In jeksterslab/ramR: Reticular Action Model (RAM) Notation
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
Eq* functions in the ramR package require
a character string input
that specifies the associations between the variables.
Each line in the character string
is a structural equation linking two variables at a time.
Defining Structural Equations
Syntax
Each line should follow the syntax below
lhs <space> op <space> rhs <space> par.label <space> par.start <\n> or <;>
The elements in the syntax are as follows:
lhs is the variable on the left-hand side,rhs is the variable on the right-hand side,op is the operation between lhs and rhs,par.label is the column of parameter label,par.start is the column of starting values for estimation
(only for functions that require it), and\n or ; are line breaks.
Each line should end with a line break.
Operations
The associations are defined by the following operations:
left-hand side measured by right-hand side,left-hand side regressed on right-hand side,left-hand side covarying with right-hand side, andleft-hand side regressed on 1 for mean structure.
par.label
Each parameter should be labeled.
The par.label should be a number for fixed parameters
and a character string for free parameters.
Equality contraints can be imposed by using the same par.label.
par.start
Numerical values as starting values for estimation.
Starting values for fixed parameters should be NA.
Starting values should be identical
for parameters constrained to be equal,
that is, parameters with the same par.label.
Line breaks
The characters \n and ; can be used as line breaks.
Each line should end with a line break.
Comments
Comments can be written after a hash (#) sign.
Confirmatory Factor Analysis Example
In the following example,
let y1 to y6 be observed variables,
eta[1] and eta[2] be latent variables,
and epsilon[1] to epsilon[6] be error terms.
Let y1 to y3 be indicators of eta[1], and
y4 to y6 of eta[2].
The first indicator of the latent variable has a factor loading of $1$.
The latent variables are correlated.
The factor loadings are labeled with lambda[*],
the intercepts are labeled with nu[*],
the error variances are labeled with theta[*],
the latent variable variances and covariances are labeled with psi[*],
and the latent variable means are constrained to zero.
You can use any string variable names and labels.
These names and labels were chosen to conform
to typical structural equation modeling notation.
The structural equations can be defined as follows.
eq <- "
# items
## loadings
eta[1] by y1 1
eta[1] by y2 lambda[y2]
eta[1] by y3 lambda[y3]
eta[2] by y4 1
eta[2] by y5 lambda[y5]
eta[2] by y6 lambda[y6]
## intercepts
y1 on 1 nu[y1]
y2 on 1 nu[y2]
y3 on 1 nu[y3]
y4 on 1 nu[y4]
y5 on 1 nu[y5]
y6 on 1 nu[y6]
# errors
## Asymmetric - fixed to 1
epsilon[1] by y1 1
epsilon[2] by y2 1
epsilon[3] by y3 1
epsilon[4] by y4 1
epsilon[5] by y5 1
epsilon[6] by y6 1
## Symmetric - error variances
epsilon[1] with epsilon[1] theta[epsilon[1]]
epsilon[2] with epsilon[2] theta[epsilon[2]]
epsilon[3] with epsilon[3] theta[epsilon[3]]
epsilon[4] with epsilon[4] theta[epsilon[4]]
epsilon[5] with epsilon[5] theta[epsilon[5]]
epsilon[6] with epsilon[6] theta[epsilon[6]]
# factors
## variances and covariances
eta[1] with eta[1] psi[eta[11]]
eta[1] with eta[2] psi[eta[12]]
eta[2] with eta[2] psi[eta[22]]
## means
eta[1] on 1 0 # There is no need to define equations for parameters constrained to zero.
eta[2] on 1 0 # The following equations were added for completeness.
