Jacob Agaloos Pesigan
#' LISREL Matrices
#'
#' Helper function to construct LISREL Matrices
#'
#' @author Ivan Jacob Agaloos Pesigan
#' @param eta_on_eta
#' Vector values of the strict lower triangle of
#' \eqn{\mathbf{B}_{m \times m}}
#' (\code{BE}),
#' that is,
#' the coefficients
#' for endogenous variables.
#' Note that lower triangle is populated by column.
#' If no value for \code{eta_on_eta} is provided,
#' it is assumed that there are no direct paths
#' between any of the endogenous variables.
#' @param eta_on_xi
#' Vector values of
#' \eqn{\boldsymbol{\Gamma}_{m \times n}}
#' coefficient matrix
#' for latent exogenous variables.
#' Note that matrix is populated by column.
#' @param PS
#' \eqn{\boldsymbol{\Psi}_{m \times m}}
#' variance-covariance of
#' \eqn{\boldsymbol{\zeta}}.
#' \eqn{\boldsymbol{\zeta}}
#' is a
#' matrix of
#' residual variances and covariances in regression equations.
#' If a vector is supplied,
#' the matrix is assummed to be a diagonal matrix
#' with diagonals equal to \code{PS}.
#' @param PH
#' \eqn{\boldsymbol{\Phi}_{n \times n}}
#' variance-covariance matrix of
#' \eqn{\boldsymbol{\xi}}.
#' If a vector is supplied,
#' the matrix is assummed to be a diagonal matrix
#' with diagonals equal to \code{PH}.
#' @param TE
#' \eqn{\boldsymbol{\Theta_{\boldsymbol{\epsilon}}}}
#' \eqn{p \times p}
#' matrix
#' of residual variances and covariances for
#' \eqn{\mathbf{y}}
#' (\eqn{\boldsymbol{\epsilon}}).
#' If a vector is supplied,
#' the matrix is assummed to be a diagonal matrix
#' with diagonals equal to \code{TE}.
#' @param TD
#' \eqn{\boldsymbol{\Theta_{\boldsymbol{\delta}}}}
#' \eqn{q \times q}
#' matrix
#' of residual variances and covariances for
#' \eqn{\mathbf{x}}
#' (\eqn{\boldsymbol{\delta}}).
#' If a vector is supplied,
#' the matrix is assummed to be a diagonal matrix
#' with diagonals equal to \code{TD}.
#' @param latent Logical.
#' If \code{TRUE}, uses the structural equations with latent variables LISREL notation,
#' If \code{FALSE}, uses the structural equations with observed variables LISREL notation.
#' @inheritParams lisrel
#' @export
lisrel_mat <- function(LY,
LX,
TE,
TD,
eta_on_eta = NULL,
eta_on_xi,
PS,
PH,
latent = TRUE) {
if (is.null(eta_on_eta)) {
m <- 1
BE <- matrix(
data = 0,
nrow = m,
ncol = m
)
} else {
BE <- vech2tri(
x = eta_on_eta,
lower = TRUE,
diag = FALSE
)
m <- nrow(BE)
}
GA <- matrix(
data = eta_on_xi,
nrow = m
)
n <- ncol(GA)
if (is.vector(PS)) {
PS <- diag(x = PS, nrow = m, ncol = m)
} else if (ncol(PS) == 1) {
PS <- diag(x = as.vector(PS), nrow = m, ncol = m)
} else {
if (isSymmetric(PS)) {
text <- paste(
"\"PS\" should have",
m,
"rows, and",
m,
"columns."
)
stop(text)
}
}
if (is.vector(PH)) {
PH <- diag(x = PH, nrow = n, ncol = n)
} else if (ncol(PH) == 1) {
PH <- diag(x = as.vector(PH), nrow = n, ncol = n)
} else {
if (isSymmetric(PH)) {
text <- paste(
"\"PH\" should have",
n,
"rows, and",
n,
"columns."
)
stop(text)
}
}
out <- list(
BE = BE,
I = diag(nrow(BE)),
GA = GA,
PS = PS,
PH = PH
)
if (latent) {
if (is.vector(TE)) {
p <- length(TE)
TE <- diag(x = TE, nrow = p, ncol = p)
} else if (ncol(TE) == 1) {
p <- nrow(TE)
TE <- diag(x = as.vector(TE), nrow = p, ncol = p)
} else {
p <- nrow(TE)
}
if (is.vector(TD)) {
q <- length(TD)
TD <- diag(x = TD, nrow = q, ncol = q)
} else if (ncol(TD) == 1) {
q <- nrow(TD)
TD <- diag(x = as.vector(TD), nrow = q, ncol = q)
} else {
q <- nrow(TD)
}
if (m != ncol(LY)) {
stop("The number of endogenous variables in \"LY\" (columns) does not match the elements in \"BE\".")
}
if (n != ncol(LX)) {
stop("The number of exogenous variables in \"LX\" (columns) does not match the elements in \"GA\".")
}
if (p != nrow(LY)) {
stop("The number of y observed indicator variables in \"LY\" (rows) does not match the elements in \"TE\".")
}
if (q != nrow(LX)) {
stop("The number of x observed indicator variables in \"LX\" (rows) does not match the elements in \"TD\".")
}
out[["LY"]] <- LY
out[["LX"]] <- LX
out[["TE"]] <- TE
out[["TD"]] <- TD
return(out)
} else {
return(out)
}
}
#' LISREL Structural Equations with Observed Variables (yy)
#'
#' Model-implied variance-covariance matrix for \eqn{\mathbf{y}} variables
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with observed variables.
