Delta Method Sampling Variance-Covariance Matrix for the Total, Direct, and Indirect Effects of X on Y Through M Over a Specific Time Interval or a Range of Time Intervals
Source:R/cTMed-delta-med.R
DeltaMed.RdThis function computes the delta method sampling variance-covariance matrix for the total, direct, and indirect effects of the independent variable \(X\) on the dependent variable \(Y\) through mediator variables \(\mathbf{m}\) over a specific time interval \(\Delta t\) or a range of time intervals using the first-order stochastic differential equation model's drift matrix \(\boldsymbol{\Phi}\).
Arguments
- phi
Numeric matrix. The drift matrix (\(\boldsymbol{\Phi}\)).
phishould 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 (\(\Delta t\)).
- 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.- 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.
Value
Returns an object
of class ctmeddelta which is a list with the following elements:
- call
Function call.
- args
Function arguments.
- fun
Function used ("DeltaMed").
- output
A list with length of
length(delta_t).
Each element in the output list has the following elements:
- delta_t
Time interval.
- jacobian
Jacobian matrix.
- est
Estimated total, direct, and indirect effects.
- vcov
Sampling variance-covariance matrix of the estimated total, direct, and indirect effects.
Details
See Total(),
Direct(), and
Indirect() for 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
$$ \mathbf{g} \left( \hat{\boldsymbol{\theta}} \right) \approx \mathcal{N} \left( \mathbf{g} \left( \boldsymbol{\theta} \right) , \mathbf{J} \hat{\mathbb{V}} \left( \hat{\boldsymbol{\theta}} \right) \mathbf{J}^{\prime} \right) . $$
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.
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(),
DeltaIndirectCentral(),
DeltaTotalCentral(),
Direct(),
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")
vcov_phi_vec <- matrix(
data = c(
0.002704274, -0.001475275, 0.000949122,
-0.001619422, 0.000885122, -0.000569404,
0.00085493, -0.000465824, 0.000297815,
-0.001475275, 0.004428442, -0.002642303,
0.000980573, -0.00271817, 0.001618805,
-0.000586921, 0.001478421, -0.000871547,
0.000949122, -0.002642303, 0.006402668,
-0.000697798, 0.001813471, -0.004043138,
0.000463086, -0.001120949, 0.002271711,
-0.001619422, 0.000980573, -0.000697798,
0.002079286, -0.001152501, 0.000753,
-0.001528701, 0.000820587, -0.000517524,
0.000885122, -0.00271817, 0.001813471,
-0.001152501, 0.00342605, -0.002075005,
0.000899165, -0.002532849, 0.001475579,
-0.000569404, 0.001618805, -0.004043138,
0.000753, -0.002075005, 0.004984032,
-0.000622255, 0.001634917, -0.003705661,
0.00085493, -0.000586921, 0.000463086,
-0.001528701, 0.000899165, -0.000622255,
0.002060076, -0.001096684, 0.000686386,
-0.000465824, 0.001478421, -0.001120949,
0.000820587, -0.002532849, 0.001634917,
-0.001096684, 0.003328692, -0.001926088,
0.000297815, -0.000871547, 0.002271711,
-0.000517524, 0.001475579, -0.003705661,
0.000686386, -0.001926088, 0.004726235
),
nrow = 9
)
# Specific time interval ----------------------------------------------------
DeltaMed(
phi = phi,
vcov_phi_vec = vcov_phi_vec,
delta_t = 1,
from = "x",
to = "y",
med = "m"
)
#>
#> Total, Direct, and Indirect Effects
#>
#> $`1`
#> interval est se z p 2.5% 97.5%
#> total 1 -0.1000 0.0329 -3.0424 0.0023 -0.1645 -0.0356
#> direct 1 -0.2675 0.0436 -6.1280 0.0000 -0.3530 -0.1819
#> indirect 1 0.1674 0.0194 8.6167 0.0000 0.1293 0.2055
