Skip to contents

This function generates a Monte Carlo method sampling distribution for the elements of the standardized matrix of lagged coefficients \(\boldsymbol{\beta}\) over a specific time interval \(\Delta t\) or a range of time intervals using the first-order stochastic differential equation model drift matrix \(\boldsymbol{\Phi}\) and process noise covariance matrix \(\boldsymbol{\Sigma}\).

Usage

MCBetaStd(
  phi,
  sigma,
  vcov_theta,
  delta_t,
  R,
  test_phi = TRUE,
  ncores = NULL,
  seed = NULL,
  tol = 0.01
)

Arguments

phi

Numeric matrix. The drift matrix (\(\boldsymbol{\Phi}\)). phi should have row and column names pertaining to the variables in the system.

sigma

Numeric matrix. The process noise covariance matrix (\(\boldsymbol{\Sigma}\)).

vcov_theta

Numeric matrix. The sampling variance-covariance matrix of \(\mathrm{vec} \left( \boldsymbol{\Phi} \right)\) and \(\mathrm{vech} \left( \boldsymbol{\Sigma} \right)\)

delta_t

Numeric. Time interval (\(\Delta t\)).

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.

Value

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

call

Function call.

args

Function arguments.

fun

Function used ("MCBetaStd").

output

A list the length of which is equal to the length of delta_t.

Each element in the output list has the following elements:

est

Estimated elements of the standardized matrix of lagged coefficients.

thetahatstar

A matrix of Monte Carlo elements of the standardized matrix of lagged coefficients.

Details

See TotalStd().

Monte Carlo Method

Let \(\boldsymbol{\theta}\) be a vector that combines \(\mathrm{vec} \left( \boldsymbol{\Phi} \right)\), that is, the elements of the \(\boldsymbol{\Phi}\) matrix in vector form sorted column-wise and \(\mathrm{vech} \left( \boldsymbol{\Sigma} \right)\), that is, the unique elements of the \(\boldsymbol{\Sigma}\) matrix in vector form sorted column-wise. Let \(\hat{\boldsymbol{\theta}}\) be a vector that combines \(\mathrm{vec} \left( \hat{\boldsymbol{\Phi}} \right)\) and \(\mathrm{vech} \left( \hat{\boldsymbol{\Sigma}} \right)\). 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