"
eq <- "
# items
## loadings
eta[1] by y1 1
eta[1] by y2 lambda[y2]
eta[1] by y3 lambda[y3]
eta[2] by y4 1
eta[2] by y5 lambda[y5]
eta[2] by y6 lambda[y6]
## intercepts
y1 on 1 nu[y1]
y2 on 1 nu[y2]
y3 on 1 nu[y3]
y4 on 1 nu[y4]
y5 on 1 nu[y5]
y6 on 1 nu[y6]
# errors
## Asymmetric - fixed to 1
epsilon[1] by y1 1
epsilon[2] by y2 1
epsilon[3] by y3 1
epsilon[4] by y4 1
epsilon[5] by y5 1
epsilon[6] by y6 1
## Symmetric - error variances
epsilon[1] with epsilon[1] theta[epsilon[1]]
epsilon[2] with epsilon[2] theta[epsilon[2]]
epsilon[3] with epsilon[3] theta[epsilon[3]]
epsilon[4] with epsilon[4] theta[epsilon[4]]
epsilon[5] with epsilon[5] theta[epsilon[5]]
epsilon[6] with epsilon[6] theta[epsilon[6]]
# factors
## variances and covariances
eta[1] with eta[1] psi[eta[11]]
eta[1] with eta[2] psi[eta[12]]
eta[2] with eta[2] psi[eta[22]]
## means
eta[1] on 1 0 # There is no need to define equations for parameters constrained to zero.
eta[2] on 1 0 # The following equations were added for completeness.
"
knitr::include_graphics("figure1-1.png")
EqParse
The EqParse() function can parse the defined equation
and setup labels for the unknown parameters.
ramR::EqParse(eq)
Symbolic
Eq2RAM
The Eq2RAM() function constructs
the RAM notation matrix representation of the model.
It also updates the parameter table to include
RAM notation information.
RAM <- ramR::Eq2RAM(eq)
RAM$par.table
cat(
"\\begin{align*}",
"\\mathrm{variables}=",
yacR::as.tex(yacR::as.ysym.mat(RAM$variables)),
"\\end{align*}",
sep = ""
)
cat(
"\\begin{align*}",
"\\mathbf{A}=",
yacR::as.tex(RAM$A),
"\\end{align*}",
sep = ""
)
cat(
"\\begin{align*}",
"\\mathbf{S}=",
yacR::as.tex(RAM$S),
"\\end{align*}",
sep = ""
)
cat(
"\\begin{align*}",
"\\mathbf{u}=",
yacR::as.tex(RAM$u),
"\\end{align*}",
sep = ""
)
cat(
"\\begin{align*}",
"\\mathbf{F}=",
yacR::as.tex(RAM$Filter),
"\\end{align*}",
sep = ""
)
Eq2Expectations
The Eq2Expectations() function constructs the
mean and covariance expectations.
Expectations <- ramR::Eq2Expectations(eq)
The covariance expectations for all the variables
are given below.
latex <- ramR::Eq2Expectations(eq, format = "tex")
cat(
"\\begin{align*}",
"\\mathbf{C}=",
latex$C,
"\\end{align*}",
sep = ""
)
The scaled/standardized $\mathbf{C}$ matrix is given below.
$$r latex$C.scaled$$
The covariance expectations for the observed variables y1 to y6
are given below.
cat(
"\\begin{align*}",
"\\mathbf{M}=",
latex$M,
"\\end{align*}",
sep = ""
)
The scaled/standardized $\mathbf{M}$ matrix is given below.
$$r latex$M.scaled$$
The mean expectations for all the variables
are given below.
cat(
"\\begin{align*}",
"\\mathbf{v}=",
latex$v,
"\\end{align*}",
sep = ""
)
The mean expectations for the observed variables y1 to y6
are given below.
cat(
"\\begin{align*}",
"\\mathbf{g}=",
latex$g,
"\\end{align*}",
sep = ""
)
Scaled RAM Matrices
RAM.scaled <- ramR::RAMScaled(
A = yacR::as.ysym(RAM$A),
S = RAM$S,
Filter = RAM$Filter,
C = Expectations$C,
C.scaled = Expectations$C.scaled
)
$$
\mathbf{A}_{\mathrm{scaled}} = r yacR::as.tex(RAM.scaled$A.scaled)
$$
$$
\mathbf{S}_{\mathrm{scaled}} = r yacR::as.tex(RAM.scaled$S.scaled)
$$
Numerical
Values can be placed on the par.label column.
