Uncertainty Quantification in CT-Med (Total, Direct, and Indirect Effects)

Author

Ivan Jacob Agaloos Pesigan

Published

February 2, 2026

Install the cTMed package

library(cTMed)

Extracting Matrices for Use in `cTMed`

After fitting, we extract the estimated drift matrix (\(\boldsymbol{\Phi}\)) and process noise covariance (\(\boldsymbol{\Sigma}\)), along with their sampling covariance matrices.

fn <- "fit-ct-var-dynr.Rds"
fit <- readRDS(fn)
summary(fit)

Loading required namespace: dynr

Coefficients:
             Estimate Std. Error t value   ci.lower   ci.upper Pr(>|t|)    
phi_1_1    -0.2150421  0.1195297  -1.799 -0.4493160  0.0192318   0.0361 *  
phi_2_1     0.7992531  0.0799146  10.001  0.6426233  0.9558830   <2e-16 ***
phi_3_1    -0.5372629  0.0900443  -5.967 -0.7137466 -0.3607793   <2e-16 ***
phi_1_2    -0.1590345  0.0980611  -1.622 -0.3512306  0.0331617   0.0525 .  
phi_2_2    -0.5277810  0.0669206  -7.887 -0.6589430 -0.3966190   <2e-16 ***
phi_3_2     0.7714035  0.0748155  10.311  0.6247679  0.9180392   <2e-16 ***
phi_1_3     0.0944037  0.0879781   1.073 -0.0780302  0.2668375   0.1417    
phi_2_3     0.0700308  0.0592130   1.183 -0.0460246  0.1860862   0.1185    
phi_3_3    -0.7375808  0.0671581 -10.983 -0.8692084 -0.6059533   <2e-16 ***
sigma_1_1   0.2161389  0.0333941   6.472  0.1506876  0.2815902   <2e-16 ***
sigma_2_1   0.0148896  0.0142932   1.042 -0.0131246  0.0429038   0.1488    
sigma_3_1  -0.0577149  0.0166399  -3.468 -0.0903285 -0.0251013   0.0003 ***
sigma_2_2   0.0602356  0.0123083   4.894  0.0361118  0.0843594   <2e-16 ***
sigma_3_2   0.0061543  0.0098537   0.625 -0.0131586  0.0254671   0.2662    
sigma_3_3   0.0852357  0.0167118   5.100  0.0524810  0.1179903   <2e-16 ***
theta_1_1   0.1958856  0.0078438  24.973  0.1805121  0.2112592   <2e-16 ***
theta_2_2   0.2008968  0.0070651  28.435  0.1870494  0.2147442   <2e-16 ***
theta_3_3   0.1933677  0.0070266  27.519  0.1795958  0.2071396   <2e-16 ***
mu0_1_1    -0.1764606  0.1990029  -0.887 -0.5664990  0.2135778   0.1877    
mu0_2_1    -0.2433819  0.2454784  -0.991 -0.7245107  0.2377469   0.1608    
mu0_3_1     0.3312270  0.2619670   1.264 -0.1822190  0.8446729   0.1031    
sigma0_1_1  0.7319468  0.2506898   2.920  0.2406039  1.2232897   0.0018 ** 
sigma0_2_1 -0.0838456  0.2213391  -0.379 -0.5176622  0.3499710   0.3524    
sigma0_3_1  0.1028722  0.2350566   0.438 -0.3578302  0.5635746   0.3308    
sigma0_2_2  1.1670776  0.3852839   3.029  0.4119350  1.9222202   0.0012 ** 
sigma0_3_2  0.0007447  0.2904693   0.003 -0.5685648  0.5700541   0.4990    
sigma0_3_3  1.3219659  0.4366714   3.027  0.4661057  2.1778261   0.0012 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

