Direct: Direct Effect of X on Y Over a Specific Time Interval
In cTMed: Continuous Time Mediation
Direct R Documentation
Direct Effect of X on Y
Over a Specific Time Interval
Description
This function computes the direct effect
of the independent variable X
on the dependent variable Y
through mediator variables \mathbf{m}
over a specific time interval \Delta t
using the first-order stochastic differential equation model's
drift matrix \boldsymbol{\Phi}.
Usage
Direct(phi, delta_t, from, to, med)
Arguments
phi
Numeric matrix.
The drift matrix (\boldsymbol{\Phi}).
phi should have row and column names
pertaining to the variables in the system.
delta_t
Numeric.
Time interval
(\Delta t).
from
Character string.
Name of the independent variable X in phi.
to
Character string.
Name of the dependent variable Y in phi.
med
Character vector.
Name/s of the mediator variable/s in phi.
Details
The direct effect
of the independent variable X
on the dependent variable Y
relative to some mediator variables \mathbf{m}
is given by
\mathrm{Direct}_{{\Delta t}_{i, j, \mathrm{\mathbf{m}}}}
=
\exp
\left(
\Delta t
\mathbf{D}
\boldsymbol{\Phi}
\mathbf{D}
\right)_{i, j}
where
\boldsymbol{\Phi} denotes the drift matrix,
\mathbf{D} a diagonal matrix
where the diagonal elements corresponding
to mediator variables \mathbf{m}
are set to zero and the rest to one,
i the row index of Y in \boldsymbol{\Phi},
j the column index of X in \boldsymbol{\Phi}, and
\Delta t the time interval.
Linear Stochastic Differential Equation Model
The measurement model is given by
\mathbf{y}_{i, t}
=
\boldsymbol{\nu}
+
\boldsymbol{\Lambda}
\boldsymbol{\eta}_{i, t}
+
\boldsymbol{\varepsilon}_{i, t},
\quad
\mathrm{with}
\quad
\boldsymbol{\varepsilon}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\boldsymbol{\Theta}
\right)
where
\mathbf{y}_{i, t},
\boldsymbol{\eta}_{i, t},
and
\boldsymbol{\varepsilon}_{i, t}
are random variables
and
\boldsymbol{\nu},
\boldsymbol{\Lambda},
and
\boldsymbol{\Theta}
are model parameters.
\mathbf{y}_{i, t}
represents a vector of observed random variables,
\boldsymbol{\eta}_{i, t}
a vector of latent random variables,
and
\boldsymbol{\varepsilon}_{i, t}
a vector of random measurement errors,
at time t and individual i.
\boldsymbol{\nu}
denotes a vector of intercepts,
\boldsymbol{\Lambda}
a matrix of factor loadings,
and
\boldsymbol{\Theta}
the covariance matrix of
\boldsymbol{\varepsilon}.
An alternative representation of the measurement error
is given by
\boldsymbol{\varepsilon}_{i, t}
=
\boldsymbol{\Theta}^{\frac{1}{2}}
\mathbf{z}_{i, t},
\quad
\mathrm{with}
\quad
\mathbf{z}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\mathbf{I}
\right)
where
\mathbf{z}_{i, t} is a vector of
independent standard normal random variables and
\left( \boldsymbol{\Theta}^{\frac{1}{2}} \right)
\left( \boldsymbol{\Theta}^{\frac{1}{2}} \right)^{\prime}
=
\boldsymbol{\Theta} .
The dynamic structure is given by
\mathrm{d} \boldsymbol{\eta}_{i, t}
=
\left(
\boldsymbol{\iota}
+
\boldsymbol{\Phi}
\boldsymbol{\eta}_{i, t}
\right)
\mathrm{d}t
+
\boldsymbol{\Sigma}^{\frac{1}{2}}
\mathrm{d}
\mathbf{W}_{i, t}
where
\boldsymbol{\iota}
is a term which is unobserved and constant over time,
\boldsymbol{\Phi}
is the drift matrix
which represents the rate of change of the solution
in the absence of any random fluctuations,
\boldsymbol{\Sigma}
is the matrix of volatility
or randomness in the process, and
\mathrm{d}\boldsymbol{W}
is a Wiener process or Brownian motion,
which represents random fluctuations.