#'
#' @details \deqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)
#' =
#' \left( \mathbf{I} - \mathbf{B} \right)^{-1}
#' \left(
#' \boldsymbol{\Gamma}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Gamma}^{T} +
#' \boldsymbol{\Psi}
#' \right)
#' \left[ \left( \mathbf{I} - \mathbf{B} \right)^{-1} \right]^{T}
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @param inv
#' The inverse of \code{I} minus \code{BE}
#' (\eqn{\left( \mathbf{I} - \mathbf{B} \right)^{-1}}).
#' @inheritParams lisrel_obs
#' @return Returns the model-implied variance-covariance matrix for
#' \eqn{\mathbf{y}}
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH}),
#' and
#' \eqn{\boldsymbol{\Psi}}
#' (\code{PS})
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_obs_yy <- function(inv,
GA,
PH,
PS) {
inv %*% (GA %*% PH %*% t(GA) + PS) %*% t(inv)
}
#' LISREL Structural Equations with Observed Variables (yx)
#'
#' Model-implied variance-covariance matrix for \eqn{\mathbf{y}} variables
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with observed variables.
#'
#' @details \deqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)
#' =
#' \left( \mathbf{I} - \mathbf{B} \right)^{-1}
#' \boldsymbol{\Gamma}
#' \boldsymbol{\Phi}
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @inheritParams lisrel_obs
#' @inheritParams lisrel_obs_yy
#' @return Returns the model-implied variance-covariance matrix for
#' \eqn{\mathbf{yx}}
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' and
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH}),
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_obs_yx <- function(inv,
GA,
PH) {
inv %*% GA %*% PH
}
#' LISREL Structural Equations with Observed Variables (xy)
#'
#' Model-implied variance-covariance matrix for \eqn{\mathbf{y}} variables
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with observed variables.
#'
#' @details \deqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)
#' =
#' \left( \mathbf{I} - \mathbf{B} \right)^{-1}
#' \left(
#' \boldsymbol{\Gamma}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Gamma}^{T} +
#' \boldsymbol{\Psi}
#' \right)
#' \left[ \left( \mathbf{I} - \mathbf{B} \right)^{-1} \right]^{T}
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @param inv The inverse of \code{I} minus \code{BE}
#' (\eqn{\left( \mathbf{I} - \mathbf{B} \right)^{-1}}).
#' @inheritParams lisrel_obs
#' @inheritParams lisrel_obs_yy
#' @return Returns the model-implied variance-covariance matrix for
#' \eqn{\mathbf{xy}}
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' and
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH}),
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_obs_xy <- function(inv,
GA,
PH) {
PH %*% t(GA) %*% t(inv)
}
#' LISREL Structural Equations with Observed Variables.
#'
#' Model-implied variance-covariance matrix
#' (\eqn{\boldsymbol{\Sigma} \left( \theta \right)})
#' using the LISREL notation
#' for structural equations with observed variables.
#'
#' @details Combines
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)}
#' (\code{\link{lisrel_obs_yy}}),
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)}
#' (\code{\link{lisrel_obs_yx}}),
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)}
#' (\code{\link{lisrel_obs_xy}}), and
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{xx}} \left( \boldsymbol{\theta} \right)}
#' (\eqn{\boldsymbol{\Phi}})
#' to produce
#' \eqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)}.
#' @inheritParams lisrel
#' @inherit lisrel references
#' @return Returns the model-implied variance-covariance matrix
#' (\eqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH}),
#' and
#' \eqn{\boldsymbol{\Psi}}
#' (\code{PS})
#' matrices.
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_obs <- function(BE, I, GA, PH, PS) {
inv <- solve(I - BE)
yy <- lisrel_obs_yy(
inv = inv,
GA = GA,
PH = PH,
PS = PS
)
yx <- lisrel_obs_yx(
inv = inv,
GA = GA,
PH = PH
)
top <- cbind(
yy,
yx
)
bottom <- cbind(
t(yx),
PH
)
rbind(
top,
bottom
)
}
#' LISREL Structural Equations with Latent Variables
#' (Factor Analysis).
#'
#' Model-implied variance-covariance matrix
#' (\eqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for factor analysis.
#'
#' @details \deqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Lambda}^{T} +
#' \boldsymbol{\Theta}
#' }
#' Note that this notation treats
#' all latent variables as
#' exogenous variables
#' \eqn{\boldsymbol{\xi}}.
#' @author Ivan Jacob Agaloos Pesigan
#' @param L
#' \eqn{\boldsymbol{\Lambda}_{q \times n}}
#' matrix of factor loadings
#' (\eqn{\boldsymbol{\lambda}}).
#' \eqn{q}
#' is the number of indicators and
#' \eqn{n}
#' is the number of latent factor variables.
#' @param TH
#' \eqn{\boldsymbol{\Theta}_{q \times q}}
#' matrix of residual variances and covariances.
#' @param PH
#' \eqn{\boldsymbol{\Phi}_{n \times n}}
#' variance-covariance matrix of
#' \eqn{\boldsymbol{\xi}}.
#' @return Returns the model-implied variance-covariance matrix
#' (\eqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)})
#' derived from the \eqn{\boldsymbol{\Lambda}},
#' \eqn{\boldsymbol{\Theta}},
#' and
#' \eqn{\boldsymbol{\Phi}}
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @examples
#' L <- matrix(
#' data = 0,
#' nrow = 9,
#' ncol = 3
#' )
#' L[1:3, 1] <- 0.76
#' L[4:6, 2] <- 0.76
#' L[7:9, 3] <- 0.76
#' PH <- matrix(
#' data = c(
#' 1, 0.50, 0.25,
#' 0.50, 1, 0.50,
#' 0.25, 0.50, 1
#' ),
#' ncol = 3
#' )
#' TH <- diag(
#' x = 0.76,
#' nrow = 9,
#' ncol = 9
#' )
#' lisrel_fa(L = L, TH = TH, PH = PH)
#' @export
lisrel_fa <- function(L,
TH,
PH) {
L %*% PH %*% t(L) + TH
}
#' LISREL Structural Equations with Latent Variables (yy).