eq <- "
# items
## loadings
eta[1] by y1 1
eta[1] by y2 1
eta[1] by y3 1
eta[2] by y4 1
eta[2] by y5 1
eta[2] by y6 1
## intercepts
y1 on 1 0
y2 on 1 0
y3 on 1 0
y4 on 1 0
y5 on 1 0
y6 on 1 0
# errors
## Asymmetric - fixed to 1
epsilon[1] by y1 1
epsilon[2] by y2 1
epsilon[3] by y3 1
epsilon[4] by y4 1
epsilon[5] by y5 1
epsilon[6] by y6 1
## Symmetric - error variances
epsilon[1] with epsilon[1] 0.50
epsilon[2] with epsilon[2] 0.50
epsilon[3] with epsilon[3] 0.50
epsilon[4] with epsilon[4] 0.50
epsilon[5] with epsilon[5] 0.50
epsilon[6] with epsilon[6] 0.50
# factors
## variances and covariances
eta[1] with eta[1] 1
eta[1] with eta[2] 0.50
eta[2] with eta[2] 1
## means
eta[1] on 1 0
eta[2] on 1 0
"
Eq2RAM
The Eq2RAM() function constructs
the RAM notation matrix representation of the model.
RAM <- ramR::Eq2RAM(eq)
cat(
"\\begin{align*}",
"\\mathrm{variables}=",
yacR::as.tex(yacR::as.ysym.mat(RAM$variables)),
"\\end{align*}",
sep = ""
)
cat(
"\\begin{align*}",
"\\mathbf{A}=",
yacR::as.tex(RAM$A),
"\\end{align*}",
sep = ""
)
cat(
"\\begin{align*}",
"\\mathbf{S}=",
yacR::as.tex(RAM$S),
"\\end{align*}",
sep = ""
)
cat(
"\\begin{align*}",
"\\mathbf{u}=",
yacR::as.tex(RAM$u),
"\\end{align*}",
sep = ""
)
cat(
"\\begin{align*}",
"\\mathbf{F}=",
yacR::as.tex(RAM$Filter),
"\\end{align*}",
sep = ""
)
Eq2Expectations
The Eq2Expectations() function constructs the
mean and covariance expectations.
Expectations <- ramR::Eq2Expectations(eq)
The covariance expectations for all the variables
are given below.
cat(
"\\begin{align*}",
"\\mathbf{C}=",
yacR::as.tex(Expectations$C),
"\\end{align*}",
sep = ""
)
The covariance expectations for the observed variables y1 to y6
are given below.
cat(
"\\begin{align*}",
"\\mathbf{M}=",
yacR::as.tex(Expectations$M),
"\\end{align*}",
sep = ""
)
The mean expectations for all the variables
are given below.
cat(
"\\begin{align*}",
"\\mathbf{v}=",
yacR::as.tex(Expectations$v),
"\\end{align*}",
sep = ""
)
The mean expectations for the observed variables y1 to y6
are given below.
cat(
"\\begin{align*}",
"\\mathbf{g}=",
yacR::as.tex(Expectations$g),
"\\end{align*}",
sep = ""
)
Scaled RAM Matrices
RAM.scaled <- ramR::RAMScaled(
A = RAM$A,
S = RAM$S,
Filter = RAM$Filter,
C = Expectations$C,
C.scaled = Expectations$C.scaled
)
$$
\mathbf{A}_{\mathrm{scaled}} = r yacR::as.tex(RAM.scaled$A.scaled)
$$
$$
\mathbf{S}_{\mathrm{scaled}} = r yacR::as.tex(RAM.scaled$S.scaled)
$$
Eq2Data
The Eq2Data() function simulates data
from the multivariate normal distribution.
