Monte Carlo Sampling Distribution of Total, Direct, and Indirect Effects of X on Y Through M Over a Specific Time Interval or a Range of Time Intervals

This function generates a Monte Carlo method sampling distribution of 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 drift matrix \(\boldsymbol{\Phi}\).

Usage

MCMed(
  phi,
  vcov_phi_vec,
  delta_t,
  from,
  to,
  med,
  R,
  test_phi = TRUE,
  ncores = NULL,
  seed = NULL
)

Arguments

phi

Numeric matrix. The drift matrix (\(\boldsymbol{\Phi}\)). phi should 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

Numeric. 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.

R

Positive integer. Number of replications.

test_phi

Logical. If test_phi = TRUE, the function tests the stability of the generated drift matrix \(\boldsymbol{\Phi}\). If the test returns FALSE, the function generates a new drift matrix \(\boldsymbol{\Phi}\) and runs the test recursively until the test returns TRUE.

ncores

Positive integer. Number of cores to use. If ncores = NULL, use a single core. Consider using multiple cores when number of replications R is a large value.

seed

Random seed.

Value

Returns an object of class ctmedmc which is a list with the following elements:

call

Function call.

args

Function arguments.

fun

Function used ("MCMed").

output

A list with length of length(delta_t).

Each element in the output list has the following elements:

est

A vector of total, direct, and indirect effects.

thetahatstar

A matrix of Monte Carlo total, direct, and indirect effects.

Details

See Total(), Direct(), and Indirect() for more details.

Monte Carlo 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 \(\mathrm{vec} \left( \hat{\boldsymbol{\Phi}} \right)\). Based on the asymptotic properties of maximum likelihood estimators, we can assume that estimators are normally distributed around the population parameters. $$ \hat{\boldsymbol{\theta}} \sim \mathcal{N} \left( \boldsymbol{\theta}, \mathbb{V} \left( \hat{\boldsymbol{\theta}} \right) \right) $$ Using this distributional assumption, a sampling distribution of \(\hat{\boldsymbol{\theta}}\) which we refer to as \(\hat{\boldsymbol{\theta}}^{\ast}\) can be generated by replacing the population parameters with sample estimates, that is, $$ \hat{\boldsymbol{\theta}}^{\ast} \sim \mathcal{N} \left( \hat{\boldsymbol{\theta}}, \hat{\mathbb{V}} \left( \hat{\boldsymbol{\theta}} \right) \right) . $$ Let \(\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right)\) be a parameter that is a function of the estimated parameters. A sampling distribution of \(\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right)\) , which we refer to as \(\mathbf{g} \left( \hat{\boldsymbol{\theta}}^{\ast} \right)\) , can be generated by using the simulated estimates to calculate \(\mathbf{g}\). The standard deviations of the simulated estimates are the standard errors. Percentiles corresponding to \(100 \left( 1 - \alpha \right) \%\) are the confidence intervals.

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(), DeltaMed(), DeltaTotalCentral(), Direct(), Indirect(), IndirectCentral(), MCBeta(), MCIndirectCentral(), MCPhi(), MCTotalCentral(), Med(), PosteriorBeta(), PosteriorIndirectCentral(), PosteriorMed(), PosteriorPhi(), PosteriorTotalCentral(), Total(), TotalCentral(), Trajectory()

Author

Ivan Jacob Agaloos Pesigan

Examples

set.seed(42)
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 ----------------------------------------------------
MCMed(
  phi = phi,
  vcov_phi_vec = vcov_phi_vec,
  delta_t = 1,
  from = "x",
  to = "y",
  med = "m",
  R = 100L # use a large value for R in actual research
)
#> 
#> Total, Direct, and Indirect Effects
#> 
#> $`1`
#>          interval     est     se   R    2.5%   97.5%
#> total           1 -0.1000 0.0342 100 -0.1666 -0.0440
#> direct          1 -0.2675 0.0440 100 -0.3567 -0.1863
#> indirect        1  0.1674 0.0201 100  0.1273  0.2006
#> 

# Range of time intervals ---------------------------------------------------
mc <- MCMed(
  phi = phi,
  vcov_phi_vec = vcov_phi_vec,
  delta_t = 1:5,
  from = "x",
  to = "y",
  med = "m",
  R = 100L # use a large value for R in actual research
)
plot(mc)