#'
#' Model-implied variance-covariance matrix for \eqn{\mathbf{y}} variables
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with latent variables.
#'
#' @details \deqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{y}}
#' \left( \mathbf{I} - \mathbf{B} \right)^{-1}
#' \left(
#' \boldsymbol{\Gamma}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Gamma}^{T} +
#' \boldsymbol{\Psi}
#' \right)
#' \left[ \left( \mathbf{I} - \mathbf{B} \right)^{-1} \right]^{T}
#' \boldsymbol{\Lambda}_{\mathbf{y}}^{T} +
#' \boldsymbol{\Theta}_{\boldsymbol{\epsilon}}
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @param inv
#' The inverse of \code{I} minus \code{BE}
#' (\eqn{\left( \mathbf{I} - \mathbf{B} \right)^{-1}}).
#' @inheritParams lisrel
#' @return Returns the model-implied variance-covariance matrix for
#' \eqn{\mathbf{y}}
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{y}}}
#' (\code{LY}),
#' \eqn{\boldsymbol{\Theta}_{\boldsymbol{\epsilon}}}
#' (\code{TE}),
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' \eqn{\boldsymbol{\Psi}}
#' (\code{PS}),
#' and
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH})
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_yy <- function(LY,
TE,
inv,
GA,
PS,
PH) {
LY %*% inv %*% (GA %*% PH %*% t(GA) + PS) %*% t(inv) %*% t(LY) + TE
}
#' LISREL Structural Equations with Latent Variables (yx).
#'
#' Model-implied variance-covariance matrix for \eqn{\mathbf{yx}} variables
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with latent variables.
#'
#' @details \deqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{y}}
#' \left( \mathbf{I} - \mathbf{B} \right)^{-1}
#' \boldsymbol{\Gamma}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Lambda}_{\mathbf{x}}^{T}
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @inheritParams lisrel
#' @inheritParams lisrel_yy
#' @return Returns the model-implied variance-covariance matrix for
#' \eqn{\mathbf{yx}}
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{y}}}
#' (\code{LY}),
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{x}}}
#' (\code{LX}),
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' and
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH})
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_yx <- function(LY,
LX,
inv,
GA,
PH) {
LY %*% inv %*% GA %*% PH %*% t(LX)
}
#' LISREL Structural Equations with Latent Variables (xy).
#'
#' Model-implied variance-covariance matrix for \eqn{\mathbf{xy}} variables
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with latent variables.
#'
#' @details \deqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{x}}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Gamma}^{T}
#' \left[ \left( \mathbf{I} - \mathbf{B} \right)^{-1} \right]^{T}
#' \boldsymbol{\Lambda}_{\mathbf{y}}^{T}
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @inheritParams lisrel
#' @inheritParams lisrel_yy
#' @return Returns the model-implied variance-covariance matrix for
#' \eqn{\mathbf{xy}}
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{y}}}
#' (\code{LY}),
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{x}}}
#' (\code{LX}),
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' and
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH})
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_xy <- function(LY,
LX,
inv,
GA,
PH) {
LX %*% PH %*% t(GA) %*% t(inv) %*% t(LY)
}
#' LISREL Structural Equations with Latent Variables (xx).
#'
#' Model-implied variance-covariance matrix for \eqn{\mathbf{x}} variables
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{xx}} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with latent variables.
#'
#' @details \deqn{\boldsymbol{\Sigma}_{\mathbf{xx}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{x}}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Lambda}_{\mathbf{y}}^{T} +
#' \boldsymbol{\Theta}_{\boldsymbol{\delta}}
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @inheritParams lisrel
#' @return Returns the model-implied variance-covariance matrix for
#' \eqn{\mathbf{x}}
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{xx}} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{x}}}
#' (\code{LX}),
#' \eqn{\boldsymbol{\Theta}_{\boldsymbol{\delta}}}
#' (\code{TD}),
#' and
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH})
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_xx <- function(LX,
TD,
PH) {
LX %*% PH %*% t(LX) + TD
}
#' LISREL Structural Equations with Latent Variables.
#'
#' Model-implied variance-covariance matrix
#' (\eqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with latent variables.
#'
#' @details Combines
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)}
#' (from \code{\link{lisrel_yy}}),
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)}
#' (from \code{\link{lisrel_yx}}),
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)}
#' (from \code{\link{lisrel_xy}}),
#' and
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{xx}} \left( \boldsymbol{\theta} \right)}
#' (from \code{\link{lisrel_xx}})
#' to produce
#' \eqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)}
#' .
#' The variance-covariance matrices are derived using the following equations
#' \deqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{y}}
#' \left( \mathbf{I} - \mathbf{B} \right)^{-1}
#' \left(
#' \boldsymbol{\Gamma}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Gamma}^{T} +
#' \boldsymbol{\Psi}
#' \right)
#' \left[ \left( \mathbf{I} - \mathbf{B} \right)^{-1} \right]^{T}
#' \boldsymbol{\Lambda}_{\mathbf{y}}^{T} +
#' \boldsymbol{\Theta}_{\boldsymbol{\epsilon}}
#' }
#' \deqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{y}}
#' \left( \mathbf{I} - \mathbf{B} \right)^{-1}
#' \boldsymbol{\Gamma}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Lambda}_{\mathbf{x}}^{T}
#' }
#' \deqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{x}}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Gamma}^{T}
#' \left[ \left( \mathbf{I} - \mathbf{B} \right)^{-1} \right]^{T}
#' \boldsymbol{\Lambda}_{\mathbf{y}}^{T}
#' }
#' \deqn{\boldsymbol{\Sigma}_{\mathbf{xx}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{x}}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Lambda}_{\mathbf{y}}^{T} +
#' \boldsymbol{\Theta}_{\boldsymbol{\delta}} .
