Total {cTMed} | R Documentation |
Total Effect Matrix Over a Specific Time Interval
Description
This function computes the total effects matrix
over a specific time interval \Delta t
using the first-order stochastic differential equation model's
drift matrix \boldsymbol{\Phi}
.
Usage
Total(phi, delta_t)
Arguments
phi |
Numeric matrix.
The drift matrix ( |
delta_t |
Numeric.
Time interval
( |
Details
The total effect matrix
over a specific time interval \Delta t
is given by
\mathrm{Total}_{\Delta t}
=
\exp
\left(
\Delta t
\boldsymbol{\Phi}
\right)
where
\boldsymbol{\Phi}
denotes the drift matrix, 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 ("Total").
- output
The matrix of total effects.
Author(s)
Ivan Jacob Agaloos Pesigan
References
Bollen, K. A. (1987). Total, direct, and indirect effects in structural equation models. Sociological Methodology, 17, 37. 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. 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. doi:10.1007/s11336-021-09767-0
See Also
Other Continuous Time Mediation Functions:
DeltaBeta()
,
DeltaBetaStd()
,
DeltaIndirectCentral()
,
DeltaMed()
,
DeltaMedStd()
,
DeltaTotalCentral()
,
Direct()
,
DirectStd()
,
ExpCov()
,
ExpMean()
,
Indirect()
,
IndirectCentral()
,
IndirectStd()
,
MCBeta()
,
MCBetaStd()
,
MCIndirectCentral()
,
MCMed()
,
MCMedStd()
,
MCPhi()
,
MCTotalCentral()
,
Med()
,
MedStd()
,
PosteriorBeta()
,
PosteriorIndirectCentral()
,
PosteriorMed()
,
PosteriorTotalCentral()
,
TotalCentral()
,
TotalStd()
,
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
Total(
phi = phi,
delta_t = delta_t
)
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)
Total(
phi = phi,
delta_t = delta_t
)