Data <- ramR::Eq2Data(eq, n = 100000)
str(Data)
cov(Data)
colMeans(Data)
jeksterslab/ramR documentation built on March 14, 2021, 9:38 a.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
Eq* functions in the ramR package require
a character string input
that specifies the associations between the variables.
Each line in the character string
is a structural equation linking two variables at a time.
Defining Structural Equations
Syntax
Each line should follow the syntax below
lhs <space> op <space> rhs <space> par.label <space> par.start <\n> or <;>
The elements in the syntax are as follows:
lhsis the variable on the left-hand side,rhsis the variable on the right-hand side,opis the operation betweenlhsandrhs,par.labelis the column of parameter label,par.startis the column of starting values for estimation (only for functions that require it), and\nor;are line breaks. Each line should end with a line break.
Operations
The associations are defined by the following operations:
left-hand sidemeasuredbyright-hand side,left-hand sideregressedonright-hand side,left-hand sidecovaryingwithright-hand side, andleft-hand sideregressedon 1for mean structure.
par.label
Each parameter should be labeled.
The par.label should be a number for fixed parameters
and a character string for free parameters.
Equality contraints can be imposed by using the same par.label.
par.start
Numerical values as starting values for estimation.
Starting values for fixed parameters should be NA.
Starting values should be identical
for parameters constrained to be equal,
that is, parameters with the same par.label.
Line breaks
The characters \n and ; can be used as line breaks.
Each line should end with a line break.
Comments
Comments can be written after a hash (#) sign.
Confirmatory Factor Analysis Example
In the following example,
let y1 to y6 be observed variables,
eta[1] and eta[2] be latent variables,
and epsilon[1] to epsilon[6] be error terms.
Let y1 to y3 be indicators of eta[1], and
y4 to y6 of eta[2].
The first indicator of the latent variable has a factor loading of $1$.
The latent variables are correlated.
The factor loadings are labeled with lambda[*],
the intercepts are labeled with nu[*],
the error variances are labeled with theta[*],
the latent variable variances and covariances are labeled with psi[*],
and the latent variable means are constrained to zero.
You can use any string variable names and labels.
These names and labels were chosen to conform
to typical structural equation modeling notation.
The structural equations can be defined as follows.
eq <- " # items ## loadings eta[1] by y1 1 eta[1] by y2 lambda[y2] eta[1] by y3 lambda[y3] eta[2] by y4 1 eta[2] by y5 lambda[y5] eta[2] by y6 lambda[y6] ## intercepts y1 on 1 nu[y1] y2 on 1 nu[y2] y3 on 1 nu[y3] y4 on 1 nu[y4] y5 on 1 nu[y5] y6 on 1 nu[y6] # errors ## Asymmetric - fixed to 1 epsilon[1] by y1 1 epsilon[2] by y2 1 epsilon[3] by y3 1 epsilon[4] by y4 1 epsilon[5] by y5 1 epsilon[6] by y6 1 ## Symmetric - error variances epsilon[1] with epsilon[1] theta[epsilon[1]] epsilon[2] with epsilon[2] theta[epsilon[2]] epsilon[3] with epsilon[3] theta[epsilon[3]] epsilon[4] with epsilon[4] theta[epsilon[4]] epsilon[5] with epsilon[5] theta[epsilon[5]] epsilon[6] with epsilon[6] theta[epsilon[6]] # factors ## variances and covariances eta[1] with eta[1] psi[eta[11]] eta[1] with eta[2] psi[eta[12]] eta[2] with eta[2] psi[eta[22]] ## means eta[1] on 1 0 # There is no need to define equations for parameters constrained to zero. eta[2] on 1 0 # The following equations were added for completeness. "