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @param LY
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{y}}}
#' \eqn{p \times m}
#' matrix of factor loadings
#' (\eqn{\boldsymbol{\lambda}}).
#' \eqn{p}
#' is the number of observed indicators
#' (\eqn{\mathbf{y}})
#' and
#' \eqn{m}
#' is the number of latent endogenous variables
#' (\eqn{\boldsymbol{\eta}}).
#' @param LX
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{x}}}
#' \eqn{q \times n}
#' matrix of factor loadings
#' (\eqn{\boldsymbol{\lambda}}).
#' \eqn{q}
#' is the number of observed indicators
#' (\eqn{\mathbf{x}})
#' and
#' \eqn{n}
#' is the number of latent exogenous variables
#' (\eqn{\boldsymbol{\xi}}).
#' @param TE
#' \eqn{\boldsymbol{\Theta_{\boldsymbol{\epsilon}}}}
#' \eqn{p \times p}
#' matrix
#' of residual variances and covariances for
#' \eqn{\mathbf{y}}
#' (\eqn{\boldsymbol{\epsilon}}).
#' @param TD
#' \eqn{\boldsymbol{\Theta_{\boldsymbol{\delta}}}}
#' \eqn{q \times q}
#' matrix
#' of residual variances and covariances for
#' \eqn{\mathbf{x}}
#' (\eqn{\boldsymbol{\delta}}).
#' @param BE
#' \eqn{\mathbf{B}_{m \times m}}
#' coefficient matrix
#' for endogenous variables.
#' @param I
#' \eqn{\mathbf{I}_{m \times m}}
#' identity matrix.
#' @param GA
#' \eqn{\boldsymbol{\Gamma}_{m \times n}}
#' coefficient matrix
#' for exogenous variables.
#' @param PS
#' \eqn{\boldsymbol{\Psi}_{m \times m}}
#' variance-covariance of
#' \eqn{\boldsymbol{\zeta}}.
#' \eqn{\boldsymbol{\zeta}}
#' is a
#' matrix of
#' residual variances and covariances in regression equations.
#' @param PH
#' \eqn{\boldsymbol{\Phi}_{n \times n}}
#' variance-covariance matrix of
#' \eqn{\boldsymbol{\xi}}.
#' @return Returns the model-implied variance-covariance matrix
#' (\eqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{y}}}
#' (\code{LY}),
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{x}}}
#' (\code{LX}),
#' \eqn{\boldsymbol{\Theta}_{\boldsymbol{\epsilon}}}
#' (\code{TE}),
#' \eqn{\boldsymbol{\Theta}_{\boldsymbol{\delta}}}
#' (\code{TD}),
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' \eqn{\boldsymbol{\Psi}}
#' (\code{PS}),
#' and
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH})
#' matrices.
#' @references
#' Bollen, K. A. (1989).
#' \emph{Structural equations with latent variables}.
#' New York: Wiley.
#'
#' Jöreskog, K. G., & Sörbom, D. (1996).
#' \emph{Lisrel 8: User's reference guide} (2nd ed.).
#' Scientific Software.
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel <- function(LY,
LX,
TE,
TD,
BE,
I,
GA,
PS,
PH) {
inv <- solve(I - BE)
yy <- lisrel_yy(
LY = LY,
inv = inv,
GA = GA,
PH = PH,
PS = PS,
TE = TE
)
yx <- lisrel_yx(
LY = LY,
inv = inv,
GA = GA,
PH = PH,
LX = LX
)
xx <- lisrel_xx(
LX = LX,
TD = TD,
PH = PH
)
top <- cbind(
yy,
yx
)
bottom <- cbind(
t(yx),
xx
)
rbind(
top,
bottom
)
}
#' Lisrel Residuals
#'
#' @inheritParams lisrel_mat
#' @inheritParams sem_fml
#' @param ... Arguments to pass to \code{\link{lisrel_mat}}.
#' @export
lisrel_residual <- function(obs,
latent = TRUE,
...) {
args <- lisrel_mat(
...,
latent = latent
)
if (latent) {
imp <- do.call(
what = lisrel,
args = args
)
} else {
imp <- do.call(
what = lisrel_obs,
args = args
)
}
obs - imp
}
jeksterslabds/jeksterslabRds documentation built on July 16, 2020, 3:41 p.m.
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Defines functions lisrel_residual lisrel lisrel_xx lisrel_xy lisrel_yx lisrel_yy lisrel_fa lisrel_obs lisrel_obs_xy lisrel_obs_yx lisrel_obs_yy lisrel_mat
Documented in lisrel lisrel_fa lisrel_mat lisrel_obs lisrel_obs_xy lisrel_obs_yx lisrel_obs_yy lisrel_residual lisrel_xx lisrel_xy lisrel_yx lisrel_yy
# LISREL Model Related Functions
# Ivan Jacob Agaloos Pesigan
#' LISREL Matrices
#'
#' Helper function to construct LISREL Matrices
#'
#' @author Ivan Jacob Agaloos Pesigan
#' @param eta_on_eta
#' Vector values of the strict lower triangle of
#' \eqn{\mathbf{B}_{m \times m}}
#' (\code{BE}),
#' that is,
#' the coefficients
#' for endogenous variables.
