MCBetaStd: Monte Carlo Sampling Distribution for the Elements of the...
In cTMed: Continuous-Time Mediation
View source: R/cTMed-mc-beta-std.R
MCBetaStd R Documentation
Monte Carlo Sampling Distribution
for the Elements of the Standardized Matrix of Lagged Coefficients
Over a Specific Time Interval
or a Range of Time Intervals
Description
This function generates a Monte Carlo method
sampling distribution
for the elements of the standardized matrix of lagged coefficients
\boldsymbol{\beta}
over a specific time interval \Delta t
or a range of time intervals
using the first-order stochastic differential equation model
drift matrix \boldsymbol{\Phi}
and process noise covariance matrix \boldsymbol{\Sigma}.
Usage
MCBetaStd(
phi,
sigma,
vcov_theta,
delta_t,
R,
test_phi = TRUE,
ncores = NULL,
seed = NULL,
tol = 0.01
)
Arguments
phi
Numeric matrix.
The drift matrix (\boldsymbol{\Phi}).
phi should have row and column names
pertaining to the variables in the system.
sigma
Numeric matrix.
The process noise covariance matrix (\boldsymbol{\Sigma}).
vcov_theta
Numeric matrix.
The sampling variance-covariance matrix of
\mathrm{vec} \left( \boldsymbol{\Phi} \right) and
\mathrm{vech} \left( \boldsymbol{\Sigma} \right)
delta_t
Numeric.
Time interval
(\Delta t).
R
Positive integer.
Number of replications.
test_phi
Logical.
If test_phi = TRUE,
the function tests the stability
of the generated drift matrix \boldsymbol{\Phi}.
If the test returns FALSE,
the function generates a new drift matrix \boldsymbol{\Phi}
and runs the test recursively
until the test returns TRUE.
ncores
Positive integer.
Number of cores to use.
If ncores = NULL,
use a single core.
Consider using multiple cores
when number of replications R
is a large value.
seed
Random seed.
tol
Numeric.
Smallest possible time interval to allow.
Details
See TotalStd().
Monte Carlo Method
Let \boldsymbol{\theta} be
a vector that combines
\mathrm{vec} \left( \boldsymbol{\Phi} \right),
that is,
the elements of the \boldsymbol{\Phi} matrix
in vector form sorted column-wise and
\mathrm{vech} \left( \boldsymbol{\Sigma} \right),
that is,
the unique elements of the \boldsymbol{\Sigma} matrix
in vector form sorted column-wise.
Let \hat{\boldsymbol{\theta}} be
a vector that combines
\mathrm{vec} \left( \hat{\boldsymbol{\Phi}} \right) and
\mathrm{vech} \left( \hat{\boldsymbol{\Sigma}} \right).
Based on the asymptotic properties of maximum likelihood estimators,
we can assume that estimators are normally distributed
around the population parameters.
\hat{\boldsymbol{\theta}}
\sim
\mathcal{N}
\left(
\boldsymbol{\theta},
\mathbb{V} \left( \hat{\boldsymbol{\theta}} \right)
\right)
Using this distributional assumption,
a sampling distribution of \hat{\boldsymbol{\theta}}
which we refer to as \hat{\boldsymbol{\theta}}^{\ast}
can be generated by replacing the population parameters
with sample estimates,
that is,
\hat{\boldsymbol{\theta}}^{\ast}
\sim
\mathcal{N}
\left(
\hat{\boldsymbol{\theta}},
\hat{\mathbb{V}} \left( \hat{\boldsymbol{\theta}} \right)
\right) .
Let
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right)
be a parameter that is a function of the estimated parameters.
A sampling distribution of
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right) ,
which we refer to as
\mathbf{g} \left( \hat{\boldsymbol{\theta}}^{\ast} \right) ,
can be generated by using the simulated estimates
to calculate
\mathbf{g}.
The standard deviations of the simulated estimates
are the standard errors.
Percentiles corresponding to
100 \left( 1 - \alpha \right) \%
are the confidence intervals.
Value
Returns an object
of class ctmedmc which is a list with the following elements:
- call
Function call.
- args
Function arguments.
- fun
Function used ("MCBetaStd").
- output
A list the length of which is equal to
the length of delta_t.
Each element in the output list has the following elements:
- est
Estimated elements of the standardized matrix
of lagged coefficients.
- thetahatstar
A matrix of Monte Carlo
elements of the standardized matrix of lagged coefficients.