Author

Ivan Jacob Agaloos Pesigan

Examples

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")
sigma <- matrix(
  data = c(
    0.24455556, 0.02201587, -0.05004762,
    0.02201587, 0.07067800, 0.01539456,
    -0.05004762, 0.01539456, 0.07553061
  ),
  nrow = 3
)
vcov_theta <- matrix(
  data = c(
    0.00843, 0.00040, -0.00151, -0.00600, -0.00033,
    0.00110, 0.00324, 0.00020, -0.00061, -0.00115,
    0.00011, 0.00015, 0.00001, -0.00002, -0.00001,
    0.00040, 0.00374, 0.00016, -0.00022, -0.00273,
    -0.00016, 0.00009, 0.00150, 0.00012, -0.00010,
    -0.00026, 0.00002, 0.00012, 0.00004, -0.00001,
    -0.00151, 0.00016, 0.00389, 0.00103, -0.00007,
    -0.00283, -0.00050, 0.00000, 0.00156, 0.00021,
    -0.00005, -0.00031, 0.00001, 0.00007, 0.00006,
    -0.00600, -0.00022, 0.00103, 0.00644, 0.00031,
    -0.00119, -0.00374, -0.00021, 0.00070, 0.00064,
    -0.00015, -0.00005, 0.00000, 0.00003, -0.00001,
    -0.00033, -0.00273, -0.00007, 0.00031, 0.00287,
    0.00013, -0.00014, -0.00170, -0.00012, 0.00006,
    0.00014, -0.00001, -0.00015, 0.00000, 0.00001,
    0.00110, -0.00016, -0.00283, -0.00119, 0.00013,
    0.00297, 0.00063, -0.00004, -0.00177, -0.00013,
    0.00005, 0.00017, -0.00002, -0.00008, 0.00001,
    0.00324, 0.00009, -0.00050, -0.00374, -0.00014,
    0.00063, 0.00495, 0.00024, -0.00093, -0.00020,
    0.00006, -0.00010, 0.00000, -0.00001, 0.00004,
    0.00020, 0.00150, 0.00000, -0.00021, -0.00170,
    -0.00004, 0.00024, 0.00214, 0.00012, -0.00002,
    -0.00004, 0.00000, 0.00006, -0.00005, -0.00001,
    -0.00061, 0.00012, 0.00156, 0.00070, -0.00012,
    -0.00177, -0.00093, 0.00012, 0.00223, 0.00004,
    -0.00002, -0.00003, 0.00001, 0.00003, -0.00013,
    -0.00115, -0.00010, 0.00021, 0.00064, 0.00006,
    -0.00013, -0.00020, -0.00002, 0.00004, 0.00057,
    0.00001, -0.00009, 0.00000, 0.00000, 0.00001,
    0.00011, -0.00026, -0.00005, -0.00015, 0.00014,
    0.00005, 0.00006, -0.00004, -0.00002, 0.00001,
    0.00012, 0.00001, 0.00000, -0.00002, 0.00000,
    0.00015, 0.00002, -0.00031, -0.00005, -0.00001,
    0.00017, -0.00010, 0.00000, -0.00003, -0.00009,
    0.00001, 0.00014, 0.00000, 0.00000, -0.00005,
    0.00001, 0.00012, 0.00001, 0.00000, -0.00015,
    -0.00002, 0.00000, 0.00006, 0.00001, 0.00000,
    0.00000, 0.00000, 0.00010, 0.00001, 0.00000,
    -0.00002, 0.00004, 0.00007, 0.00003, 0.00000,
    -0.00008, -0.00001, -0.00005, 0.00003, 0.00000,
    -0.00002, 0.00000, 0.00001, 0.00005, 0.00001,
    -0.00001, -0.00001, 0.00006, -0.00001, 0.00001,
    0.00001, 0.00004, -0.00001, -0.00013, 0.00001,
    0.00000, -0.00005, 0.00000, 0.00001, 0.00012
  ),
  nrow = 15
)

# Specific time interval ----------------------------------------------------
MCBetaStd(
  phi = phi,
  sigma = sigma,
  vcov_theta = vcov_theta,
  delta_t = 1,
  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%
#> from x to x        1  0.6998 0.0493 100  0.5952  0.8043
#> from x to m        1  0.3888 0.0275 100  0.3294  0.4295
#> from x to y        1 -0.1069 0.0353 100 -0.1865 -0.0383
#> from m to x        1  0.0000 0.0501 100 -0.0907  0.0962
#> from m to m        1  0.5999 0.0307 100  0.5578  0.6694
#> from m to y        1  0.5494 0.0394 100  0.4636  0.6343
#> from y to x        1  0.0000 0.0358 100 -0.0736  0.0636
#> from y to m        1  0.0000 0.0217 100 -0.0488  0.0367
#> from y to y        1  0.5001 0.0286 100  0.4538  0.5538
#> 

# Range of time intervals ---------------------------------------------------
mc <- MCBetaStd(
  phi = phi,
  sigma = sigma,
  vcov_theta = vcov_theta,
  delta_t = 1:5,
  R = 100L # use a large value for R in actual research
)
plot(mc)










