Standardized Total, Direct, and Indirect Effects in Continuous-Time Mediation Model
Ivan Jacob Agaloos Pesigan
2026-02-05
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 -0.351839 0.036416 -9.662 -0.423213 -0.280465 <2e-16 ***
#> phi_2_1 0.744282 0.021777 34.177 0.701599 0.786964 <2e-16 ***
#> phi_3_1 -0.458680 0.023534 -19.490 -0.504806 -0.412554 <2e-16 ***
#> phi_1_2 0.017311 0.031705 0.546 -0.044829 0.079451 0.2925
#> phi_2_2 -0.488821 0.019277 -25.358 -0.526602 -0.451039 <2e-16 ***
#> phi_3_2 0.726800 0.020871 34.824 0.685894 0.767706 <2e-16 ***
#> phi_1_3 -0.023814 0.024025 -0.991 -0.070903 0.023275 0.1608
#> phi_2_3 -0.009810 0.014718 -0.667 -0.038657 0.019036 0.2525
#> phi_3_3 -0.688334 0.016040 -42.913 -0.719773 -0.656896 <2e-16 ***
#> sigma_1_1 0.242180 0.006794 35.646 0.228864 0.255496 <2e-16 ***
#> sigma_2_1 0.023273 0.002545 9.146 0.018285 0.028261 <2e-16 ***
#> sigma_3_1 -0.050574 0.002749 -18.395 -0.055963 -0.045186 <2e-16 ***
#> sigma_2_2 0.070722 0.001907 37.093 0.066985 0.074458 <2e-16 ***
#> sigma_3_2 0.014987 0.001381 10.854 0.012281 0.017694 <2e-16 ***
#> sigma_3_3 0.072376 0.002099 34.475 0.068261 0.076491 <2e-16 ***
#> theta_1_1 0.198861 0.001170 169.909 0.196567 0.201155 <2e-16 ***
#> theta_2_2 0.199520 0.001000 199.500 0.197560 0.201480 <2e-16 ***
#> theta_3_3 0.201172 0.001016 198.052 0.199181 0.203162 <2e-16 ***
#> mu0_1_1 0.006324 0.111110 0.057 -0.211447 0.224095 0.4773
#> mu0_2_1 -0.042530 0.114320 -0.372 -0.266593 0.181533 0.3549
#> mu0_3_1 0.130043 0.102109 1.274 -0.070086 0.330172 0.1014
#> sigma0_1_1 1.150287 0.168811 6.814 0.819425 1.481150 <2e-16 ***
#> sigma0_2_1 0.413648 0.133495 3.099 0.152003 0.675293 0.0010 ***
#> sigma0_3_1 0.225993 0.123478 1.830 -0.016019 0.468006 0.0336 *
#> sigma0_2_2 1.221957 0.182233 6.705 0.864787 1.579128 <2e-16 ***
#> sigma0_3_2 0.235327 0.117629 2.001 0.004779 0.465875 0.0227 *
#> sigma0_3_3 0.962594 0.142152 6.772 0.683981 1.241207 <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 -0.3518392 0.01731083 -0.023814339
#> m 0.7442816 -0.48882067 -0.009810166
#> y -0.4586796 0.72679980 -0.688334177
# Process noise covariance matrix
sigma
#> [,1] [,2] [,3]
#> [1,] 0.24218026 0.02327296 -0.05057416
#> [2,] 0.02327296 0.07072156 0.01498732
#> [3,] -0.05057416 0.01498732 0.07237598
# Sampling variance-covariance matrix
vcov_theta
#> phi_1_1 phi_2_1 phi_3_1 phi_1_2 phi_2_2
#> phi_1_1 1.326121e-03 9.158296e-05 -2.258193e-04 -1.108000e-03 -8.829144e-05
#> phi_2_1 9.158296e-05 4.742430e-04 -3.596064e-06 -6.566903e-05 -4.021299e-04
#> phi_3_1 -2.258193e-04 -3.596064e-06 5.538551e-04 1.845433e-04 1.077730e-05
#> phi_1_2 -1.108000e-03 -6.566903e-05 1.845433e-04 1.005190e-03 7.060966e-05
#> phi_2_2 -8.829144e-05 -4.021299e-04 1.077730e-05 7.060966e-05 3.715859e-04
#> phi_3_2 1.994853e-04 -3.470399e-06 -4.716662e-04 -1.780693e-04 -3.561584e-06
#> phi_1_3 7.414387e-04 3.913166e-05 -1.151977e-04 -6.965449e-04 -4.344659e-05
#> phi_2_3 6.974956e-05 2.704005e-04 -1.452237e-05 -5.931610e-05 -2.595864e-04
#> phi_3_3 -1.424724e-04 7.176532e-06 3.208390e-04 1.318148e-04 -2.963173e-06
#> sigma_1_1 -1.982932e-04 -2.184773e-05 3.456919e-05 1.543789e-04 1.849510e-05
