Delta Method Sampling Variance-Covariance Matrix for the Standardized Total Effect Centrality Over a Specific Time Interval or a Range of Time Intervals
Source:R/cTMed-delta-total-central-std.R
DeltaTotalCentralStd.RdThis function computes the delta method sampling variance-covariance matrix for the standardized 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 \(\boldsymbol{\Phi}\) and process noise covariance matrix \(\boldsymbol{\Sigma}\).
Usage
DeltaTotalCentralStd(
phi,
sigma,
vcov_theta,
delta_t,
sigma_diag = FALSE,
ncores = NULL,
tol = 0.001
)Arguments
- phi
Numeric matrix. The drift matrix (\(\boldsymbol{\Phi}\)).
phishould 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
Vector of positive numbers. Time interval (\(\Delta t\)).
- sigma_diag
Logical. If
sigma_diag = TRUE, treat \(\boldsymbol{\Sigma}\) as a diagonal matrix.- ncores
Positive integer. Number of cores to use. If
ncores = NULL, use a single core. Consider using multiple cores when the length ofdelta_tis long.- tol
Numeric. Smallest possible time interval to allow.
Value
Returns an object
of class ctmeddelta which is a list with the following elements:
- call
Function call.
- args
Function arguments.
- fun
Function used ("DeltaTotalCentralStd").
- output
A list of length
length(delta_t).
Each element in the output list has the following elements:
- delta_t
Time interval.
- jacobian
Jacobian matrix.
- est
Estimated standardized total effect centrality.
- vcov
Sampling variance-covariance matrix of estimated standardized total effect centrality.
Details
See TotalCentralStd() for more details.
Delta 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 \(\mathrm{vech} \left( \hat{\boldsymbol{\Sigma}} \right)\). 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
$$ \mathbf{g} \left( \hat{\boldsymbol{\theta}} \right) \approx \mathcal{N} \left( \mathbf{g} \left( \boldsymbol{\theta} \right) , \mathbf{J} \hat{\mathbb{V}} \left( \hat{\boldsymbol{\theta}} \right) \mathbf{J}^{\prime} \right) . $$
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
Pesigan, I. J. A., Russell, M. A., & Chow, S.-M. (2025). Inferences and effect sizes for direct, indirect, and total effects in continuous-time mediation models. Psychological Methods. doi:10.1037/met0000779
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:
BootBeta(),
BootBetaStd(),
BootDirectCentral(),
BootDirectCentralStd(),
BootIndirectCentral(),
BootIndirectCentralStd(),
BootMed(),
BootMedStd(),
BootTotalCentral(),
BootTotalCentralStd(),
DeltaBeta(),
DeltaBetaStd(),
DeltaDirectCentral(),
DeltaDirectCentralStd(),
DeltaIndirectCentral(),
DeltaMed(),
DeltaMedStd(),
DeltaTotalCentral(),
Direct(),
DirectCentral(),
DirectCentralStd(),
DirectStd(),
Indirect(),
IndirectCentral(),
IndirectCentralStd(),
IndirectStd(),
MCBeta(),
MCBetaStd(),
MCDirectCentral(),
MCDirectCentralStd(),
MCIndirectCentral(),
MCIndirectCentralStd(),
MCMed(),
MCMedStd(),
MCPhi(),
MCPhiSigma(),
MCTotalCentral(),
MCTotalCentralStd(),
Med(),
MedStd(),
PosteriorBeta(),
PosteriorBetaStd(),
PosteriorDirectCentral(),
PosteriorDirectCentralStd(),
PosteriorIndirectCentral(),
PosteriorIndirectCentralStd(),
PosteriorMed(),
PosteriorMedStd(),
PosteriorTotalCentral(),
PosteriorTotalCentralStd(),
Total(),
TotalCentral(),
TotalCentralStd(),
