Standardized Total, Direct, and Indirect Effects in Continuous-Time Mediation Model
Ivan Jacob Agaloos Pesigan
2025-08-25
Source:vignettes/med-std.Rmd
med-std.RmdThe cTMed package offers tools for estimating and
quantifying uncertainty in standardized total, direct, and indirect
effects within continuous-time mediation models across various time
intervals using the delta and Monte Carlo methods. To implement these
approaches, estimates from a continuous-time vector autoregressive
(CT-VAR) model are required, particularly the drift matrix, and process
noise covariance matrix with the corresponding sampling
variance-covariance matrix. For guidance on fitting CT-VAR models using
the dynr or OpenMx packages, refer to Fit
the Continuous-Time Vector Autoregressive Model Using the dynr
Package and Fit
the Continuous-Time Vector Autoregressive Model Using the OpenMx
Package, respectively.
summary(fit)
#> Coefficients:
#> Estimate Std. Error t value ci.lower ci.upper Pr(>|t|)
#> phi_1_1 -0.3518738 0.0345599 -10.182 -0.4196100 -0.2841377 <2e-16 ***
#> phi_2_1 0.7442754 0.0213025 34.938 0.7025233 0.7860275 <2e-16 ***
#> phi_3_1 -0.4586681 0.0224194 -20.459 -0.5026093 -0.4147269 <2e-16 ***
#> phi_1_2 0.0173316 0.0304068 0.570 -0.0422645 0.0769278 0.2843
#> phi_2_2 -0.4888138 0.0188377 -25.949 -0.5257350 -0.4518927 <2e-16 ***
#> phi_3_2 0.7267855 0.0198254 36.659 0.6879284 0.7656426 <2e-16 ***
#> phi_1_3 -0.0238213 0.0233767 -1.019 -0.0696387 0.0219962 0.1541
#> phi_2_3 -0.0098181 0.0144220 -0.681 -0.0380848 0.0184485 0.2480
#> phi_3_3 -0.6883228 0.0152967 -44.998 -0.7183038 -0.6583418 <2e-16 ***
#> sigma_1_1 0.2421847 0.0064484 37.557 0.2295461 0.2548232 <2e-16 ***
#> sigma_2_1 0.0232727 0.0025183 9.241 0.0183369 0.0282084 <2e-16 ***
#> sigma_3_1 -0.0505753 0.0026989 -18.739 -0.0558649 -0.0452856 <2e-16 ***
#> sigma_2_2 0.0707197 0.0018899 37.420 0.0670155 0.0744238 <2e-16 ***
#> sigma_3_2 0.0149893 0.0013541 11.070 0.0123354 0.0176433 <2e-16 ***
#> sigma_3_3 0.0723762 0.0020868 34.683 0.0682861 0.0764663 <2e-16 ***
#> theta_1_1 0.1988610 0.0011588 171.608 0.1965898 0.2011323 <2e-16 ***
#> theta_2_2 0.1995204 0.0009996 199.609 0.1975613 0.2014795 <2e-16 ***
#> theta_3_3 0.2011716 0.0010145 198.288 0.1991832 0.2031601 <2e-16 ***
#> mu0_1_1 0.0062149 0.1219710 0.051 -0.2328439 0.2452737 0.4797
#> mu0_2_1 -0.0425006 0.1208284 -0.352 -0.2793200 0.1943187 0.3625
#> mu0_3_1 0.1300648 0.1041761 1.249 -0.0741167 0.3342462 0.1059
#> sigma0_1_1 1.1504264 0.1810442 6.354 0.7955863 1.5052665 <2e-16 ***
#> sigma0_2_1 0.4136629 0.1418008 2.917 0.1357384 0.6915875 0.0018 **
#> sigma0_3_1 0.2260142 0.1187792 1.903 -0.0067889 0.4588172 0.0285 *
#> sigma0_2_2 1.2218038 0.1985814 6.153 0.8325914 1.6110161 <2e-16 ***
#> sigma0_3_2 0.2353104 0.1267133 1.857 -0.0130432 0.4836640 0.0317 *
#> sigma0_3_3 0.9626701 0.1594441 6.038 0.6501655 1.2751747 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> -2 log-likelihood value at convergence = 429365.49
#> AIC = 429419.49
#> BIC = 429676.34