#' Note that lower triangle is populated by column.
#' If no value for \code{eta_on_eta} is provided,
#' it is assumed that there are no direct paths
#' between any of the endogenous variables.
#' @param eta_on_xi
#' Vector values of
#' \eqn{\boldsymbol{\Gamma}_{m \times n}}
#' coefficient matrix
#' for latent exogenous variables.
#' Note that matrix is populated by column.
#' @param PS
#' \eqn{\boldsymbol{\Psi}_{m \times m}}
#' variance-covariance of
#' \eqn{\boldsymbol{\zeta}}.
#' \eqn{\boldsymbol{\zeta}}
#' is a
#' matrix of
#' residual variances and covariances in regression equations.
#' If a vector is supplied,
#' the matrix is assummed to be a diagonal matrix
#' with diagonals equal to \code{PS}.
#' @param PH
#' \eqn{\boldsymbol{\Phi}_{n \times n}}
#' variance-covariance matrix of
#' \eqn{\boldsymbol{\xi}}.
#' If a vector is supplied,
#' the matrix is assummed to be a diagonal matrix
#' with diagonals equal to \code{PH}.
#' @param TE
#' \eqn{\boldsymbol{\Theta_{\boldsymbol{\epsilon}}}}
#' \eqn{p \times p}
#' matrix
#' of residual variances and covariances for
#' \eqn{\mathbf{y}}
#' (\eqn{\boldsymbol{\epsilon}}).
#' If a vector is supplied,
#' the matrix is assummed to be a diagonal matrix
#' with diagonals equal to \code{TE}.
#' @param TD
#' \eqn{\boldsymbol{\Theta_{\boldsymbol{\delta}}}}
#' \eqn{q \times q}
#' matrix
#' of residual variances and covariances for
#' \eqn{\mathbf{x}}
#' (\eqn{\boldsymbol{\delta}}).
#' If a vector is supplied,
#' the matrix is assummed to be a diagonal matrix
#' with diagonals equal to \code{TD}.
#' @param latent Logical.
#' If \code{TRUE}, uses the structural equations with latent variables LISREL notation,
#' If \code{FALSE}, uses the structural equations with observed variables LISREL notation.
#' @inheritParams lisrel
#' @export
lisrel_mat <- function(LY,
LX,
TE,
TD,
eta_on_eta = NULL,
eta_on_xi,
PS,
PH,
latent = TRUE) {
if (is.null(eta_on_eta)) {
m <- 1
BE <- matrix(
data = 0,
nrow = m,
ncol = m
)
} else {
BE <- vech2tri(
x = eta_on_eta,
lower = TRUE,
diag = FALSE
)
m <- nrow(BE)
}
GA <- matrix(
data = eta_on_xi,
nrow = m
)
n <- ncol(GA)
if (is.vector(PS)) {
PS <- diag(x = PS, nrow = m, ncol = m)
} else if (ncol(PS) == 1) {
PS <- diag(x = as.vector(PS), nrow = m, ncol = m)
} else {
if (isSymmetric(PS)) {
text <- paste(
"\"PS\" should have",
m,
"rows, and",
m,
"columns."
)
stop(text)
}
}
if (is.vector(PH)) {
PH <- diag(x = PH, nrow = n, ncol = n)
} else if (ncol(PH) == 1) {
PH <- diag(x = as.vector(PH), nrow = n, ncol = n)
} else {
if (isSymmetric(PH)) {
text <- paste(
"\"PH\" should have",
n,
"rows, and",
n,
"columns."
)
stop(text)
}
}
out <- list(
BE = BE,
I = diag(nrow(BE)),
GA = GA,
PS = PS,
PH = PH
)
if (latent) {
if (is.vector(TE)) {
p <- length(TE)
TE <- diag(x = TE, nrow = p, ncol = p)
} else if (ncol(TE) == 1) {
p <- nrow(TE)
TE <- diag(x = as.vector(TE), nrow = p, ncol = p)
} else {
p <- nrow(TE)
}
if (is.vector(TD)) {
q <- length(TD)
TD <- diag(x = TD, nrow = q, ncol = q)
} else if (ncol(TD) == 1) {
q <- nrow(TD)
TD <- diag(x = as.vector(TD), nrow = q, ncol = q)
} else {
q <- nrow(TD)
}
if (m != ncol(LY)) {
stop("The number of endogenous variables in \"LY\" (columns) does not match the elements in \"BE\".")
}
if (n != ncol(LX)) {
stop("The number of exogenous variables in \"LX\" (columns) does not match the elements in \"GA\".")
}
if (p != nrow(LY)) {
stop("The number of y observed indicator variables in \"LY\" (rows) does not match the elements in \"TE\".")
}
if (q != nrow(LX)) {
stop("The number of x observed indicator variables in \"LX\" (rows) does not match the elements in \"TD\".")
}
out[["LY"]] <- LY
out[["LX"]] <- LX
out[["TE"]] <- TE
out[["TD"]] <- TD
return(out)
} else {
return(out)
}
}
#' LISREL Structural Equations with Observed Variables (yy)
#'
#' Model-implied variance-covariance matrix for \eqn{\mathbf{y}} variables
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with observed variables.
#'
#' @details \deqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)
#' =
#' \left( \mathbf{I} - \mathbf{B} \right)^{-1}
#' \left(
#' \boldsymbol{\Gamma}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Gamma}^{T} +
#' \boldsymbol{\Psi}
#' \right)
#' \left[ \left( \mathbf{I} - \mathbf{B} \right)^{-1} \right]^{T}
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @param inv
#' The inverse of \code{I} minus \code{BE}
#' (\eqn{\left( \mathbf{I} - \mathbf{B} \right)^{-1}}).