#>
# Range of time intervals ---------------------------------------------------
delta <- DeltaMed(
phi = phi,
vcov_phi_vec = vcov_phi_vec,
delta_t = 1:5,
from = "x",
to = "y",
med = "m"
)
plot(delta)
# Methods -------------------------------------------------------------------
# DeltaMed has a number of methods including
# print, summary, confint, and plot
print(delta)
#>
#> Total, Direct, and Indirect Effects
#>
#> $`1`
#> interval est se z p 2.5% 97.5%
#> total 1 -0.1000 0.0329 -3.0424 0.0023 -0.1645 -0.0356
#> direct 1 -0.2675 0.0436 -6.1280 0.0000 -0.3530 -0.1819
#> indirect 1 0.1674 0.0194 8.6167 0.0000 0.1293 0.2055
#>
#> $`2`
#> interval est se z p 2.5% 97.5%
#> total 2 0.0799 0.0370 2.1603 0.0307 0.0074 0.1524
#> direct 2 -0.3209 0.0512 -6.2642 0.0000 -0.4213 -0.2205
#> indirect 2 0.4008 0.0389 10.3027 0.0000 0.3246 0.4771
#>
#> $`3`
#> interval est se z p 2.5% 97.5%
#> total 3 0.2508 0.0366 6.8606 0 0.1792 0.3225
#> direct 3 -0.2914 0.0485 -6.0055 0 -0.3866 -0.1963
#> indirect 3 0.5423 0.0493 11.0007 0 0.4456 0.6389
#>
#> $`4`
#> interval est se z p 2.5% 97.5%
#> total 4 0.3449 0.0350 9.8570 0 0.2763 0.4135
#> direct 4 -0.2374 0.0433 -5.4828 0 -0.3223 -0.1525
#> indirect 4 0.5823 0.0548 10.6249 0 0.4749 0.6897
#>
#> $`5`
#> interval est se z p 2.5% 97.5%
#> total 5 0.3693 0.0342 10.8008 0 0.3022 0.4363
#> direct 5 -0.1828 0.0376 -4.8614 0 -0.2566 -0.1091
#> indirect 5 0.5521 0.0571 9.6771 0 0.4403 0.6639
#>
summary(delta)
#> effect interval est se z p 2.5%
#> 1 total 1 -0.1000384 0.03288098 -3.042439 2.346691e-03 -0.164483903
#> 2 direct 1 -0.2674539 0.04364490 -6.127952 8.901719e-10 -0.352996324
#> 3 indirect 1 0.1674155 0.01942918 8.616704 6.890881e-18 0.129335016
#> 4 total 2 0.0799008 0.03698524 2.160343 3.074615e-02 0.007411056
#> 5 direct 2 -0.3209035 0.05122836 -6.264177 3.747993e-10 -0.421309236
#> 6 indirect 2 0.4008043 0.03890291 10.302682 6.852356e-25 0.324556004
#> 7 total 3 0.2508138 0.03655859 6.860599 6.857257e-12 0.179160303
#> 8 direct 3 -0.2914426 0.04852944 -6.005480 1.907663e-09 -0.386558557
#> 9 indirect 3 0.5422564 0.04929290 11.000699 3.791795e-28 0.445644109
#> 10 total 4 0.3449279 0.03499303 9.857046 6.390162e-23 0.276342831
#> 11 direct 4 -0.2373900 0.04329747 -5.482768 4.187214e-08 -0.322251508
#> 12 indirect 4 0.5823179 0.05480696 10.624891 2.282791e-26 0.474898266
#> 13 total 5 0.3692538 0.03418778 10.800755 3.413856e-27 0.302246977
#> 14 direct 5 -0.1828447 0.03761138 -4.861420 1.165464e-06 -0.256561662
#> 15 indirect 5 0.5520985 0.05705207 9.677098 3.772724e-22 0.440278497
#> 97.5%
#> 1 -0.03559285
#> 2 -0.18191145
#> 3 0.20549601
#> 4 0.15239055
#> 5 -0.22049777
#> 6 0.47705260
#> 7 0.32246735
#> 8 -0.19632665
#> 9 0.63886875
#> 10 0.41351299
#> 11 -0.15252853
#> 12 0.68973759
#> 13 0.43626060
#> 14 -0.10912777
#> 15 0.66391851
confint(delta, level = 0.95)
#> effect interval 2.5 % 97.5 %
#> 1 total 1 -0.164483903 -0.03559285
#> 2 direct 1 -0.352996324 -0.18191145
#> 3 indirect 1 0.129335016 0.20549601
#> 4 total 2 0.007411056 0.15239055
#> 5 direct 2 -0.421309236 -0.22049777
#> 6 indirect 2 0.324556004 0.47705260
#> 7 total 3 0.179160303 0.32246735
#> 8 direct 3 -0.386558557 -0.19632665
#> 9 indirect 3 0.445644109 0.63886875
#> 10 total 4 0.276342831 0.41351299
#> 11 direct 4 -0.322251508 -0.15252853
#> 12 indirect 4 0.474898266 0.68973759
#> 13 total 5 0.302246977 0.43626060
#> 14 direct 5 -0.256561662 -0.10912777
#> 15 indirect 5 0.440278497 0.66391851
plot(delta)