# Methods -------------------------------------------------------------------
# MCMed has a number of methods including
# print, summary, confint, and plot
print(mc)
#> 
#> Total, Direct, and Indirect Effects
#> 
#> $`1`
#>          interval     est     se   R    2.5%   97.5%
#> total           1 -0.1000 0.0340 100 -0.1648 -0.0320
#> direct          1 -0.2675 0.0471 100 -0.3397 -0.1738
#> indirect        1  0.1674 0.0201 100  0.1239  0.1982
#> 
#> $`2`
#>          interval     est     se   R    2.5%   97.5%
#> total           2  0.0799 0.0352 100  0.0110  0.1542
#> direct          2 -0.3209 0.0555 100 -0.4178 -0.2099
#> indirect        2  0.4008 0.0413 100  0.3090  0.4663
#> 
#> $`3`
#>          interval     est     se   R    2.5%   97.5%
#> total           3  0.2508 0.0337 100  0.1884  0.3100
#> direct          3 -0.2914 0.0524 100 -0.3863 -0.1918
#> indirect        3  0.5423 0.0523 100  0.4351  0.6286
#> 
#> $`4`
#>          interval     est     se   R    2.5%   97.5%
#> total           4  0.3449 0.0324 100  0.2823  0.3989
#> direct          4 -0.2374 0.0464 100 -0.3272 -0.1538
#> indirect        4  0.5823 0.0575 100  0.4668  0.6888
#> 
#> $`5`
#>          interval     est     se   R    2.5%   97.5%
#> total           5  0.3693 0.0325 100  0.3091  0.4402
#> direct          5 -0.1828 0.0402 100 -0.2636 -0.1132
#> indirect        5  0.5521 0.0594 100  0.4404  0.6682
#> 
summary(mc)
#>      effect interval        est         se   R        2.5%      97.5%
#> 1     total        1 -0.1000384 0.03395899 100 -0.16476025 -0.0320114
#> 2    direct        1 -0.2674539 0.04709743 100 -0.33969758 -0.1737522
#> 3  indirect        1  0.1674155 0.02007626 100  0.12388402  0.1982442
#> 4     total        2  0.0799008 0.03520320 100  0.01098553  0.1541882
#> 5    direct        2 -0.3209035 0.05552653 100 -0.41779752 -0.2098965
#> 6  indirect        2  0.4008043 0.04125368 100  0.30898862  0.4662702
#> 7     total        3  0.2508138 0.03368900 100  0.18841270  0.3099660
#> 8    direct        3 -0.2914426 0.05239901 100 -0.38632534 -0.1918376
#> 9  indirect        3  0.5422564 0.05233930 100  0.43511469  0.6285914
#> 10    total        4  0.3449279 0.03244313 100  0.28231133  0.3989232
#> 11   direct        4 -0.2373900 0.04644744 100 -0.32719487 -0.1538166
#> 12 indirect        4  0.5823179 0.05746986 100  0.46682812  0.6888180
#> 13    total        5  0.3692538 0.03254938 100  0.30907920  0.4401860
#> 14   direct        5 -0.1828447 0.04019984 100 -0.26357998 -0.1131929
#> 15 indirect        5  0.5520985 0.05939646 100  0.44035670  0.6682153
confint(mc, level = 0.95)
#>      effect interval       2.5 %     97.5 %
#> 1     total        1 -0.16476025 -0.0320114
#> 2    direct        1 -0.33969758 -0.1737522
#> 3  indirect        1  0.12388402  0.1982442
#> 4     total        2  0.01098553  0.1541882
#> 5    direct        2 -0.41779752 -0.2098965
#> 6  indirect        2  0.30898862  0.4662702
#> 7     total        3  0.18841270  0.3099660
#> 8    direct        3 -0.38632534 -0.1918376
#> 9  indirect        3  0.43511469  0.6285914
#> 10    total        4  0.28231133  0.3989232
#> 11   direct        4 -0.32719487 -0.1538166
#> 12 indirect        4  0.46682812  0.6888180
#> 13    total        5  0.30907920  0.4401860
#> 14   direct        5 -0.26357998 -0.1131929
#> 15 indirect        5  0.44035670  0.6682153