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 ----------------------------------------------------
DeltaTotalCentralStd(
phi = phi,
sigma = sigma,
vcov_theta = vcov_theta,
delta_t = 1
)
#> Call:
#> DeltaTotalCentralStd(phi = phi, sigma = sigma, vcov_theta = vcov_theta,
#> delta_t = 1)
#>
#> Total Effect Centrality
#> variable interval est se z p 2.5% 97.5%
#> 1 x 1 0.2819 0.0438 6.4424 0 0.1962 0.3677
#> 2 m 1 0.5494 0.0599 9.1793 0 0.4321 0.6667
#> 3 y 1 0.0000 0.0554 0.0000 1 -0.1086 0.1086
# Range of time intervals ---------------------------------------------------
delta <- DeltaTotalCentralStd(
phi = phi,
sigma = sigma,
vcov_theta = vcov_theta,
delta_t = 1:5
)
plot(delta)
# Methods -------------------------------------------------------------------
# DeltaTotalCentralStd has a number of methods including
# print, summary, confint, and plot
print(delta)
#> Call:
#> DeltaTotalCentralStd(phi = phi, sigma = sigma, vcov_theta = vcov_theta,
#> delta_t = 1:5)
#>
#> Total Effect Centrality
#> variable interval est se z p 2.5% 97.5%
#> 1 x 1 0.2819 0.0438 6.4424 0.0000 0.1962 0.3677
#> 2 m 1 0.5494 0.0599 9.1793 0.0000 0.4321 0.6667
#> 3 y 1 0.0000 0.0554 0.0000 1.0000 -0.1086 0.1086
#> 4 x 2 0.5907 0.0543 10.8715 0.0000 0.4842 0.6972
#> 5 m 2 0.6044 0.0744 8.1245 0.0000 0.4586 0.7502
#> 6 y 2 0.0000 0.0792 0.0000 1.0000 -0.1553 0.1553
#> 7 x 3 0.7616 0.0638 11.9328 0.0000 0.6365 0.8866
#> 8 m 3 0.4999 0.0833 5.9990 0.0000 0.3366 0.6633
#> 9 y 3 0.0000 0.0837 0.0000 1.0000 -0.1640 0.1640
#> 10 x 4 0.7979 0.0738 10.8167 0.0000 0.6533 0.9425
#> 11 m 4 0.3686 0.0899 4.1016 0.0000 0.1925 0.5448
#> 12 y 4 0.0000 0.0775 0.0000 1.0000 -0.1520 0.1520
#> 13 x 5 0.7453 0.0823 9.0536 0.0000 0.5840 0.9067
#> 14 m 5 0.2555 0.0925 2.7625 0.0057 0.0742 0.4368
#> 15 y 5 0.0000 0.0667 0.0000 1.0000 -0.1308 0.1308
summary(delta)
#> Call:
#> DeltaTotalCentralStd(phi = phi, sigma = sigma, vcov_theta = vcov_theta,
#> delta_t = 1:5)
#>
#> Total Effect Centrality
#> variable interval est se z p 2.5% 97.5%
#> 1 x 1 0.2819 0.0438 6.4424 0.0000 0.1962 0.3677
#> 2 m 1 0.5494 0.0599 9.1793 0.0000 0.4321 0.6667
#> 3 y 1 0.0000 0.0554 0.0000 1.0000 -0.1086 0.1086
#> 4 x 2 0.5907 0.0543 10.8715 0.0000 0.4842 0.6972
#> 5 m 2 0.6044 0.0744 8.1245 0.0000 0.4586 0.7502
#> 6 y 2 0.0000 0.0792 0.0000 1.0000 -0.1553 0.1553
#> 7 x 3 0.7616 0.0638 11.9328 0.0000 0.6365 0.8866
#> 8 m 3 0.4999 0.0833 5.9990 0.0000 0.3366 0.6633
#> 9 y 3 0.0000 0.0837 0.0000 1.0000 -0.1640 0.1640
#> 10 x 4 0.7979 0.0738 10.8167 0.0000 0.6533 0.9425
#> 11 m 4 0.3686 0.0899 4.1016 0.0000 0.1925 0.5448
#> 12 y 4 0.0000 0.0775 0.0000 1.0000 -0.1520 0.1520
#> 13 x 5 0.7453 0.0823 9.0536 0.0000 0.5840 0.9067
#> 14 m 5 0.2555 0.0925 2.7625 0.0057 0.0742 0.4368
#> 15 y 5 0.0000 0.0667 0.0000 1.0000 -0.1308 0.1308
confint(delta, level = 0.95)
#> variable interval 2.5 % 97.5 %
#> 1 x 1 0.19615900 0.3677027
#> 2 m 1 0.43211639 0.6667453
#> 3 y 1 -0.10858822 0.1085882
#> 4 x 2 0.48422186 0.6972162
#> 5 m 2 0.45856163 0.7501522
#> 6 y 2 -0.15529725 0.1552973
#> 7 x 3 0.63646957 0.8866394
#> 8 m 3 0.33660726 0.6632909
#> 9 y 3 -0.16397530 0.1639753
#> 10 x 4 0.65330788 0.9424560
#> 11 m 4 0.19247845 0.5447742
#> 12 y 4 -0.15197006 0.1519701
#> 13 x 5 0.58398441 0.9066927
#> 14 m 5 0.07422134 0.4367724
#> 15 y 5 -0.13075816 0.1307582
plot(delta)