#' @inheritParams lisrel_obs
#' @return Returns the model-implied variance-covariance matrix for
#' \eqn{\mathbf{y}}
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH}),
#' and
#' \eqn{\boldsymbol{\Psi}}
#' (\code{PS})
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_obs_yy <- function(inv,
GA,
PH,
PS) {
inv %*% (GA %*% PH %*% t(GA) + PS) %*% t(inv)
}
#' LISREL Structural Equations with Observed Variables (yx)
#'
#' Model-implied variance-covariance matrix for \eqn{\mathbf{y}} variables
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with observed variables.
#'
#' @details \deqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)
#' =
#' \left( \mathbf{I} - \mathbf{B} \right)^{-1}
#' \boldsymbol{\Gamma}
#' \boldsymbol{\Phi}
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @inheritParams lisrel_obs
#' @inheritParams lisrel_obs_yy
#' @return Returns the model-implied variance-covariance matrix for
#' \eqn{\mathbf{yx}}
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' and
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH}),
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_obs_yx <- function(inv,
GA,
PH) {
inv %*% GA %*% PH
}
#' LISREL Structural Equations with Observed Variables (xy)
#'
#' Model-implied variance-covariance matrix for \eqn{\mathbf{y}} variables
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with observed variables.
#'
#' @details \deqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)
#' =
#' \left( \mathbf{I} - \mathbf{B} \right)^{-1}
#' \left(
#' \boldsymbol{\Gamma}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Gamma}^{T} +
#' \boldsymbol{\Psi}
#' \right)
#' \left[ \left( \mathbf{I} - \mathbf{B} \right)^{-1} \right]^{T}
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @param inv The inverse of \code{I} minus \code{BE}
#' (\eqn{\left( \mathbf{I} - \mathbf{B} \right)^{-1}}).
#' @inheritParams lisrel_obs
#' @inheritParams lisrel_obs_yy
#' @return Returns the model-implied variance-covariance matrix for
#' \eqn{\mathbf{xy}}
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' and
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH}),
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_obs_xy <- function(inv,
GA,
PH) {
PH %*% t(GA) %*% t(inv)
}
#' LISREL Structural Equations with Observed Variables.
#'
#' Model-implied variance-covariance matrix
#' (\eqn{\boldsymbol{\Sigma} \left( \theta \right)})
#' using the LISREL notation
#' for structural equations with observed variables.
#'
#' @details Combines
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)}
#' (\code{\link{lisrel_obs_yy}}),
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)}
#' (\code{\link{lisrel_obs_yx}}),
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)}
#' (\code{\link{lisrel_obs_xy}}), and
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{xx}} \left( \boldsymbol{\theta} \right)}
#' (\eqn{\boldsymbol{\Phi}})
#' to produce
#' \eqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)}.
#' @inheritParams lisrel
#' @inherit lisrel references
#' @return Returns the model-implied variance-covariance matrix
#' (\eqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH}),
#' and
#' \eqn{\boldsymbol{\Psi}}
#' (\code{PS})
#' matrices.
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_obs <- function(BE, I, GA, PH, PS) {
inv <- solve(I - BE)
yy <- lisrel_obs_yy(
inv = inv,
GA = GA,
PH = PH,
PS = PS
)
yx <- lisrel_obs_yx(
inv = inv,
GA = GA,
PH = PH
)
top <- cbind(
yy,
yx
)
bottom <- cbind(
t(yx),
PH
)
rbind(
top,
bottom
)
}
#' LISREL Structural Equations with Latent Variables
#' (Factor Analysis).
#'
#' Model-implied variance-covariance matrix
#' (\eqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for factor analysis.
#'
#' @details \deqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Lambda}^{T} +
#' \boldsymbol{\Theta}
#' }
#' Note that this notation treats
#' all latent variables as
#' exogenous variables
#' \eqn{\boldsymbol{\xi}}.
#' @author Ivan Jacob Agaloos Pesigan
#' @param L
#' \eqn{\boldsymbol{\Lambda}_{q \times n}}
#' matrix of factor loadings
#' (\eqn{\boldsymbol{\lambda}}).
#' \eqn{q}
#' is the number of indicators and
#' \eqn{n}
#' is the number of latent factor variables.
#' @param TH
#' \eqn{\boldsymbol{\Theta}_{q \times q}}
#' matrix of residual variances and covariances.
#' @param PH
#' \eqn{\boldsymbol{\Phi}_{n \times n}}
#' variance-covariance matrix of
#' \eqn{\boldsymbol{\xi}}.
#' @return Returns the model-implied variance-covariance matrix
#' (\eqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)})
#' derived from the \eqn{\boldsymbol{\Lambda}},
#' \eqn{\boldsymbol{\Theta}},
#' and
#' \eqn{\boldsymbol{\Phi}}
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @examples
#' L <- matrix(
#' data = 0,
#' nrow = 9,
#' ncol = 3
#' )
#' L[1:3, 1] <- 0.76
#' L[4:6, 2] <- 0.76
#' L[7:9, 3] <- 0.76
#' PH <- matrix(
#' data = c(
#' 1, 0.50, 0.25,
#' 0.50, 1, 0.50,
#' 0.25, 0.50, 1
#' ),
#' ncol = 3
#' )
#' TH <- diag(
#' x = 0.76,
#' nrow = 9,
#' ncol = 9
#' )
#' lisrel_fa(L = L, TH = TH, PH = PH)
#' @export
lisrel_fa <- function(L,
TH,
PH) {
L %*% PH %*% t(L) + TH
}
#' LISREL Structural Equations with Latent Variables (yy).
