Total, Direct, and Indirect Effects in Continuous-Time Mediation Model
Ivan Jacob Agaloos Pesigan
2025-02-24
Source:vignettes/med.Rmd
med.RmdThe cTMed package offers tools for estimating and
quantifying uncertainty in 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 its 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 -0.351837 0.040477 -8.692 -0.431170 -0.272503 <2e-16 ***
#> phi_2_1 0.744281 0.022684 32.810 0.699821 0.788741 <2e-16 ***
#> phi_3_1 -0.458683 0.023307 -19.680 -0.504364 -0.413001 <2e-16 ***
#> phi_1_2 0.017299 0.035160 0.492 -0.051614 0.086211 0.3114
#> phi_2_2 -0.488816 0.019999 -24.442 -0.528013 -0.449619 <2e-16 ***
#> phi_3_2 0.726799 0.020673 35.157 0.686280 0.767318 <2e-16 ***
#> phi_1_3 -0.023825 0.026208 -0.909 -0.075191 0.027542 0.1817
#> phi_2_3 -0.009800 0.015134 -0.648 -0.039463 0.019863 0.2586
#> phi_3_3 -0.688346 0.015868 -43.379 -0.719448 -0.657245 <2e-16 ***
#> sigma_1_1 0.242175 0.007291 33.216 0.227885 0.256465 <2e-16 ***
#> sigma_2_1 0.023273 0.002647 8.792 0.018084 0.028461 <2e-16 ***
#> sigma_3_1 -0.050572 0.002740 -18.459 -0.055941 -0.045202 <2e-16 ***
#> sigma_2_2 0.070721 0.001916 36.921 0.066967 0.074476 <2e-16 ***
#> sigma_3_2 0.014990 0.001380 10.865 0.012286 0.017695 <2e-16 ***
#> sigma_3_3 0.072377 0.002106 34.361 0.068249 0.076505 <2e-16 ***
#> theta_1_1 0.198862 0.001189 167.270 0.196532 0.201192 <2e-16 ***
#> theta_2_2 0.199520 0.001001 199.355 0.197558 0.201481 <2e-16 ***
#> theta_3_3 0.201171 0.001017 197.798 0.199178 0.203165 <2e-16 ***
#> mu0_1_1 0.006365 0.118695 0.054 -0.226274 0.239004 0.4786
#> mu0_2_1 -0.042579 0.113165 -0.376 -0.264378 0.179220 0.3534
#> mu0_3_1 0.130157 0.102344 1.272 -0.070434 0.330747 0.1017
#> sigma0_1_1 1.150287 0.205378 5.601 0.747754 1.552820 <2e-16 ***
#> sigma0_2_1 0.413683 0.134865 3.067 0.149352 0.678015 0.0011 **
#> sigma0_3_1 0.225909 0.118800 1.902 -0.006935 0.458753 0.0286 *
#> sigma0_2_2 1.221862 0.198459 6.157 0.832890 1.610835 <2e-16 ***
#> sigma0_3_2 0.235429 0.125869 1.870 -0.011271 0.482128 0.0307 *
#> sigma0_3_3 0.962525 0.150708 6.387 0.667142 1.257908 <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")
vcov_phi_vec <- vcov(fit)[phi_varnames, phi_varnames]
# Drift matrix
phi
#> x m y
#> x -0.3518368 0.01729864 -0.02382471
#> m 0.7442809 -0.48881584 -0.00979981
#> y -0.4586826 0.72679902 -0.68834647
# Sampling variance-covariance matrix
vcov_phi_vec
#> phi_1_1 phi_2_1 phi_3_1 phi_1_2 phi_2_2
#> phi_1_1 1.638395e-03 4.205720e-05 -2.365513e-04 -1.376185e-03 -4.400313e-05
#> phi_2_1 4.205720e-05 5.145749e-04 1.901381e-05 -2.143907e-05 -4.359631e-04
#> phi_3_1 -2.365513e-04 1.901381e-05 5.432299e-04 1.936645e-04 -9.386709e-06
#> phi_1_2 -1.376185e-03 -2.143907e-05 1.936645e-04 1.236241e-03 3.126359e-05
#> phi_2_2 -4.400313e-05 -4.359631e-04 -9.386709e-06 3.126359e-05 3.999593e-04
#> phi_3_2 2.074921e-04 -2.517298e-05 -4.623548e-04 -1.848726e-04 1.576373e-05
#> phi_1_3 9.257184e-04 8.251254e-06 -1.204207e-04 -8.555641e-04 -1.597444e-05
#> phi_2_3 3.762530e-05 2.928343e-04 7.109899e-07 -3.090243e-05 -2.783699e-04
#> phi_3_3 -1.487146e-04 2.380476e-05 3.133472e-04 1.369958e-04 -1.775455e-05
#> phi_3_2 phi_1_3 phi_2_3 phi_3_3
#> phi_1_1 2.074921e-04 9.257184e-04 3.762530e-05 -1.487146e-04
#> phi_2_1 -2.517298e-05 8.251254e-06 2.928343e-04 2.380476e-05
#> phi_3_1 -4.623548e-04 -1.204207e-04 7.109899e-07 3.133472e-04
#> phi_1_2 -1.848726e-04 -8.555641e-04 -3.090243e-05 1.369958e-04
#> phi_2_2 1.576373e-05 -1.597444e-05 -2.783699e-04 -1.775455e-05
#> phi_3_2 4.273789e-04 1.199960e-04 -4.260080e-06 -3.007907e-04
#> phi_1_3 1.199960e-04 6.868542e-04 2.386256e-05 -1.071758e-04
#> phi_2_3 -4.260080e-06 2.386256e-05 2.290520e-04 8.181839e-06
#> phi_3_3 -3.007907e-04 -1.071758e-04 8.181839e-06 2.518045e-04In 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 <- DeltaMed(
phi = phi,
vcov_phi_vec = vcov_phi_vec,
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 1.23162 secs
plot(delta)
Monte Carlo Method
start <- Sys.time()
mc <- MCMed(
phi = phi,
vcov_phi_vec = vcov_phi_vec,
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 16.84551 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
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