phi_varnames <- c(
"phi_1_1",
"phi_2_1",
"phi_3_1",
"phi_1_2",
"phi_2_2",
"phi_3_2",
"phi_1_3",
"phi_2_3",
"phi_3_3"
)
phi <- matrix(
data = coef(fit)[phi_varnames],
nrow = 3,
ncol = 3
)
colnames(phi) <- rownames(phi) <- c("x", "m", "y")
sigma_varnames <- c(
"sigma_1_1", "sigma_2_1", "sigma_3_1",
"sigma_2_1", "sigma_2_2", "sigma_3_2",
"sigma_3_1", "sigma_3_2", "sigma_3_3"
)
sigma <- matrix(
data = coef(fit)[sigma_varnames],
nrow = 3,
ncol = 3
)
theta_varnames <- c(
phi_varnames,
"sigma_1_1", "sigma_2_1", "sigma_3_1",
"sigma_2_2", "sigma_3_2",
"sigma_3_3"
)
vcov_theta <- vcov(fit)[theta_varnames, theta_varnames]
# Drift matrix
phi
#> x m y
#> x -0.3518738 0.01733165 -0.023821271
#> m 0.7442754 -0.48881381 -0.009818147
#> y -0.4586681 0.72678550 -0.688322785
# Process noise covariance matrix
sigma
#> [,1] [,2] [,3]
#> [1,] 0.24218466 0.02327267 -0.05057526
#> [2,] 0.02327267 0.07071965 0.01498933
#> [3,] -0.05057526 0.01498933 0.07237617
# Sampling variance-covariance matrix
vcov_theta
#> phi_1_1 phi_2_1 phi_3_1 phi_1_2 phi_2_2
#> phi_1_1 1.194386e-03 9.434520e-05 -1.832722e-04 -1.004483e-03 -8.964831e-05
#> phi_2_1 9.434520e-05 4.537958e-04 -4.543459e-06 -6.913945e-05 -3.836601e-04
#> phi_3_1 -1.832722e-04 -4.543459e-06 5.026282e-04 1.483225e-04 1.026409e-05
#> phi_1_2 -1.004483e-03 -6.913945e-05 1.483225e-04 9.245720e-04 7.282409e-05
#> phi_2_2 -8.964831e-05 -3.836601e-04 1.026409e-05 7.282409e-05 3.548578e-04
#> phi_3_2 1.607647e-04 -2.804753e-06 -4.250983e-04 -1.450161e-04 -2.919274e-06
#> phi_1_3 6.774418e-04 4.293486e-05 -9.128814e-05 -6.468942e-04 -4.622427e-05
#> phi_2_3 7.077153e-05 2.572020e-04 -1.302596e-05 -6.094418e-05 -2.476033e-04
#> phi_3_3 -1.143637e-04 6.496454e-06 2.868236e-04 1.076101e-04 -3.253344e-06
#> sigma_1_1 -1.729920e-04 -2.237575e-05 2.852008e-05 1.340106e-04 1.876584e-05
#> sigma_2_1 1.322993e-05 -3.283686e-05 -5.521064e-06 -1.446626e-05 2.438253e-05
#> sigma_3_1 1.541527e-05 4.680862e-06 -3.927416e-05 -9.108262e-06 -3.735018e-06
#> sigma_2_2 1.607424e-06 1.345058e-05 1.017099e-06 -1.221604e-06 -1.378053e-05
#> sigma_3_2 -1.924164e-06 1.728793e-06 7.082798e-06 2.066926e-06 2.530022e-07
#> sigma_3_3 6.711228e-09 -9.215972e-07 4.134147e-06 -8.068151e-07 5.584090e-07
#> phi_3_2 phi_1_3 phi_2_3 phi_3_3 sigma_1_1
#> phi_1_1 1.607647e-04 6.774418e-04 7.077153e-05 -1.143637e-04 -1.729920e-04
#> phi_2_1 -2.804753e-06 4.293486e-05 2.572020e-04 6.496454e-06 -2.237575e-05
#> phi_3_1 -4.250983e-04 -9.128814e-05 -1.302596e-05 2.868236e-04 2.852008e-05
#> phi_1_2 -1.450161e-04 -6.468942e-04 -6.094418e-05 1.076101e-04 1.340106e-04
#> phi_2_2 -2.919274e-06 -4.622427e-05 -2.476033e-04 -3.253344e-06 1.876584e-05
#> phi_3_2 3.930470e-04 9.435689e-05 8.776401e-06 -2.762247e-04 -2.291365e-05
#> phi_1_3 9.435689e-05 5.464686e-04 4.564710e-05 -8.808095e-05 -8.158519e-05
#> phi_2_3 8.776401e-06 4.564710e-05 2.079953e-04 -1.947278e-06 -1.283264e-05
#> phi_3_3 -2.762247e-04 -8.808095e-05 -1.947278e-06 2.339894e-04 1.465593e-05