#> sigma_2_1 1.502203e-05 -3.437997e-05 -6.620570e-06 -1.591184e-05 2.574007e-05
#> sigma_3_1 1.794495e-05 4.674418e-06 -4.279970e-05 -1.117672e-05 -3.814434e-06
#> sigma_2_2 4.361454e-07 1.424515e-05 1.010140e-06 -1.579752e-07 -1.451605e-05
#> sigma_3_2 -3.741407e-06 1.607279e-06 8.491226e-06 3.585007e-06 4.201294e-07
#> sigma_3_3 -1.043117e-06 -8.607286e-07 3.442663e-06 7.089937e-08 4.858637e-07
#> phi_3_2 phi_1_3 phi_2_3 phi_3_3 sigma_1_1
#> phi_1_1 1.994853e-04 7.414387e-04 6.974956e-05 -1.424724e-04 -1.982932e-04
#> phi_2_1 -3.470399e-06 3.913166e-05 2.704005e-04 7.176532e-06 -2.184773e-05
#> phi_3_1 -4.716662e-04 -1.151977e-04 -1.452237e-05 3.208390e-04 3.456919e-05
#> phi_1_2 -1.780693e-04 -6.965449e-04 -5.931610e-05 1.318148e-04 1.543789e-04
#> phi_2_2 -3.561584e-06 -4.344659e-05 -2.595864e-04 -2.963173e-06 1.849510e-05
#> phi_3_2 4.355884e-04 1.163975e-04 1.028072e-05 -3.075120e-04 -2.844219e-05
#> phi_1_3 1.163975e-04 5.772234e-04 4.373490e-05 -1.045143e-04 -9.445101e-05
#> phi_2_3 1.028072e-05 4.373490e-05 2.166149e-04 -2.958830e-06 -1.266847e-05
#> phi_3_3 -3.075120e-04 -1.045143e-04 -2.958830e-06 2.572924e-04 1.859927e-05
#> sigma_1_1 -2.844219e-05 -9.445101e-05 -1.266847e-05 1.859927e-05 4.616001e-05
#> sigma_2_1 6.267263e-06 1.038566e-05 -1.506230e-05 -4.238014e-06 -3.795706e-07
#> sigma_3_1 3.275583e-05 1.446600e-06 2.154429e-06 -1.897455e-05 -5.585381e-06
#> sigma_2_2 -1.392458e-06 1.948765e-08 9.194261e-06 1.027499e-06 -1.692509e-07
#> sigma_3_2 -8.374628e-06 -2.326402e-06 -2.643065e-06 5.086543e-06 3.054065e-07
#> sigma_3_3 6.621115e-07 1.340588e-06 1.970612e-07 -5.788873e-06 6.193436e-07
#> sigma_2_1 sigma_3_1 sigma_2_2 sigma_3_2 sigma_3_3
#> phi_1_1 1.502203e-05 1.794495e-05 4.361454e-07 -3.741407e-06 -1.043117e-06
#> phi_2_1 -3.437997e-05 4.674418e-06 1.424515e-05 1.607279e-06 -8.607286e-07
#> phi_3_1 -6.620570e-06 -4.279970e-05 1.010140e-06 8.491226e-06 3.442663e-06
#> phi_1_2 -1.591184e-05 -1.117672e-05 -1.579752e-07 3.585007e-06 7.089937e-08
#> phi_2_2 2.574007e-05 -3.814434e-06 -1.451605e-05 4.201294e-07 4.858637e-07
#> phi_3_2 6.267263e-06 3.275583e-05 -1.392458e-06 -8.374628e-06 6.621115e-07
#> phi_1_3 1.038566e-05 1.446600e-06 1.948765e-08 -2.326402e-06 1.340588e-06
#> phi_2_3 -1.506230e-05 2.154429e-06 9.194261e-06 -2.643065e-06 1.970612e-07
#> phi_3_3 -4.238014e-06 -1.897455e-05 1.027499e-06 5.086543e-06 -5.788873e-06
#> sigma_1_1 -3.795706e-07 -5.585381e-06 -1.692509e-07 3.054065e-07 6.193436e-07
#> sigma_2_1 6.475588e-06 1.002384e-07 -4.387889e-07 -9.708718e-07 3.360567e-08
#> sigma_3_1 1.002384e-07 7.558692e-06 9.058849e-08 -3.163361e-07 -1.947016e-06
#> sigma_2_2 -4.387889e-07 9.058849e-08 3.635137e-06 3.523268e-08 -4.509608e-08
#> sigma_3_2 -9.708718e-07 -3.163361e-07 3.523268e-08 1.906780e-06 4.773520e-08
#> sigma_3_3 3.360567e-08 -1.947016e-06 -4.509608e-08 4.773520e-08 4.407331e-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 0.8466625 secs
plot(delta)
Monte 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 15.01979 mins
plot(mc)
References
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
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. https://doi.org/10.1037/met0000779
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