M: Matrix of Covariance Expectations of Observed Variables...

In jeksterslab/ramR: Reticular Action Model (RAM) Notation

Description

Derives the matrix of covariance expectations of observed variables \mathbf{M}.

Usage

M(A, S, Filter = NULL, check = TRUE, ...)

## Default S3 method:
M(A, S, Filter = NULL, check = TRUE, ...)

## S3 method for class 'yac_symbol'
M(
  A,
  S,
  Filter = NULL,
  check = TRUE,
  exe = TRUE,
  R = FALSE,
  format = "ysym",
  simplify = FALSE,
  ...
)

Arguments

A	t by t matrix \mathbf{A}. Asymmetric paths (single-headed arrows), such as regression coefficients and factor loadings.
S	t by t numeric matrix \mathbf{S}. Symmetric paths (double-headed arrows), such as variances and covariances.
Filter	p by t numeric matrix \mathbf{F}. Filter matrix used to select observed variables.
check	Logical. If check = TRUE do some preprocessing with input matrices using CheckRAMMatrices().
...	...
exe	Logical. If exe = TRUE, executes the resulting yacas expression. If exe = FALSE, returns the resulting yacas expression as a character string. If exe = FALSE, the arguments str, ysym, simplify, and tex, are ignored.
R	Logical. If R = TRUE, returns symbolic result as an R expression. If R = FALSE, returns symbolic result as "ysym", "str", or "tex" depending of format.
format	Character string. Only used when R = FALSE. If format = "ysym", returns symbolic result as yac_symbol. If format = "str", returns symbolic result as a characetr string. If format = "tex", returns symbolic result as LaTeX math.
simplify	Logical. Simplify symbolic results.

Details

The matrix of covariance expectations for given variables \mathbf{M} as a function of Reticular Action Model (RAM) matrices is given by

\mathbf{M} = \mathbf{F} ≤ft( \mathbf{I} - \mathbf{A} \right)^{-1} \mathbf{S} ≤ft[ ≤ft( \mathbf{I} - \mathbf{A} \right)^{-1} \right]^{\mathsf{T}} \mathbf{F}^{\mathsf{T}} \\ = \mathbf{F} \mathbf{E} \mathbf{S} \mathbf{E}^{\mathsf{T}} \mathbf{F}^{\mathsf{T}} \\ = \mathbf{F} \mathbf{C} \mathbf{F}^{\mathsf{T}}

where

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

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

• \mathbf{I}_{t \times t} represents an identity matrix,

• \mathbf{F}_{p \times t} represents the filter matrix used to select the observed variables,

• p number of observed variables,

• q number of latent variables, and

• t number of observed and latent variables, that is p + q .

Value

\mathbf{M} = \mathbf{F} \mathbf{C} \mathbf{F}^{\mathsf{T}}

Author(s)

Ivan Jacob Agaloos Pesigan

References

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. https://doi.org/10.1111/j.2044-8317.1984.tb00802.x

See Also

Other RAM matrices functions: C(), Expectations(), E(), IminusA(), RAMScaled(), S(), g(), u(), v()

Examples

# Numeric -----------------------------------------------------------
# This is a numerical example for the model
# y = alpha + beta * x + e
# y = 0 + 1 * x + e
#--------------------------------------------------------------------

A <- S <- matrixR::ZeroMatrix(3)
A[1, ] <- c(0, 1, 1)
diag(S) <- c(0, 0.25, 1)
colnames(A) <- rownames(A) <- c("y", "x", "e")
Filter <- diag(2)
Filter <- cbind(Filter, 0)
colnames(Filter) <- c("y", "x", "e")
M(A, S, Filter)
# Symbolic ----------------------------------------------------------
# This is a symbolic example for the model
# y = alpha + beta * x + e
# y = 0 + 1 * x + e
#--------------------------------------------------------------------

A <- S <- matrixR::ZeroMatrix(3)
A[1, ] <- c(0, "beta", 1)
diag(S) <- c(0, "sigmax2", "sigmae2")
M(Ryacas::ysym(A), S, Filter)
M(Ryacas::ysym(A), S, Filter, format = "str")
M(Ryacas::ysym(A), S, Filter, format = "tex")
M(Ryacas::ysym(A), S, Filter, R = TRUE)

# Assigning values to symbols

beta <- 1
sigmax2 <- 0.25
sigmae2 <- 1

M(Ryacas::ysym(A), S, Filter)
M(Ryacas::ysym(A), S, Filter, format = "str")
M(Ryacas::ysym(A), S, Filter, format = "tex")
M(Ryacas::ysym(A), S, Filter, R = TRUE)
eval(M(Ryacas::ysym(A), S, Filter, R = TRUE))

jeksterslab/ramR documentation built on March 14, 2021, 9:38 a.m.