# Methods -------------------------------------------------------------------
# MCBetaStd 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%
#> from x to x        1  0.6998 0.0458 100  0.6098  0.7961
#> from x to m        1  0.3888 0.0277 100  0.3403  0.4361
#> from x to y        1 -0.1069 0.0337 100 -0.1713 -0.0391
#> from m to x        1  0.0000 0.0541 100 -0.1146  0.1092
#> from m to m        1  0.5999 0.0323 100  0.5396  0.6545
#> from m to y        1  0.5494 0.0359 100  0.4864  0.6221
#> from y to x        1  0.0000 0.0416 100 -0.0700  0.0842
#> from y to m        1  0.0000 0.0229 100 -0.0385  0.0466
#> from y to y        1  0.5001 0.0265 100  0.4555  0.5554
#> 
#> $`2`
#>             interval    est     se   R    2.5%  97.5%
#> from x to x        2 0.4897 0.0530 100  0.4098 0.6038
#> from x to m        2 0.5053 0.0376 100  0.4387 0.5810
#> from x to y        2 0.0854 0.0385 100  0.0165 0.1550
#> from m to x        2 0.0000 0.0620 100 -0.1285 0.1200
#> from m to m        2 0.3599 0.0482 100  0.2711 0.4316
#> from m to y        2 0.6044 0.0409 100  0.5299 0.6692
#> from y to x        2 0.0000 0.0502 100 -0.0870 0.1036
#> from y to m        2 0.0000 0.0364 100 -0.0627 0.0748
#> from y to y        2 0.2501 0.0307 100  0.1958 0.3056
#> 
#> $`3`
#>             interval    est     se   R    2.5%  97.5%
#> from x to x        3 0.3427 0.0528 100  0.2622 0.4434
#> from x to m        3 0.4936 0.0427 100  0.4220 0.5822
#> from x to y        3 0.2680 0.0363 100  0.1941 0.3368
#> from m to x        3 0.0000 0.0597 100 -0.1136 0.1074
#> from m to m        3 0.2159 0.0565 100  0.1054 0.3108
#> from m to y        3 0.4999 0.0450 100  0.4214 0.5896
#> from y to x        3 0.0000 0.0460 100 -0.0806 0.0938
#> from y to m        3 0.0000 0.0438 100 -0.0759 0.0835
#> from y to y        3 0.1251 0.0288 100  0.0732 0.1724
#> 
#> $`4`
#>             interval    est     se   R    2.5%  97.5%
#> from x to x        4 0.2398 0.0514 100  0.1597 0.3290
#> from x to m        4 0.4293 0.0454 100  0.3589 0.5204
#> from x to y        4 0.3686 0.0358 100  0.3038 0.4534
#> from m to x        4 0.0000 0.0555 100 -0.0927 0.1129
#> from m to m        4 0.1295 0.0593 100  0.0053 0.2421
#> from m to y        4 0.3686 0.0495 100  0.2854 0.4576
#> from y to x        4 0.0000 0.0380 100 -0.0671 0.0769
#> from y to m        4 0.0000 0.0452 100 -0.0817 0.0864
#> from y to y        4 0.0625 0.0304 100  0.0085 0.1241
#> 
#> $`5`
#>             interval    est     se   R    2.5%  97.5%
#> from x to x        5 0.1678 0.0499 100  0.0819 0.2714
#> from x to m        5 0.3508 0.0468 100  0.2665 0.4416
#> from x to y        5 0.3946 0.0385 100  0.3385 0.4786
#> from m to x        5 0.0000 0.0502 100 -0.0711 0.1098
#> from m to m        5 0.0777 0.0586 100 -0.0316 0.2029
#> from m to y        5 0.2555 0.0518 100  0.1590 0.3509
#> from y to x        5 0.0000 0.0299 100 -0.0516 0.0589
#> from y to m        5 0.0000 0.0425 100 -0.0761 0.0825
#> from y to y        5 0.0313 0.0342 100 -0.0316 0.0922
#> 
summary(mc)
#>         effect interval         est         se   R         2.5%       97.5%
#> 1  from x to x        1  0.69977250 0.04584153 100  0.609800263  0.79613691
#> 2  from x to m        1  0.38882458 0.02774146 100  0.340301435  0.43607954
#> 3  from x to y        1 -0.10689374 0.03373783 100 -0.171343235 -0.03910697
#> 4  from m to x        1  0.00000000 0.05413487 100 -0.114554864  0.10917299
#> 5  from m to m        1  0.59989538 0.03228086 100  0.539629131  0.65446006
#> 6  from m to y        1  0.54943087 0.03593218 100  0.486375503  0.62210286
#> 7  from y to x        1  0.00000000 0.04158096 100 -0.069978350  0.08419140
#> 8  from y to m        1  0.00000000 0.02287789 100 -0.038467706  0.04661579
