Expectations: Expectations from the Reticular Action Model (RAM) Matrices
In jeksterslab/ramR: Reticular Action Model (RAM) Notation

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

Derives the mean and covariance expectations from the Reticular Action Model (RAM) matrices.

Expectations(A, S, u = NULL, Filter = NULL, check = TRUE, ...)

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

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

`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.
`u`	vector of length `t` or `t by 1` matrix. Mean structure parameters.
`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()`.
`...`	...
`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.

The vector of expected values \mathbf{v} as a function of Reticular Action Model (RAM) matrices is given by

\mathbf{v} = ≤ft( \mathbf{I} - \mathbf{A} \right)^{\mathsf{T}} \mathbf{u} \\ = \mathbf{E} \mathbf{u}

The vector of expected values of observed variables \mathbf{g} as a function of Reticular Action Model (RAM) matrices is given by

\mathbf{g} = \mathbf{F} ≤ft( \mathbf{I} - \mathbf{A} \right)^{\mathsf{T}} \mathbf{u} \\ = \mathbf{F} \mathbf{E} \mathbf{u} \\ = \mathbf{F} \mathbf{v}

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

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

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}}

The matrix of scaled/standardized covariance expectations \mathbf{C}_{\mathrm{scaled}} (also known as correlations) is given by

\mathbf{C}_{\mathrm{scaled}} = \mathbf{D}^{-1} \mathbf{C} \mathbf{D}^{-1}

The matrix of scaled/standardized covariance expectations for given variables \mathbf{M}_{\mathrm{scaled}} (also known as correlations) is given by

\mathbf{M}_{\mathrm{scaled}} = \mathbf{F} \mathbf{C}_{\mathrm{scaled}} \mathbf{F}^{\mathsf{T}}

where

\mathbf{u}_{t \times 1} vector of parameters for the mean structure,
\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{F}_{p \times t} represents the filter matrix used to select the observed variables,
\mathbf{I}_{t \times t} represents an identity matrix,
\mathbf{D}_{t \times t} represents a diagonal matrix who diagonal elements are the square root of the diagonal elements of \mathbf{C},
p number of observed variables,
q number of latent variables, and
t number of observed and latent variables, that is p + q .

Returns a list with the following elements

v: t by 1 matrix \mathbf{v} of expected values.
g: p by 1 matrix \mathbf{g} of expected values of observed variables.
C: t by t matrix \mathbf{C} of expected covariances.
M: p by p matrix \mathbf{M} of expected covariances of observed variables.
C.scaled: t by t matrix \mathbf{C} of scaled/standardized expected covariances (also known as correlations).
M.scaled: p by p matrix \mathbf{M} of scaled/standardized expected covariances (also known as correlations) of observed variables.

Ivan Jacob Agaloos Pesigan

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

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

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

# Numeric -----------------------------------------------------------
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")
u <- c(0.00, 0.50, 0.00)
Filter <- diag(2)
Filter <- cbind(Filter, 0)
colnames(Filter) <- c("y", "x", "e")
Expectations(A, S, u, Filter)

# Symbolic ----------------------------------------------------------
A <- S <- matrixR::ZeroMatrix(3)
A[1, ] <- c(0, "beta", 1)
diag(S) <- c(0, "sigmax2", "sigmae2")
u <- c("alpha", "mux", 0)
Expectations(Ryacas::ysym(A), S, u, Filter, R = FALSE, format = "ysym")
Expectations(Ryacas::ysym(A), S, u, Filter, R = FALSE, format = "str")
Expectations(Ryacas::ysym(A), S, u, Filter, R = FALSE, format = "tex")
Expectations(Ryacas::ysym(A), S, u, Filter, R = TRUE)

# Assigning values to symbols

alpha <- 0
beta <- 1
sigmax2 <- 0.25
sigmae2 <- 1
mux <- 0.50

Expectations(Ryacas::ysym(A), S, u, Filter, R = FALSE, format = "ysym")
Expectations(Ryacas::ysym(A), S, u, Filter, R = FALSE, format = "str")
Expectations(Ryacas::ysym(A), S, u, Filter, R = FALSE, format = "tex")
(Expectations <- Expectations(Ryacas::ysym(A), S, u, Filter, R = TRUE))
eval(Expectations$v)
eval(Expectations$g)
eval(Expectations$C)
eval(Expectations$M)
eval(Expectations$C.scaled)
eval(Expectations$M.scaled)