Value
Returns an object
of class ctmedeffect which is a list with the following elements:
- call
Function call.
- args
Function arguments.
- fun
Function used ("Direct").
- output
The direct effect.
Author(s)
Ivan Jacob Agaloos Pesigan
References
Bollen, K. A. (1987).
Total, direct, and indirect effects in structural equation models.
Sociological Methodology, 17, 37.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.2307/271028")}
Deboeck, P. R., & Preacher, K. J. (2015).
No need to be discrete:
A method for continuous time mediation analysis.
Structural Equation Modeling: A Multidisciplinary Journal, 23 (1), 61–75.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/10705511.2014.973960")}
Ryan, O., & Hamaker, E. L. (2021).
Time to intervene:
A continuous-time approach to network analysis and centrality.
Psychometrika, 87 (1), 214–252.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11336-021-09767-0")}
See Also
Other Continuous Time Mediation Functions:
DeltaBeta(),
DeltaIndirectCentral(),
DeltaMed(),
DeltaTotalCentral(),
Indirect(),
IndirectCentral(),
MCBeta(),
MCIndirectCentral(),
MCMed(),
MCPhi(),
MCTotalCentral(),
Med(),
PosteriorBeta(),
PosteriorIndirectCentral(),
PosteriorMed(),
PosteriorPhi(),
PosteriorTotalCentral(),
Total(),
TotalCentral(),
Trajectory()
Examples
phi <- matrix(
data = c(
-0.357, 0.771, -0.450,
0.0, -0.511, 0.729,
0, 0, -0.693
),
nrow = 3
)
colnames(phi) <- rownames(phi) <- c("x", "m", "y")
delta_t <- 1
Direct(
phi = phi,
delta_t = delta_t,
from = "x",
to = "y",
med = "m"
)
phi <- matrix(
data = c(
-6, 5.5, 0, 0,
1.25, -2.5, 5.9, -7.3,
0, 0, -6, 2.5,
5, 0, 0, -6
),
nrow = 4
)
colnames(phi) <- rownames(phi) <- paste0("y", 1:4)
Direct(
phi = phi,
delta_t = delta_t,
from = "y2",
to = "y4",
med = c("y1", "y3")
)
cTMed documentation built on Oct. 21, 2024, 5:08 p.m.
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| Direct | R Documentation |
Direct Effect of X on Y Over a Specific Time Interval
Description
This function computes the direct effect
of the independent variable X
on the dependent variable Y
through mediator variables \mathbf{m}
over a specific time interval \Delta t
using the first-order stochastic differential equation model's
drift matrix \boldsymbol{\Phi}.
Usage
Direct(phi, delta_t, from, to, med)
Arguments
phi |
Numeric matrix.
The drift matrix ( |
delta_t |
Numeric.
Time interval
( |
from |
Character string.
Name of the independent variable |
to |
Character string.
Name of the dependent variable |
med |
Character vector.
Name/s of the mediator variable/s in |
Details
The direct effect
of the independent variable X
on the dependent variable Y
relative to some mediator variables \mathbf{m}
is given by
\mathrm{Direct}_{{\Delta t}_{i, j, \mathrm{\mathbf{m}}}}
=
\exp
\left(
\Delta t
\mathbf{D}
\boldsymbol{\Phi}
\mathbf{D}
\right)_{i, j}
where
\boldsymbol{\Phi} denotes the drift matrix,
\mathbf{D} a diagonal matrix
where the diagonal elements corresponding
to mediator variables \mathbf{m}
are set to zero and the rest to one,
i the row index of Y in \boldsymbol{\Phi},
j the column index of X in \boldsymbol{\Phi}, and
\Delta t the time interval.
Linear Stochastic Differential Equation Model
The measurement model is given by
\mathbf{y}_{i, t}
=
\boldsymbol{\nu}
+
\boldsymbol{\Lambda}
\boldsymbol{\eta}_{i, t}
+
\boldsymbol{\varepsilon}_{i, t},
\quad
\mathrm{with}
\quad
\boldsymbol{\varepsilon}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\boldsymbol{\Theta}
\right)
where
\mathbf{y}_{i, t},
\boldsymbol{\eta}_{i, t},
and
\boldsymbol{\varepsilon}_{i, t}
are random variables
and
\boldsymbol{\nu},
\boldsymbol{\Lambda},
and
\boldsymbol{\Theta}
are model parameters.