#> 9  from y to y        1  0.50007360 0.02653763 100  0.455466431  0.55535984
#> 10 from x to x        2  0.48968155 0.05301609 100  0.409792609  0.60379349
#> 11 from x to m        2  0.50534282 0.03761473 100  0.438703968  0.58099242
#> 12 from x to y        2  0.08537619 0.03845986 100  0.016520970  0.15495342
#> 13 from m to x        2  0.00000000 0.06195130 100 -0.128501620  0.12002161
#> 14 from m to m        2  0.35987447 0.04819742 100  0.271065357  0.43159309
#> 15 from m to y        2  0.60435691 0.04086121 100  0.529851093  0.66921661
#> 16 from y to x        2  0.00000000 0.05016101 100 -0.087029210  0.10356125
#> 17 from y to m        2  0.00000000 0.03635905 100 -0.062692060  0.07479334
#> 18 from y to y        2  0.25007360 0.03074378 100  0.195824888  0.30557324
#> 19 from x to x        3  0.34266568 0.05281164 100  0.262187269  0.44344399
#> 20 from x to m        3  0.49355305 0.04266416 100  0.421977255  0.58218446
#> 21 from x to y        3  0.26800143 0.03626586 100  0.194064568  0.33680333
#> 22 from m to x        3  0.00000000 0.05970868 100 -0.113592271  0.10738078
#> 23 from m to m        3  0.21588703 0.05650342 100  0.105419527  0.31077230
#> 24 from m to y        3  0.49994907 0.04499194 100  0.421397024  0.58962597
#> 25 from y to x        3  0.00000000 0.04599441 100 -0.080624702  0.09382282
#> 26 from y to m        3  0.00000000 0.04375317 100 -0.075852552  0.08350084
#> 27 from y to y        3  0.12505520 0.02884224 100  0.073229218  0.17243310
#> 28 from x to x        4  0.23978802 0.05135378 100  0.159681273  0.32901570
#> 29 from x to m        4  0.42931704 0.04535012 100  0.358920308  0.52043922
#> 30 from x to y        4  0.36856490 0.03575807 100  0.303759800  0.45341980
#> 31 from m to x        4  0.00000000 0.05551758 100 -0.092728769  0.11293315
#> 32 from m to m        4  0.12950963 0.05928525 100  0.005256487  0.24206241
#> 33 from m to y        4  0.36862633 0.04953581 100  0.285419794  0.45755739
#> 34 from y to x        4  0.00000000 0.03800484 100 -0.067122703  0.07692317
#> 35 from y to m        4  0.00000000 0.04523747 100 -0.081739936  0.08638891
#> 36 from y to y        4  0.06253681 0.03037461 100  0.008546347  0.12408688
#> 37 from x to x        5  0.16779706 0.04985991 100  0.081882464  0.27142605
#> 38 from x to m        5  0.35078078 0.04675633 100  0.266472539  0.44158762
#> 39 from x to y        5  0.39455777 0.03849245 100  0.338452030  0.47862426
#> 40 from m to x        5  0.00000000 0.05015623 100 -0.071091485  0.10982758
#> 41 from m to m        5  0.07769223 0.05857726 100 -0.031648320  0.20294748
#> 42 from m to y        5  0.25549689 0.05182686 100  0.159033085  0.35086532
#> 43 from y to x        5  0.00000000 0.02987096 100 -0.051647520  0.05887924
#> 44 from y to m        5  0.00000000 0.04248226 100 -0.076134392  0.08247566
#> 45 from y to y        5  0.03127301 0.03419857 100 -0.031634742  0.09222349
confint(mc, level = 0.95)
#>         effect interval       2.5 %      97.5 %
#> 1  from x to x        1  0.60980026  0.79613691
#> 2  from x to m        1  0.34030143  0.43607954
#> 3  from x to y        1 -0.17134324 -0.03910697
#> 4  from x to x        2  0.40979261  0.60379349
#> 5  from x to m        2  0.43870397  0.58099242
#> 6  from x to y        2  0.01652097  0.15495342
#> 7  from x to x        3  0.26218727  0.44344399
#> 8  from x to m        3  0.42197725  0.58218446
#> 9  from x to y        3  0.19406457  0.33680333
#> 10 from x to x        4  0.15968127  0.32901570
#> 11 from x to m        4  0.35892031  0.52043922
#> 12 from x to y        4  0.30375980  0.45341980
#> 13 from x to x        5  0.08188246  0.27142605
#> 14 from x to m        5  0.26647254  0.44158762
#> 15 from x to y        5  0.33845203  0.47862426
plot(mc)