eq <- " # items ## loadings eta[1] by y1 1 eta[1] by y2 lambda[y2] eta[1] by y3 lambda[y3] eta[2] by y4 1 eta[2] by y5 lambda[y5] eta[2] by y6 lambda[y6] ## intercepts y1 on 1 nu[y1] y2 on 1 nu[y2] y3 on 1 nu[y3] y4 on 1 nu[y4] y5 on 1 nu[y5] y6 on 1 nu[y6] # errors ## Asymmetric - fixed to 1 epsilon[1] by y1 1 epsilon[2] by y2 1 epsilon[3] by y3 1 epsilon[4] by y4 1 epsilon[5] by y5 1 epsilon[6] by y6 1 ## Symmetric - error variances epsilon[1] with epsilon[1] theta[epsilon[1]] epsilon[2] with epsilon[2] theta[epsilon[2]] epsilon[3] with epsilon[3] theta[epsilon[3]] epsilon[4] with epsilon[4] theta[epsilon[4]] epsilon[5] with epsilon[5] theta[epsilon[5]] epsilon[6] with epsilon[6] theta[epsilon[6]] # factors ## variances and covariances eta[1] with eta[1] psi[eta[11]] eta[1] with eta[2] psi[eta[12]] eta[2] with eta[2] psi[eta[22]] ## means eta[1] on 1 0 # There is no need to define equations for parameters constrained to zero. eta[2] on 1 0 # The following equations were added for completeness. "
knitr::include_graphics("figure1-1.png")
EqParse
The EqParse() function can parse the defined equation
and setup labels for the unknown parameters.
ramR::EqParse(eq)
Symbolic
Eq2RAM
The Eq2RAM() function constructs
the RAM notation matrix representation of the model.
It also updates the parameter table to include
RAM notation information.
RAM <- ramR::Eq2RAM(eq)
RAM$par.table
cat( "\\begin{align*}", "\\mathrm{variables}=", yacR::as.tex(yacR::as.ysym.mat(RAM$variables)), "\\end{align*}", sep = "" )
cat( "\\begin{align*}", "\\mathbf{A}=", yacR::as.tex(RAM$A), "\\end{align*}", sep = "" )
cat( "\\begin{align*}", "\\mathbf{S}=", yacR::as.tex(RAM$S), "\\end{align*}", sep = "" )
cat( "\\begin{align*}", "\\mathbf{u}=", yacR::as.tex(RAM$u), "\\end{align*}", sep = "" )
cat( "\\begin{align*}", "\\mathbf{F}=", yacR::as.tex(RAM$Filter), "\\end{align*}", sep = "" )
Eq2Expectations
The Eq2Expectations() function constructs the
mean and covariance expectations.
Expectations <- ramR::Eq2Expectations(eq)
The covariance expectations for all the variables are given below.
latex <- ramR::Eq2Expectations(eq, format = "tex") cat( "\\begin{align*}", "\\mathbf{C}=", latex$C, "\\end{align*}", sep = "" )
The scaled/standardized $\mathbf{C}$ matrix is given below.
$$r latex$C.scaled$$
The covariance expectations for the observed variables y1 to y6
are given below.
cat( "\\begin{align*}", "\\mathbf{M}=", latex$M, "\\end{align*}", sep = "" )
The scaled/standardized $\mathbf{M}$ matrix is given below.
$$r latex$M.scaled$$
The mean expectations for all the variables are given below.
cat( "\\begin{align*}", "\\mathbf{v}=", latex$v, "\\end{align*}", sep = "" )
The mean expectations for the observed variables y1 to y6
are given below.
cat( "\\begin{align*}", "\\mathbf{g}=", latex$g, "\\end{align*}", sep = "" )
Scaled RAM Matrices
RAM.scaled <- ramR::RAMScaled( A = yacR::as.ysym(RAM$A), S = RAM$S, Filter = RAM$Filter, C = Expectations$C, C.scaled = Expectations$C.scaled )
$$
\mathbf{A}_{\mathrm{scaled}} = r yacR::as.tex(RAM.scaled$A.scaled)
$$
$$
\mathbf{S}_{\mathrm{scaled}} = r yacR::as.tex(RAM.scaled$S.scaled)
$$
Numerical
Values can be placed on the par.label column.