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:
BootBeta(),
BootBetaStd(),
BootIndirectCentral(),
BootMed(),
BootMedStd(),
BootTotalCentral(),
DeltaBeta(),
DeltaBetaStd(),
DeltaIndirectCentral(),
DeltaMed(),
DeltaMedStd(),
DeltaTotalCentral(),
Direct(),
DirectStd(),
Indirect(),
IndirectCentral(),
IndirectStd(),
MCBeta(),
MCIndirectCentral(),
MCMed(),
MCMedStd(),
MCPhi(),
MCPhiSigma(),
MCTotalCentral(),
Med(),
MedStd(),
PosteriorBeta(),
PosteriorIndirectCentral(),
PosteriorMed(),
PosteriorTotalCentral(),
Total(),
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")
sigma <- matrix(
data = c(
0.24455556, 0.02201587, -0.05004762,
0.02201587, 0.07067800, 0.01539456,
-0.05004762, 0.01539456, 0.07553061
),
nrow = 3
)
vcov_theta <- matrix(
data = c(
0.00843, 0.00040, -0.00151, -0.00600, -0.00033,
0.00110, 0.00324, 0.00020, -0.00061, -0.00115,
0.00011, 0.00015, 0.00001, -0.00002, -0.00001,
0.00040, 0.00374, 0.00016, -0.00022, -0.00273,
-0.00016, 0.00009, 0.00150, 0.00012, -0.00010,
-0.00026, 0.00002, 0.00012, 0.00004, -0.00001,
-0.00151, 0.00016, 0.00389, 0.00103, -0.00007,
-0.00283, -0.00050, 0.00000, 0.00156, 0.00021,
-0.00005, -0.00031, 0.00001, 0.00007, 0.00006,
-0.00600, -0.00022, 0.00103, 0.00644, 0.00031,
-0.00119, -0.00374, -0.00021, 0.00070, 0.00064,
-0.00015, -0.00005, 0.00000, 0.00003, -0.00001,
-0.00033, -0.00273, -0.00007, 0.00031, 0.00287,
0.00013, -0.00014, -0.00170, -0.00012, 0.00006,
0.00014, -0.00001, -0.00015, 0.00000, 0.00001,
0.00110, -0.00016, -0.00283, -0.00119, 0.00013,
0.00297, 0.00063, -0.00004, -0.00177, -0.00013,
0.00005, 0.00017, -0.00002, -0.00008, 0.00001,
0.00324, 0.00009, -0.00050, -0.00374, -0.00014,
0.00063, 0.00495, 0.00024, -0.00093, -0.00020,
0.00006, -0.00010, 0.00000, -0.00001, 0.00004,
0.00020, 0.00150, 0.00000, -0.00021, -0.00170,
-0.00004, 0.00024, 0.00214, 0.00012, -0.00002,
-0.00004, 0.00000, 0.00006, -0.00005, -0.00001,
-0.00061, 0.00012, 0.00156, 0.00070, -0.00012,
-0.00177, -0.00093, 0.00012, 0.00223, 0.00004,
-0.00002, -0.00003, 0.00001, 0.00003, -0.00013,
-0.00115, -0.00010, 0.00021, 0.00064, 0.00006,
-0.00013, -0.00020, -0.00002, 0.00004, 0.00057,
0.00001, -0.00009, 0.00000, 0.00000, 0.00001,
0.00011, -0.00026, -0.00005, -0.00015, 0.00014,
0.00005, 0.00006, -0.00004, -0.00002, 0.00001,
0.00012, 0.00001, 0.00000, -0.00002, 0.00000,
0.00015, 0.00002, -0.00031, -0.00005, -0.00001,
0.00017, -0.00010, 0.00000, -0.00003, -0.00009,
0.00001, 0.00014, 0.00000, 0.00000, -0.00005,
0.00001, 0.00012, 0.00001, 0.00000, -0.00015,
-0.00002, 0.00000, 0.00006, 0.00001, 0.00000,
0.00000, 0.00000, 0.00010, 0.00001, 0.00000,
-0.00002, 0.00004, 0.00007, 0.00003, 0.00000,
-0.00008, -0.00001, -0.00005, 0.00003, 0.00000,
-0.00002, 0.00000, 0.00001, 0.00005, 0.00001,
-0.00001, -0.00001, 0.00006, -0.00001, 0.00001,
0.00001, 0.00004, -0.00001, -0.00013, 0.00001,
0.00000, -0.00005, 0.00000, 0.00001, 0.00012
),
nrow = 15
)
# Specific time interval ----------------------------------------------------
MCBetaStd(
phi = phi,
sigma = sigma,
vcov_theta = vcov_theta,
delta_t = 1,
R = 100L # use a large value for R in actual research
)
# Range of time intervals ---------------------------------------------------
mc <- MCBetaStd(
phi = phi,
sigma = sigma,
vcov_theta = vcov_theta,
delta_t = 1:5,
R = 100L # use a large value for R in actual research
)
plot(mc)
# Methods -------------------------------------------------------------------
# MCBetaStd has a number of methods including
# print, summary, confint, and plot
print(mc)
summary(mc)
confint(mc, level = 0.95)
plot(mc)
cTMed documentation built on Nov. 5, 2025, 7:18 p.m.