#> sigma_1_1 -2.291365e-05 -8.158519e-05 -1.283264e-05 1.465593e-05 4.158154e-05
#> sigma_2_1 5.290129e-06 9.403917e-06 -1.409233e-05 -3.513111e-06 6.272750e-09
#> sigma_3_1 2.962004e-05 2.206458e-07 2.046372e-06 -1.675644e-05 -5.206212e-06
#> sigma_2_2 -1.398553e-06 8.112480e-07 8.663912e-06 9.962591e-07 -3.547992e-07
#> sigma_3_2 -7.103547e-06 -1.323634e-06 -2.461897e-06 4.199434e-06 4.766463e-08
#> sigma_3_3 -2.118570e-08 1.882929e-06 1.338675e-07 -5.198197e-06 4.762538e-07
#> sigma_2_1 sigma_3_1 sigma_2_2 sigma_3_2 sigma_3_3
#> phi_1_1 1.322993e-05 1.541527e-05 1.607424e-06 -1.924164e-06 6.711228e-09
#> phi_2_1 -3.283686e-05 4.680862e-06 1.345058e-05 1.728793e-06 -9.215972e-07
#> phi_3_1 -5.521064e-06 -3.927416e-05 1.017099e-06 7.082798e-06 4.134147e-06
#> phi_1_2 -1.446626e-05 -9.108262e-06 -1.221604e-06 2.066926e-06 -8.068151e-07
#> phi_2_2 2.438253e-05 -3.735018e-06 -1.378053e-05 2.530022e-07 5.584090e-07
#> phi_3_2 5.290129e-06 2.962004e-05 -1.398553e-06 -7.103547e-06 -2.118570e-08
#> phi_1_3 9.403917e-06 2.206458e-07 8.112480e-07 -1.323634e-06 1.882929e-06
#> phi_2_3 -1.409233e-05 2.046372e-06 8.663912e-06 -2.461897e-06 1.338675e-07
#> phi_3_3 -3.513111e-06 -1.675644e-05 9.962591e-07 4.199434e-06 -5.198197e-06
#> sigma_1_1 6.272750e-09 -5.206212e-06 -3.547992e-07 4.766463e-08 4.762538e-07
#> sigma_2_1 6.341788e-06 4.599255e-08 -3.676482e-07 -9.306099e-07 6.122437e-08
#> sigma_3_1 4.599255e-08 7.283856e-06 8.207137e-08 -2.042095e-07 -1.970287e-06
#> sigma_2_2 -3.676482e-07 8.207137e-08 3.571711e-06 7.140861e-08 -3.991608e-08
#> sigma_3_2 -9.306099e-07 -2.042095e-07 7.140861e-08 1.833491e-06 7.177255e-08
#> sigma_3_3 6.122437e-08 -1.970287e-06 -3.991608e-08 7.177255e-08 4.354813e-06In this example, we aim to calculate the total, direct, and indirect
effects of x on y, mediated through
m, over time intervals ranging from 0 to 10.
# time intervals
delta_t <- seq(from = 0, to = 10, length.out = 1000)Delta Method
library(cTMed)
start <- Sys.time()
delta <- DeltaMedStd(
phi = phi,
sigma = sigma,
vcov_theta = vcov_theta,
delta_t = delta_t,
from = "x",
to = "y",
med = "m",
ncores = parallel::detectCores() # use multiple cores
)
end <- Sys.time()
elapsed <- end - start
elapsed
#> Time difference of 1.411824 secs
plot(delta)
#> NULLMonte Carlo Method
start <- Sys.time()
mc <- MCMedStd(
phi = phi,
sigma = sigma,
vcov_theta = vcov_theta,
delta_t = delta_t,
from = "x",
to = "y",
med = "m",
R = 20000L,
ncores = parallel::detectCores() # use multiple cores
)
end <- Sys.time()
elapsed <- end - start
elapsed
#> Time difference of 17.67883 mins
plot(mc)
#> NULLReferences
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. https://doi.org/10.1080/10705511.2014.973960
R Core Team. (2024). R: A language and environment for
statistical computing. R Foundation for Statistical Computing. https://www.R-project.org/
Ryan, O., & Hamaker, E. L. (2021). Time to intervene: A
continuous-time approach to network analysis and centrality.
Psychometrika, 87(1), 214–252. https://doi.org/10.1007/s11336-021-09767-0