DeltaTotalCentral: Delta Method Sampling Variance-Covariance Matrix for the...
In cTMed: Continuous-Time Mediation
View source: R/cTMed-delta-total-central.R
DeltaTotalCentral R Documentation
Delta Method Sampling Variance-Covariance Matrix
for the Total Effect Centrality
Over a Specific Time Interval
or a Range of Time Intervals
Description
This function computes the delta method
sampling variance-covariance matrix
for the total effect centrality
over a specific time interval \Delta t
or a range of time intervals
using the first-order stochastic differential equation model's
drift matrix \boldsymbol{\Phi}.
Usage
DeltaTotalCentral(phi, vcov_phi_vec, delta_t, ncores = 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.
vcov_phi_vec
Numeric matrix.
The sampling variance-covariance matrix of
\mathrm{vec} \left( \boldsymbol{\Phi} \right).
delta_t
Vector of positive numbers.
Time interval
(\Delta t).
ncores
Positive integer.
Number of cores to use.
If ncores = NULL,
use a single core.
Consider using multiple cores
when the length of delta_t is long.
tol
Numeric.
Smallest possible time interval to allow.
Details
See TotalCentral() more details.
Delta Method
Let \boldsymbol{\theta} be
\mathrm{vec} \left( \boldsymbol{\Phi} \right),
that is,
the elements of the \boldsymbol{\Phi} matrix
in vector form sorted column-wise.
Let \hat{\boldsymbol{\theta}} be
\mathrm{vec} \left( \hat{\boldsymbol{\Phi}} \right).
By the multivariate central limit theory,
the function \mathbf{g}
using \hat{\boldsymbol{\theta}} as input
can be expressed as:
\sqrt{n}
\left(
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right)
-
\mathbf{g} \left( \boldsymbol{\theta} \right)
\right)
\xrightarrow[]{
\mathrm{D}
}
\mathcal{N}
\left(
0,
\mathbf{J}
\boldsymbol{\Gamma}
\mathbf{J}^{\prime}
\right)
where \mathbf{J} is the matrix of first-order derivatives
of the function \mathbf{g}
with respect to the elements of \boldsymbol{\theta}
and
\boldsymbol{\Gamma}
is the asymptotic variance-covariance matrix of
\hat{\boldsymbol{\theta}}.
From the former,
we can derive the distribution of
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right) as follows:
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right)
\approx
\mathcal{N}
\left(
\mathbf{g} \left( \boldsymbol{\theta} \right)
,
n^{-1}
\mathbf{J}
\boldsymbol{\Gamma}
\mathbf{J}^{\prime}
\right)
The uncertainty associated with the estimator
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right)
is, therefore, given by
n^{-1} \mathbf{J} \boldsymbol{\Gamma} \mathbf{J}^{\prime} .
When \boldsymbol{\Gamma} is unknown,
by substitution,
we can use
the estimated sampling variance-covariance matrix of
\hat{\boldsymbol{\theta}},
that is,
\hat{\mathbb{V}} \left( \hat{\boldsymbol{\theta}} \right)
for n^{-1} \boldsymbol{\Gamma}.
Therefore,
the sampling variance-covariance matrix of
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right)
is given by
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right)
\approx
\mathcal{N}
\left(
\mathbf{g} \left( \boldsymbol{\theta} \right)
,
\mathbf{J}
\hat{\mathbb{V}} \left( \hat{\boldsymbol{\theta}} \right)
\mathbf{J}^{\prime}
\right) .
Value
Returns an object
of class ctmeddelta which is a list with the following elements:
- call
Function call.
- args
Function arguments.
- fun
Function used ("DeltaTotalCentral").
- 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:
- delta_t
Time interval.
- jacobian
Jacobian matrix.
- est
Estimated total effect centrality.
- vcov
Sampling variance-covariance matrix of
estimated total effect centrality.