-2 log-likelihood value at convergence = 8608.96
AIC = 8662.96
BIC = 8814.18

coefs <- coef(fit)
vcovs <- vcov(fit)
phi_names <- 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"
)
sigma_names <- 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_vech_names <- c(
  "sigma_1_1", "sigma_2_1", "sigma_3_1",
  "sigma_2_2", "sigma_3_2",
  "sigma_3_3"
)
theta_names <- 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",
  "sigma_1_1", "sigma_2_1", "sigma_3_1",
  "sigma_2_2", "sigma_3_2",
  "sigma_3_3"
)
phi <- matrix(
  data = coefs[phi_names],
  nrow = 3,
  ncol = 3
)
colnames(phi) <- rownames(phi) <- c("x", "m", "y")
sigma <- matrix(
  data = coefs[sigma_names],
  nrow = 3,
  ncol = 3
)
theta <- coefs[theta_names]
vcov_phi_vec <- vcovs[phi_names, phi_names]
vcov_sigma_vech <- vcovs[sigma_vech_names, sigma_vech_names]
vcov_theta <- vcovs[theta_names, theta_names]

Estimated Drift Matrix with Corresponding Sampling Covariance Matrix

phi

           x          m           y
x -0.2150421 -0.1590345  0.09440367
m  0.7992531 -0.5277810  0.07003080
y -0.5372629  0.7714035 -0.73758084

vcov_phi_vec

              phi_1_1       phi_2_1       phi_3_1       phi_1_2       phi_2_2
phi_1_1  0.0142873522  0.0006079799 -0.0030038849 -0.0097797943 -4.790491e-04
phi_2_1  0.0006079799  0.0063863490 -0.0004996187 -0.0001104026 -4.577211e-03
phi_3_1 -0.0030038849 -0.0004996187  0.0081079793  0.0018766011  4.121789e-04
phi_1_2 -0.0097797943 -0.0001104026  0.0018766011  0.0096159753  2.870124e-04
phi_2_2 -0.0004790491 -0.0045772109  0.0004121789  0.0002870124  4.478369e-03
phi_3_2  0.0020607367  0.0002166850 -0.0056919613 -0.0020116829 -3.373929e-04
phi_1_3  0.0066272153 -0.0000384865 -0.0010966857 -0.0065404706 -5.471564e-05
phi_2_3  0.0003941462  0.0031845277 -0.0003412820 -0.0003010944 -3.150796e-03
phi_3_3 -0.0014572097 -0.0001184132  0.0039009020  0.0014237474  1.646252e-04
              phi_3_2       phi_1_3       phi_2_3       phi_3_3
phi_1_1  0.0020607367  6.627215e-03  0.0003941462 -0.0014572097
phi_2_1  0.0002166850 -3.848650e-05  0.0031845277 -0.0001184132
phi_3_1 -0.0056919613 -1.096686e-03 -0.0003412820  0.0039009020
phi_1_2 -0.0020116829 -6.540471e-03 -0.0003010944  0.0014237474
phi_2_2 -0.0003373929 -5.471564e-05 -0.0031507962  0.0001646252
phi_3_2  0.0055973556  1.189036e-03  0.0003301834 -0.0038753551
phi_1_3  0.0011890355  7.740141e-03  0.0002868850 -0.0016934695
phi_2_3  0.0003301834  2.868850e-04  0.0035061835 -0.0003437688
phi_3_3 -0.0038753551 -1.693470e-03 -0.0003437688  0.0045102156

Process Noise Covariance Matrix with Corresponding Sampling Covariance Matrix

sigma

            [,1]        [,2]         [,3]
[1,]  0.21613889 0.014889581 -0.057714887
[2,]  0.01488958 0.060235625  0.006154261
[3,] -0.05771489 0.006154261  0.085235668

vcov_sigma_vech

              sigma_1_1     sigma_2_1     sigma_3_1     sigma_2_2     sigma_3_2
sigma_1_1  1.115167e-03  4.004670e-05 -1.981839e-04  1.254314e-05 -1.364776e-05
sigma_2_1  4.004670e-05  2.042965e-04 -1.184950e-05  1.534016e-05 -4.509985e-05
sigma_3_1 -1.981839e-04 -1.184950e-05  2.768867e-04 -4.248350e-06  1.226255e-05
sigma_2_2  1.254314e-05  1.534016e-05 -4.248350e-06  1.514936e-04 -8.509441e-06
sigma_3_2 -1.364776e-05 -4.509985e-05  1.226255e-05 -8.509441e-06  9.709484e-05
sigma_3_3  3.967862e-05  7.242787e-06 -1.124499e-04  5.840479e-07 -1.087946e-05
              sigma_3_3
sigma_1_1  3.967862e-05
sigma_2_1  7.242787e-06
sigma_3_1 -1.124499e-04
sigma_2_2  5.840479e-07
sigma_3_2 -1.087946e-05
sigma_3_3  2.792858e-04