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View source: R/cTMed-mc-beta-std.R
| MCBetaStd | R Documentation |
Monte Carlo Sampling Distribution for the Elements of the Standardized Matrix of Lagged Coefficients Over a Specific Time Interval or a Range of Time Intervals
Description
This function generates a Monte Carlo method
sampling distribution
for the elements of the standardized matrix of lagged coefficients
\boldsymbol{\beta}
over a specific time interval \Delta t
or a range of time intervals
using the first-order stochastic differential equation model
drift matrix \boldsymbol{\Phi}
and process noise covariance matrix \boldsymbol{\Sigma}.
Usage
MCBetaStd(
phi,
sigma,
vcov_theta,
delta_t,
R,
test_phi = TRUE,
ncores = NULL,
seed = NULL,
tol = 0.01
)
Arguments
phi |
Numeric matrix.
The drift matrix ( |
sigma |
Numeric matrix.
The process noise covariance matrix ( |
vcov_theta |
Numeric matrix.
The sampling variance-covariance matrix of
|
delta_t |
Numeric.
Time interval
( |
R |
Positive integer. Number of replications. |
test_phi |
Logical.
If |
ncores |
Positive integer.
Number of cores to use.
If |
seed |
Random seed. |
tol |
Numeric. Smallest possible time interval to allow. |
Details
See TotalStd().
Monte Carlo Method
Let \boldsymbol{\theta} be
a vector that combines
\mathrm{vec} \left( \boldsymbol{\Phi} \right),
that is,
the elements of the \boldsymbol{\Phi} matrix
in vector form sorted column-wise and
\mathrm{vech} \left( \boldsymbol{\Sigma} \right),
that is,
the unique elements of the \boldsymbol{\Sigma} matrix
in vector form sorted column-wise.
Let \hat{\boldsymbol{\theta}} be
a vector that combines
\mathrm{vec} \left( \hat{\boldsymbol{\Phi}} \right) and
\mathrm{vech} \left( \hat{\boldsymbol{\Sigma}} \right).
Based on the asymptotic properties of maximum likelihood estimators,
we can assume that estimators are normally distributed
around the population parameters.
\hat{\boldsymbol{\theta}}
\sim
\mathcal{N}
\left(
\boldsymbol{\theta},
\mathbb{V} \left( \hat{\boldsymbol{\theta}} \right)
\right)
Using this distributional assumption,
a sampling distribution of \hat{\boldsymbol{\theta}}
which we refer to as \hat{\boldsymbol{\theta}}^{\ast}
can be generated by replacing the population parameters
with sample estimates,
that is,
\hat{\boldsymbol{\theta}}^{\ast}
\sim
\mathcal{N}
\left(
\hat{\boldsymbol{\theta}},
\hat{\mathbb{V}} \left( \hat{\boldsymbol{\theta}} \right)
\right) .
Let
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right)
be a parameter that is a function of the estimated parameters.
A sampling distribution of
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right) ,
which we refer to as
\mathbf{g} \left( \hat{\boldsymbol{\theta}}^{\ast} \right) ,
can be generated by using the simulated estimates
to calculate
\mathbf{g}.
The standard deviations of the simulated estimates
are the standard errors.
Percentiles corresponding to
100 \left( 1 - \alpha \right) \%
are the confidence intervals.
Value
Returns an object
of class ctmedmc which is a list with the following elements:
- call
Function call.
- args
Function arguments.
- fun
Function used ("MCBetaStd").
- output
A list the length of which is equal to the length of
delta_t.
Each element in the output list has the following elements:
- est
Estimated elements of the standardized matrix of lagged coefficients.
- thetahatstar
A matrix of Monte Carlo elements of the standardized matrix of lagged coefficients.