\mathbf{y}_{i, t}
represents a vector of observed random variables,
\boldsymbol{\eta}_{i, t}
a vector of latent random variables,
and
\boldsymbol{\varepsilon}_{i, t}
a vector of random measurement errors,
at time t and individual i.
\boldsymbol{\nu}
denotes a vector of intercepts,
\boldsymbol{\Lambda}
a matrix of factor loadings,
and
\boldsymbol{\Theta}
the covariance matrix of
\boldsymbol{\varepsilon}.
An alternative representation of the measurement error is given by
\boldsymbol{\varepsilon}_{i, t}
=
\boldsymbol{\Theta}^{\frac{1}{2}}
\mathbf{z}_{i, t},
\quad
\mathrm{with}
\quad
\mathbf{z}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\mathbf{I}
\right)
where
\mathbf{z}_{i, t} is a vector of
independent standard normal random variables and
\left( \boldsymbol{\Theta}^{\frac{1}{2}} \right)
\left( \boldsymbol{\Theta}^{\frac{1}{2}} \right)^{\prime}
=
\boldsymbol{\Theta} .
The dynamic structure is given by
\mathrm{d} \boldsymbol{\eta}_{i, t}
=
\left(
\boldsymbol{\iota}
+
\boldsymbol{\Phi}
\boldsymbol{\eta}_{i, t}
\right)
\mathrm{d}t
+
\boldsymbol{\Sigma}^{\frac{1}{2}}
\mathrm{d}
\mathbf{W}_{i, t}
where
\boldsymbol{\iota}
is a term which is unobserved and constant over time,
\boldsymbol{\Phi}
is the drift matrix
which represents the rate of change of the solution
in the absence of any random fluctuations,
\boldsymbol{\Sigma}
is the matrix of volatility
or randomness in the process, and
\mathrm{d}\boldsymbol{W}
is a Wiener process or Brownian motion,
which represents random fluctuations.
Value
Returns an object
of class ctmedeffect which is a list with the following elements:
- call
Function call.
- args
Function arguments.
- fun
Function used ("Direct").
- output
The direct effect.
Author(s)
Ivan Jacob Agaloos Pesigan
References
Bollen, K. A. (1987). Total, direct, and indirect effects in structural equation models. Sociological Methodology, 17, 37. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.2307/271028")}
Deboeck, P. R., & Preacher, K. J. (2015). No need to be discrete: A method for continuous time mediation analysis. Structural Equation Modeling: A Multidisciplinary Journal, 23 (1), 61–75. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/10705511.2014.973960")}
Ryan, O., & Hamaker, E. L. (2021). Time to intervene: A continuous-time approach to network analysis and centrality. Psychometrika, 87 (1), 214–252. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11336-021-09767-0")}
See Also
Other Continuous Time Mediation Functions:
DeltaBeta(),
DeltaIndirectCentral(),
DeltaMed(),
DeltaTotalCentral(),
Indirect(),
IndirectCentral(),
MCBeta(),
MCIndirectCentral(),
MCMed(),
MCPhi(),
MCTotalCentral(),
Med(),
PosteriorBeta(),
PosteriorIndirectCentral(),
PosteriorMed(),
PosteriorPhi(),
PosteriorTotalCentral(),
Total(),
TotalCentral(),
Trajectory()
Examples
phi <- matrix(
data = c(
-0.357, 0.771, -0.450,
0.0, -0.511, 0.729,
0, 0, -0.693
),
nrow = 3
)
colnames(phi) <- rownames(phi) <- c("x", "m", "y")
delta_t <- 1
Direct(
phi = phi,
delta_t = delta_t,
from = "x",
to = "y",
med = "m"
)
phi <- matrix(
data = c(
-6, 5.5, 0, 0,
1.25, -2.5, 5.9, -7.3,
0, 0, -6, 2.5,
5, 0, 0, -6
),
nrow = 4
)
colnames(phi) <- rownames(phi) <- paste0("y", 1:4)
Direct(
phi = phi,
delta_t = delta_t,
from = "y2",
to = "y4",
med = c("y1", "y3")
)
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Note that we can't provide technical support on individual packages. You should contact the package authors for that.
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