Estimated Drift Matrix and Process Noise Covariance Matrix with Corresponding Sampling Covariance Matrix

theta

     phi_1_1      phi_2_1      phi_3_1      phi_1_2      phi_2_2      phi_3_2 
-0.215042116  0.799253150 -0.537262944 -0.159034458 -0.527781028  0.771403519 
     phi_1_3      phi_2_3      phi_3_3    sigma_1_1    sigma_2_1    sigma_3_1 
 0.094403674  0.070030797 -0.737580843  0.216138890  0.014889581 -0.057714887 
   sigma_2_2    sigma_3_2    sigma_3_3 
 0.060235625  0.006154261  0.085235668

vcov_theta

                phi_1_1       phi_2_1       phi_3_1       phi_1_2       phi_2_2
phi_1_1    1.428735e-02  6.079799e-04 -3.003885e-03 -9.779794e-03 -4.790491e-04
phi_2_1    6.079799e-04  6.386349e-03 -4.996187e-04 -1.104026e-04 -4.577211e-03
phi_3_1   -3.003885e-03 -4.996187e-04  8.107979e-03  1.876601e-03  4.121789e-04
phi_1_2   -9.779794e-03 -1.104026e-04  1.876601e-03  9.615975e-03  2.870124e-04
phi_2_2   -4.790491e-04 -4.577211e-03  4.121789e-04  2.870124e-04  4.478369e-03
phi_3_2    2.060737e-03  2.166850e-04 -5.691961e-03 -2.011683e-03 -3.373929e-04
phi_1_3    6.627215e-03 -3.848650e-05 -1.096686e-03 -6.540471e-03 -5.471564e-05
phi_2_3    3.941462e-04  3.184528e-03 -3.412820e-04 -3.010944e-04 -3.150796e-03
phi_3_3   -1.457210e-03 -1.184132e-04  3.900902e-03  1.423747e-03  1.646252e-04
sigma_1_1 -2.333538e-03 -3.147503e-04  5.590589e-04  1.229505e-03  1.558647e-04
sigma_2_1  1.860527e-04 -4.867247e-04 -7.184730e-05 -2.533846e-04  2.364259e-04
sigma_3_1  2.316315e-04  7.797280e-05 -6.592973e-04 -3.085935e-05 -2.232206e-05
sigma_2_2 -3.241493e-05  1.690887e-04  3.090526e-05  2.853755e-05 -2.312286e-04
sigma_3_2 -5.186949e-06  3.832628e-05  9.180382e-05  2.027968e-05  3.420616e-05
sigma_3_3 -3.757908e-05 -4.591419e-05  8.377694e-05 -7.921819e-07  1.697090e-05
                phi_3_2       phi_1_3       phi_2_3       phi_3_3     sigma_1_1
phi_1_1    2.060737e-03  6.627215e-03  3.941462e-04 -1.457210e-03 -2.333538e-03
phi_2_1    2.166850e-04 -3.848650e-05  3.184528e-03 -1.184132e-04 -3.147503e-04
phi_3_1   -5.691961e-03 -1.096686e-03 -3.412820e-04  3.900902e-03  5.590589e-04
phi_1_2   -2.011683e-03 -6.540471e-03 -3.010944e-04  1.423747e-03  1.229505e-03
phi_2_2   -3.373929e-04 -5.471564e-05 -3.150796e-03  1.646252e-04  1.558647e-04
phi_3_2    5.597356e-03  1.189036e-03  3.301834e-04 -3.875355e-03 -2.910012e-04
phi_1_3    1.189036e-03  7.740141e-03  2.868850e-04 -1.693470e-03 -5.863538e-04
phi_2_3    3.301834e-04  2.868850e-04  3.506184e-03 -3.437688e-04 -7.760723e-05
phi_3_3   -3.875355e-03 -1.693470e-03 -3.437688e-04  4.510216e-03  1.448105e-04
sigma_1_1 -2.910012e-04 -5.863538e-04 -7.760723e-05  1.448105e-04  1.115167e-03