#'
#' Model-implied variance-covariance matrix for \eqn{\mathbf{y}} variables
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with latent variables.
#'
#' @details \deqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{y}}
#' \left( \mathbf{I} - \mathbf{B} \right)^{-1}
#' \left(
#' \boldsymbol{\Gamma}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Gamma}^{T} +
#' \boldsymbol{\Psi}
#' \right)
#' \left[ \left( \mathbf{I} - \mathbf{B} \right)^{-1} \right]^{T}
#' \boldsymbol{\Lambda}_{\mathbf{y}}^{T} +
#' \boldsymbol{\Theta}_{\boldsymbol{\epsilon}}
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @param inv
#' The inverse of \code{I} minus \code{BE}
#' (\eqn{\left( \mathbf{I} - \mathbf{B} \right)^{-1}}).
#' @inheritParams lisrel
#' @return Returns the model-implied variance-covariance matrix for
#' \eqn{\mathbf{y}}
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{y}}}
#' (\code{LY}),
#' \eqn{\boldsymbol{\Theta}_{\boldsymbol{\epsilon}}}
#' (\code{TE}),
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' \eqn{\boldsymbol{\Psi}}
#' (\code{PS}),
#' and
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH})
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_yy <- function(LY,
TE,
inv,
GA,
PS,
PH) {
LY %*% inv %*% (GA %*% PH %*% t(GA) + PS) %*% t(inv) %*% t(LY) + TE
}
#' LISREL Structural Equations with Latent Variables (yx).
#'
#' Model-implied variance-covariance matrix for \eqn{\mathbf{yx}} variables
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with latent variables.
#'
#' @details \deqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{y}}
#' \left( \mathbf{I} - \mathbf{B} \right)^{-1}
#' \boldsymbol{\Gamma}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Lambda}_{\mathbf{x}}^{T}
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @inheritParams lisrel
#' @inheritParams lisrel_yy
#' @return Returns the model-implied variance-covariance matrix for
#' \eqn{\mathbf{yx}}
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{y}}}
#' (\code{LY}),
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{x}}}
#' (\code{LX}),
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' and
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH})
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_yx <- function(LY,
LX,
inv,
GA,
PH) {
LY %*% inv %*% GA %*% PH %*% t(LX)
}
#' LISREL Structural Equations with Latent Variables (xy).
#'
#' Model-implied variance-covariance matrix for \eqn{\mathbf{xy}} variables
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with latent variables.
#'
#' @details \deqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{x}}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Gamma}^{T}
#' \left[ \left( \mathbf{I} - \mathbf{B} \right)^{-1} \right]^{T}
#' \boldsymbol{\Lambda}_{\mathbf{y}}^{T}
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @inheritParams lisrel
#' @inheritParams lisrel_yy
#' @return Returns the model-implied variance-covariance matrix for
#' \eqn{\mathbf{xy}}
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{y}}}
#' (\code{LY}),
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{x}}}
#' (\code{LX}),
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' and
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH})
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_xy <- function(LY,
LX,
inv,
GA,
PH) {
LX %*% PH %*% t(GA) %*% t(inv) %*% t(LY)
}
#' LISREL Structural Equations with Latent Variables (xx).
#'
#' Model-implied variance-covariance matrix for \eqn{\mathbf{x}} variables
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{xx}} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with latent variables.
#'
#' @details \deqn{\boldsymbol{\Sigma}_{\mathbf{xx}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{x}}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Lambda}_{\mathbf{y}}^{T} +
#' \boldsymbol{\Theta}_{\boldsymbol{\delta}}
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @inheritParams lisrel
#' @return Returns the model-implied variance-covariance matrix for
#' \eqn{\mathbf{x}}
#' (\eqn{\boldsymbol{\Sigma}_{\mathbf{xx}} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{x}}}
#' (\code{LX}),
#' \eqn{\boldsymbol{\Theta}_{\boldsymbol{\delta}}}
#' (\code{TD}),
#' and
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH})
#' matrices.
#' @inherit lisrel references
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel_xx <- function(LX,
TD,
PH) {
LX %*% PH %*% t(LX) + TD
}
#' LISREL Structural Equations with Latent Variables.
#'
#' Model-implied variance-covariance matrix
#' (\eqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)})
#' using the LISREL notation
#' for structural equations with latent variables.
#'
#' @details Combines
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)}
#' (from \code{\link{lisrel_yy}}),
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)}
#' (from \code{\link{lisrel_yx}}),
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)}
#' (from \code{\link{lisrel_xy}}),
#' and
#' \eqn{\boldsymbol{\Sigma}_{\mathbf{xx}} \left( \boldsymbol{\theta} \right)}
#' (from \code{\link{lisrel_xx}})
#' to produce
#' \eqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)}
#' .