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(),
Direct(),
DirectStd(),
Indirect(),
IndirectCentral(),
IndirectStd(),
MCBeta(),
MCBetaStd(),
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")
vcov_phi_vec <- matrix(
data = c(
0.00843, 0.00040, -0.00151,
-0.00600, -0.00033, 0.00110,
0.00324, 0.00020, -0.00061,
0.00040, 0.00374, 0.00016,
-0.00022, -0.00273, -0.00016,
0.00009, 0.00150, 0.00012,
-0.00151, 0.00016, 0.00389,
0.00103, -0.00007, -0.00283,
-0.00050, 0.00000, 0.00156,
-0.00600, -0.00022, 0.00103,
0.00644, 0.00031, -0.00119,
-0.00374, -0.00021, 0.00070,
-0.00033, -0.00273, -0.00007,
0.00031, 0.00287, 0.00013,
-0.00014, -0.00170, -0.00012,
0.00110, -0.00016, -0.00283,
-0.00119, 0.00013, 0.00297,
0.00063, -0.00004, -0.00177,
0.00324, 0.00009, -0.00050,
-0.00374, -0.00014, 0.00063,
0.00495, 0.00024, -0.00093,
0.00020, 0.00150, 0.00000,
-0.00021, -0.00170, -0.00004,
0.00024, 0.00214, 0.00012,
-0.00061, 0.00012, 0.00156,
0.00070, -0.00012, -0.00177,
-0.00093, 0.00012, 0.00223
),
nrow = 9
)
# Specific time interval ----------------------------------------------------
DeltaTotalCentral(
phi = phi,
vcov_phi_vec = vcov_phi_vec,
delta_t = 1
)
# Range of time intervals ---------------------------------------------------
delta <- DeltaTotalCentral(
phi = phi,
vcov_phi_vec = vcov_phi_vec,
delta_t = 1:5
)
plot(delta)
# Methods -------------------------------------------------------------------
# DeltaTotalCentral has a number of methods including
# print, summary, confint, and plot
print(delta)
summary(delta)
confint(delta, level = 0.95)
plot(delta)
cTMed documentation built on Nov. 5, 2025, 7:18 p.m.
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View source: R/cTMed-delta-total-central.R
| DeltaTotalCentral | R Documentation |
Delta Method Sampling Variance-Covariance Matrix for the Total Effect Centrality Over a Specific Time Interval or a Range of Time Intervals
Description
This function computes the delta method
sampling variance-covariance matrix
for the total effect centrality
over a specific time interval \Delta t
or a range of time intervals
using the first-order stochastic differential equation model's
drift matrix \boldsymbol{\Phi}.
Usage
DeltaTotalCentral(phi, vcov_phi_vec, delta_t, ncores = NULL, tol = 0.01)
Arguments
phi |
Numeric matrix.
The drift matrix ( |
vcov_phi_vec |
Numeric matrix.
The sampling variance-covariance matrix of
|
delta_t |
Vector of positive numbers.
Time interval
( |
ncores |
Positive integer.
Number of cores to use.
If |
tol |
Numeric. Smallest possible time interval to allow. |
Details
See TotalCentral() more details.
Delta Method
Let \boldsymbol{\theta} be
\mathrm{vec} \left( \boldsymbol{\Phi} \right),
that is,
the elements of the \boldsymbol{\Phi} matrix
in vector form sorted column-wise.
Let \hat{\boldsymbol{\theta}} be
\mathrm{vec} \left( \hat{\boldsymbol{\Phi}} \right).
By the multivariate central limit theory,
the function \mathbf{g}
using \hat{\boldsymbol{\theta}} as input
can be expressed as:
\sqrt{n}
\left(
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right)
-
\mathbf{g} \left( \boldsymbol{\theta} \right)
\right)
\xrightarrow[]{
\mathrm{D}
}
\mathcal{N}
\left(
0,
\mathbf{J}
\boldsymbol{\Gamma}
\mathbf{J}^{\prime}
\right)
where \mathbf{J} is the matrix of first-order derivatives
of the function \mathbf{g}
with respect to the elements of \boldsymbol{\theta}
and
\boldsymbol{\Gamma}
is the asymptotic variance-covariance matrix of
\hat{\boldsymbol{\theta}}.
