MCMed: 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

Description

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,
  tol = 0.01
)

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.

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.
tol: Numeric. Smallest possible time interval to allow.

Author

Ivan Jacob Agaloos Pesigan

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. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1007/s11336-021-09767-0")

Examples

Run this code

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(
    0.00843, 0.00040, -0.00151,
    -0.00600, -0.00033, 0.00110,
    0.00324, 0.00020, -0.00061,
    0.00040, 0.00374, 0.00016,
    -0.00022, -0.00273, -0.00016,
    0.00009, 0.00150, 0.00012,
    -0.00151, 0.00016, 0.00389,
    0.00103, -0.00007, -0.00283,
    -0.00050, 0.00000, 0.00156,
    -0.00600, -0.00022, 0.00103,
    0.00644, 0.00031, -0.00119,
    -0.00374, -0.00021, 0.00070,
    -0.00033, -0.00273, -0.00007,
    0.00031, 0.00287, 0.00013,
    -0.00014, -0.00170, -0.00012,
    0.00110, -0.00016, -0.00283,
    -0.00119, 0.00013, 0.00297,
    0.00063, -0.00004, -0.00177,
    0.00324, 0.00009, -0.00050,
    -0.00374, -0.00014, 0.00063,
    0.00495, 0.00024, -0.00093,
    0.00020, 0.00150, 0.00000,
    -0.00021, -0.00170, -0.00004,
    0.00024, 0.00214, 0.00012,
    -0.00061, 0.00012, 0.00156,
    0.00070, -0.00012, -0.00177,
    -0.00093, 0.00012, 0.00223
  ),
  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
)

# 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)
summary(mc)
confint(mc, level = 0.95)

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