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:
BootBeta(),
BootBetaStd(),
BootIndirectCentral(),
BootMed(),
BootMedStd(),
BootTotalCentral(),
DeltaBeta(),
DeltaBetaStd(),
DeltaIndirectCentral(),
DeltaMed(),
DeltaMedStd(),
DeltaTotalCentral(),
Direct(),
DirectStd(),
Indirect(),
IndirectCentral(),
IndirectStd(),
MCBeta(),
MCIndirectCentral(),
MCMed(),
MCMedStd(),
MCPhi(),
MCPhiSigma(),
MCTotalCentral(),
Med(),
MedStd(),
PosteriorBeta(),
PosteriorIndirectCentral(),
PosteriorMed(),
PosteriorTotalCentral(),
Total(),
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")
sigma <- matrix(
data = c(
0.24455556, 0.02201587, -0.05004762,
0.02201587, 0.07067800, 0.01539456,
-0.05004762, 0.01539456, 0.07553061
),
nrow = 3
)
vcov_theta <- matrix(
data = c(
0.00843, 0.00040, -0.00151, -0.00600, -0.00033,
0.00110, 0.00324, 0.00020, -0.00061, -0.00115,
0.00011, 0.00015, 0.00001, -0.00002, -0.00001,
0.00040, 0.00374, 0.00016, -0.00022, -0.00273,
-0.00016, 0.00009, 0.00150, 0.00012, -0.00010,
-0.00026, 0.00002, 0.00012, 0.00004, -0.00001,
-0.00151, 0.00016, 0.00389, 0.00103, -0.00007,
-0.00283, -0.00050, 0.00000, 0.00156, 0.00021,
-0.00005, -0.00031, 0.00001, 0.00007, 0.00006,
-0.00600, -0.00022, 0.00103, 0.00644, 0.00031,
-0.00119, -0.00374, -0.00021, 0.00070, 0.00064,
-0.00015, -0.00005, 0.00000, 0.00003, -0.00001,
-0.00033, -0.00273, -0.00007, 0.00031, 0.00287,
0.00013, -0.00014, -0.00170, -0.00012, 0.00006,
0.00014, -0.00001, -0.00015, 0.00000, 0.00001,
0.00110, -0.00016, -0.00283, -0.00119, 0.00013,
0.00297, 0.00063, -0.00004, -0.00177, -0.00013,
0.00005, 0.00017, -0.00002, -0.00008, 0.00001,
0.00324, 0.00009, -0.00050, -0.00374, -0.00014,
0.00063, 0.00495, 0.00024, -0.00093, -0.00020,
0.00006, -0.00010, 0.00000, -0.00001, 0.00004,
0.00020, 0.00150, 0.00000, -0.00021, -0.00170,
-0.00004, 0.00024, 0.00214, 0.00012, -0.00002,
-0.00004, 0.00000, 0.00006, -0.00005, -0.00001,
-0.00061, 0.00012, 0.00156, 0.00070, -0.00012,
-0.00177, -0.00093, 0.00012, 0.00223, 0.00004,
-0.00002, -0.00003, 0.00001, 0.00003, -0.00013,
-0.00115, -0.00010, 0.00021, 0.00064, 0.00006,
-0.00013, -0.00020, -0.00002, 0.00004, 0.00057,
0.00001, -0.00009, 0.00000, 0.00000, 0.00001,
0.00011, -0.00026, -0.00005, -0.00015, 0.00014,
0.00005, 0.00006, -0.00004, -0.00002, 0.00001,
0.00012, 0.00001, 0.00000, -0.00002, 0.00000,
0.00015, 0.00002, -0.00031, -0.00005, -0.00001,
0.00017, -0.00010, 0.00000, -0.00003, -0.00009,
0.00001, 0.00014, 0.00000, 0.00000, -0.00005,
0.00001, 0.00012, 0.00001, 0.00000, -0.00015,
-0.00002, 0.00000, 0.00006, 0.00001, 0.00000,
0.00000, 0.00000, 0.00010, 0.00001, 0.00000,
-0.00002, 0.00004, 0.00007, 0.00003, 0.00000,
-0.00008, -0.00001, -0.00005, 0.00003, 0.00000,
-0.00002, 0.00000, 0.00001, 0.00005, 0.00001,
-0.00001, -0.00001, 0.00006, -0.00001, 0.00001,
0.00001, 0.00004, -0.00001, -0.00013, 0.00001,
0.00000, -0.00005, 0.00000, 0.00001, 0.00012
),
nrow = 15
)
# Specific time interval ----------------------------------------------------
MCBetaStd(
phi = phi,
sigma = sigma,
vcov_theta = vcov_theta,
delta_t = 1,
R = 100L # use a large value for R in actual research
)
# Range of time intervals ---------------------------------------------------
mc <- MCBetaStd(
phi = phi,
sigma = sigma,
vcov_theta = vcov_theta,
delta_t = 1:5,
R = 100L # use a large value for R in actual research
)
plot(mc)
# Methods -------------------------------------------------------------------
# MCBetaStd has a number of methods including
# print, summary, confint, and plot
print(mc)
summary(mc)
confint(mc, level = 0.95)
plot(mc)
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