sigma_2_1  7.670478e-05  1.414938e-04 -1.119311e-04 -3.768986e-05  4.004670e-05
sigma_3_1  3.251735e-04 -1.831107e-04 -2.155920e-05 -9.609252e-05 -1.981839e-04
sigma_2_2 -3.096602e-05 -1.275493e-05  1.359967e-04  1.458799e-05  1.254314e-05
sigma_3_2 -1.288004e-04 -1.380574e-06 -1.015506e-04  6.378117e-05 -1.364776e-05
sigma_3_3  7.878627e-05  5.620888e-05  1.958059e-05 -2.839006e-04  3.967862e-05
              sigma_2_1     sigma_3_1     sigma_2_2     sigma_3_2     sigma_3_3
phi_1_1    1.860527e-04  2.316315e-04 -3.241493e-05 -5.186949e-06 -3.757908e-05
phi_2_1   -4.867247e-04  7.797280e-05  1.690887e-04  3.832628e-05 -4.591419e-05
phi_3_1   -7.184730e-05 -6.592973e-04  3.090526e-05  9.180382e-05  8.377694e-05
phi_1_2   -2.533846e-04 -3.085935e-05  2.853755e-05  2.027968e-05 -7.921819e-07
phi_2_2    2.364259e-04 -2.232206e-05 -2.312286e-04  3.420616e-05  1.697090e-05
phi_3_2    7.670478e-05  3.251735e-04 -3.096602e-05 -1.288004e-04  7.878627e-05
phi_1_3    1.414938e-04 -1.831107e-04 -1.275493e-05 -1.380574e-06  5.620888e-05
phi_2_3   -1.119311e-04 -2.155920e-05  1.359967e-04 -1.015506e-04  1.958059e-05
phi_3_3   -3.768986e-05 -9.609252e-05  1.458799e-05  6.378117e-05 -2.839006e-04
sigma_1_1  4.004670e-05 -1.981839e-04  1.254314e-05 -1.364776e-05  3.967862e-05
sigma_2_1  2.042965e-04 -1.184950e-05  1.534016e-05 -4.509985e-05  7.242787e-06
sigma_3_1 -1.184950e-05  2.768867e-04 -4.248350e-06  1.226255e-05 -1.124499e-04
sigma_2_2  1.534016e-05 -4.248350e-06  1.514936e-04 -8.509441e-06  5.840479e-07
sigma_3_2 -4.509985e-05  1.226255e-05 -8.509441e-06  9.709484e-05 -1.087946e-05
sigma_3_3  7.242787e-06 -1.124499e-04  5.840479e-07 -1.087946e-05  2.792858e-04

delta_t <- seq(from = 0.01, to = 10, length.out = 1000)

Unstandardized

Delta Method

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
)

plot(delta)

Monte Carlo Method

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
)

plot(mc)

Standardized

Delta Method

delta_std <- 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
)

plot(delta_std)

Monte Carlo Method

mc_std <- 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
)

Warning in cleanup(mc.cleanup): unable to terminate child: No such process

plot(mc_std)

Install the cTMed package

Extracting Matrices for Use in cTMed

Estimated Drift Matrix with Corresponding Sampling Covariance Matrix

Process Noise Covariance Matrix with Corresponding Sampling Covariance Matrix

Estimated Drift Matrix and Process Noise Covariance Matrix with Corresponding Sampling Covariance Matrix

Unstandardized

Delta Method

Monte Carlo Method

Standardized

Delta Method

Monte Carlo Method

Extracting Matrices for Use in `cTMed`