From the former,
we can derive the distribution of
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right) as follows:
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right)
\approx
\mathcal{N}
\left(
\mathbf{g} \left( \boldsymbol{\theta} \right)
,
n^{-1}
\mathbf{J}
\boldsymbol{\Gamma}
\mathbf{J}^{\prime}
\right)
The uncertainty associated with the estimator
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right)
is, therefore, given by
n^{-1} \mathbf{J} \boldsymbol{\Gamma} \mathbf{J}^{\prime} .
When \boldsymbol{\Gamma} is unknown,
by substitution,
we can use
the estimated sampling variance-covariance matrix of
\hat{\boldsymbol{\theta}},
that is,
\hat{\mathbb{V}} \left( \hat{\boldsymbol{\theta}} \right)
for n^{-1} \boldsymbol{\Gamma}.
Therefore,
the sampling variance-covariance matrix of
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right)
is given by
\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right)
\approx
\mathcal{N}
\left(
\mathbf{g} \left( \boldsymbol{\theta} \right)
,
\mathbf{J}
\hat{\mathbb{V}} \left( \hat{\boldsymbol{\theta}} \right)
\mathbf{J}^{\prime}
\right) .
Value
Returns an object
of class ctmeddelta which is a list with the following elements:
- call
Function call.
- args
Function arguments.
- fun
Function used ("DeltaTotalCentral").
- 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:
- delta_t
Time interval.
- jacobian
Jacobian matrix.
- est
Estimated total effect centrality.
- vcov
Sampling variance-covariance matrix of estimated total effect centrality.
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(),
Direct(),
DirectStd(),
Indirect(),
IndirectCentral(),
IndirectStd(),
MCBeta(),
MCBetaStd(),
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")
vcov_phi_vec <- matrix(
data = c(
0.00843, 0.00040, -0.00151,
-0.00600, -0.00033, 0.00110,
0.00324, 0.00020, -0.00061,
0.00040, 0.00374, 0.00016,
-0.00022, -0.00273, -0.00016,
0.00009, 0.00150, 0.00012,
-0.00151, 0.00016, 0.00389,
0.00103, -0.00007, -0.00283,
-0.00050, 0.00000, 0.00156,
-0.00600, -0.00022, 0.00103,
0.00644, 0.00031, -0.00119,
-0.00374, -0.00021, 0.00070,
-0.00033, -0.00273, -0.00007,
0.00031, 0.00287, 0.00013,
-0.00014, -0.00170, -0.00012,
0.00110, -0.00016, -0.00283,
-0.00119, 0.00013, 0.00297,
0.00063, -0.00004, -0.00177,
0.00324, 0.00009, -0.00050,
-0.00374, -0.00014, 0.00063,
0.00495, 0.00024, -0.00093,
0.00020, 0.00150, 0.00000,
-0.00021, -0.00170, -0.00004,
0.00024, 0.00214, 0.00012,
-0.00061, 0.00012, 0.00156,
0.00070, -0.00012, -0.00177,
-0.00093, 0.00012, 0.00223
),
nrow = 9
)
# Specific time interval ----------------------------------------------------
DeltaTotalCentral(
phi = phi,
vcov_phi_vec = vcov_phi_vec,
delta_t = 1
)
# Range of time intervals ---------------------------------------------------
delta <- DeltaTotalCentral(
phi = phi,
vcov_phi_vec = vcov_phi_vec,
delta_t = 1:5
)
plot(delta)
# Methods -------------------------------------------------------------------
# DeltaTotalCentral has a number of methods including
# print, summary, confint, and plot
print(delta)
summary(delta)
confint(delta, level = 0.95)
plot(delta)
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