#' The variance-covariance matrices are derived using the following equations
#' \deqn{\boldsymbol{\Sigma}_{\mathbf{yy}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{y}}
#' \left( \mathbf{I} - \mathbf{B} \right)^{-1}
#' \left(
#' \boldsymbol{\Gamma}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Gamma}^{T} +
#' \boldsymbol{\Psi}
#' \right)
#' \left[ \left( \mathbf{I} - \mathbf{B} \right)^{-1} \right]^{T}
#' \boldsymbol{\Lambda}_{\mathbf{y}}^{T} +
#' \boldsymbol{\Theta}_{\boldsymbol{\epsilon}}
#' }
#' \deqn{\boldsymbol{\Sigma}_{\mathbf{yx}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{y}}
#' \left( \mathbf{I} - \mathbf{B} \right)^{-1}
#' \boldsymbol{\Gamma}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Lambda}_{\mathbf{x}}^{T}
#' }
#' \deqn{\boldsymbol{\Sigma}_{\mathbf{xy}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{x}}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Gamma}^{T}
#' \left[ \left( \mathbf{I} - \mathbf{B} \right)^{-1} \right]^{T}
#' \boldsymbol{\Lambda}_{\mathbf{y}}^{T}
#' }
#' \deqn{\boldsymbol{\Sigma}_{\mathbf{xx}} \left( \boldsymbol{\theta} \right)
#' =
#' \boldsymbol{\Lambda}_{\mathbf{x}}
#' \boldsymbol{\Phi}
#' \boldsymbol{\Lambda}_{\mathbf{y}}^{T} +
#' \boldsymbol{\Theta}_{\boldsymbol{\delta}} .
#' }
#' @author Ivan Jacob Agaloos Pesigan
#' @param LY
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{y}}}
#' \eqn{p \times m}
#' matrix of factor loadings
#' (\eqn{\boldsymbol{\lambda}}).
#' \eqn{p}
#' is the number of observed indicators
#' (\eqn{\mathbf{y}})
#' and
#' \eqn{m}
#' is the number of latent endogenous variables
#' (\eqn{\boldsymbol{\eta}}).
#' @param LX
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{x}}}
#' \eqn{q \times n}
#' matrix of factor loadings
#' (\eqn{\boldsymbol{\lambda}}).
#' \eqn{q}
#' is the number of observed indicators
#' (\eqn{\mathbf{x}})
#' and
#' \eqn{n}
#' is the number of latent exogenous variables
#' (\eqn{\boldsymbol{\xi}}).
#' @param TE
#' \eqn{\boldsymbol{\Theta_{\boldsymbol{\epsilon}}}}
#' \eqn{p \times p}
#' matrix
#' of residual variances and covariances for
#' \eqn{\mathbf{y}}
#' (\eqn{\boldsymbol{\epsilon}}).
#' @param TD
#' \eqn{\boldsymbol{\Theta_{\boldsymbol{\delta}}}}
#' \eqn{q \times q}
#' matrix
#' of residual variances and covariances for
#' \eqn{\mathbf{x}}
#' (\eqn{\boldsymbol{\delta}}).
#' @param BE
#' \eqn{\mathbf{B}_{m \times m}}
#' coefficient matrix
#' for endogenous variables.
#' @param I
#' \eqn{\mathbf{I}_{m \times m}}
#' identity matrix.
#' @param GA
#' \eqn{\boldsymbol{\Gamma}_{m \times n}}
#' coefficient matrix
#' for exogenous variables.
#' @param PS
#' \eqn{\boldsymbol{\Psi}_{m \times m}}
#' variance-covariance of
#' \eqn{\boldsymbol{\zeta}}.
#' \eqn{\boldsymbol{\zeta}}
#' is a
#' matrix of
#' residual variances and covariances in regression equations.
#' @param PH
#' \eqn{\boldsymbol{\Phi}_{n \times n}}
#' variance-covariance matrix of
#' \eqn{\boldsymbol{\xi}}.
#' @return Returns the model-implied variance-covariance matrix
#' (\eqn{\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)})
#' derived from the
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{y}}}
#' (\code{LY}),
#' \eqn{\boldsymbol{\Lambda}_{\mathbf{x}}}
#' (\code{LX}),
#' \eqn{\boldsymbol{\Theta}_{\boldsymbol{\epsilon}}}
#' (\code{TE}),
#' \eqn{\boldsymbol{\Theta}_{\boldsymbol{\delta}}}
#' (\code{TD}),
#' \eqn{\mathbf{B}}
#' (\code{BE}),
#' \eqn{\mathbf{I}}
#' (\code{I}),
#' \eqn{\boldsymbol{\Gamma}}
#' (\code{GA}),
#' \eqn{\boldsymbol{\Psi}}
#' (\code{PS}),
#' and
#' \eqn{\boldsymbol{\Phi}}
#' (\code{PH})
#' matrices.
#' @references
#' Bollen, K. A. (1989).
#' \emph{Structural equations with latent variables}.
#' New York: Wiley.
#'
#' Jöreskog, K. G., & Sörbom, D. (1996).
#' \emph{Lisrel 8: User's reference guide} (2nd ed.).
#' Scientific Software.
#' @family SEM notation functions
#' @keywords matrix lisrel
#' @export
lisrel <- function(LY,
LX,
TE,
TD,
BE,
I,
GA,
PS,
PH) {
inv <- solve(I - BE)
yy <- lisrel_yy(
LY = LY,
inv = inv,
GA = GA,
PH = PH,
PS = PS,
TE = TE
)
yx <- lisrel_yx(
LY = LY,
inv = inv,
GA = GA,
PH = PH,
LX = LX
)
xx <- lisrel_xx(
LX = LX,
TD = TD,
PH = PH
)
top <- cbind(
yy,
yx
)
bottom <- cbind(
t(yx),
xx
)
rbind(
top,
bottom
)
}
#' Lisrel Residuals
#'
#' @inheritParams lisrel_mat
#' @inheritParams sem_fml
#' @param ... Arguments to pass to \code{\link{lisrel_mat}}.
#' @export
lisrel_residual <- function(obs,
latent = TRUE,
...) {
args <- lisrel_mat(
...,
latent = latent
)
if (latent) {
imp <- do.call(
what = lisrel,
args = args
)
} else {
imp <- do.call(
what = lisrel_obs,
args = args
)
}
obs - imp
}
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