eq <- " # items ## loadings eta[1] by y1 1 eta[1] by y2 1 eta[1] by y3 1 eta[2] by y4 1 eta[2] by y5 1 eta[2] by y6 1 ## intercepts y1 on 1 0 y2 on 1 0 y3 on 1 0 y4 on 1 0 y5 on 1 0 y6 on 1 0 # errors ## Asymmetric - fixed to 1 epsilon[1] by y1 1 epsilon[2] by y2 1 epsilon[3] by y3 1 epsilon[4] by y4 1 epsilon[5] by y5 1 epsilon[6] by y6 1 ## Symmetric - error variances epsilon[1] with epsilon[1] 0.50 epsilon[2] with epsilon[2] 0.50 epsilon[3] with epsilon[3] 0.50 epsilon[4] with epsilon[4] 0.50 epsilon[5] with epsilon[5] 0.50 epsilon[6] with epsilon[6] 0.50 # factors ## variances and covariances eta[1] with eta[1] 1 eta[1] with eta[2] 0.50 eta[2] with eta[2] 1 ## means eta[1] on 1 0 eta[2] on 1 0 "
Eq2RAM
The Eq2RAM() function constructs
the RAM notation matrix representation of the model.
RAM <- ramR::Eq2RAM(eq)
cat( "\\begin{align*}", "\\mathrm{variables}=", yacR::as.tex(yacR::as.ysym.mat(RAM$variables)), "\\end{align*}", sep = "" )
cat( "\\begin{align*}", "\\mathbf{A}=", yacR::as.tex(RAM$A), "\\end{align*}", sep = "" )
cat( "\\begin{align*}", "\\mathbf{S}=", yacR::as.tex(RAM$S), "\\end{align*}", sep = "" )
cat( "\\begin{align*}", "\\mathbf{u}=", yacR::as.tex(RAM$u), "\\end{align*}", sep = "" )
cat( "\\begin{align*}", "\\mathbf{F}=", yacR::as.tex(RAM$Filter), "\\end{align*}", sep = "" )
Eq2Expectations
The Eq2Expectations() function constructs the
mean and covariance expectations.
Expectations <- ramR::Eq2Expectations(eq)
The covariance expectations for all the variables are given below.
cat( "\\begin{align*}", "\\mathbf{C}=", yacR::as.tex(Expectations$C), "\\end{align*}", sep = "" )
The covariance expectations for the observed variables y1 to y6
are given below.
cat( "\\begin{align*}", "\\mathbf{M}=", yacR::as.tex(Expectations$M), "\\end{align*}", sep = "" )
The mean expectations for all the variables are given below.
cat( "\\begin{align*}", "\\mathbf{v}=", yacR::as.tex(Expectations$v), "\\end{align*}", sep = "" )
The mean expectations for the observed variables y1 to y6
are given below.
cat( "\\begin{align*}", "\\mathbf{g}=", yacR::as.tex(Expectations$g), "\\end{align*}", sep = "" )
Scaled RAM Matrices
RAM.scaled <- ramR::RAMScaled( A = RAM$A, S = RAM$S, Filter = RAM$Filter, C = Expectations$C, C.scaled = Expectations$C.scaled )
$$
\mathbf{A}_{\mathrm{scaled}} = r yacR::as.tex(RAM.scaled$A.scaled)
$$
$$
\mathbf{S}_{\mathrm{scaled}} = r yacR::as.tex(RAM.scaled$S.scaled)
$$
Eq2Data
The Eq2Data() function simulates data
from the multivariate normal distribution.
Data <- ramR::Eq2Data(eq, n = 100000) str(Data) cov(Data) colMeans(Data)
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