Bootstrap Sampling Distribution for the Elements of the Standardized Matrix of Lagged Coefficients Over a Specific Time Interval or a Range of Time Intervals
Source:R/cTMed-boot-beta-std.R
BootBetaStd.RdThis function generates a bootstrap 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 \(\boldsymbol{\Phi}\).
Arguments
- phi
List of numeric matrices. Each element of the list is a bootstrap estimate of the drift matrix (\(\boldsymbol{\Phi}\)).
- sigma
List of numeric matrices. Each element of the list is a bootstrap estimate of the process noise covariance matrix (\(\boldsymbol{\Sigma}\)).
- phi_hat
Numeric matrix. The estimated drift matrix (\(\hat{\boldsymbol{\Phi}}\)) from the original data set.
phi_hatshould have row and column names pertaining to the variables in the system.- sigma_hat
Numeric matrix. The estimated process noise covariance matrix (\(\hat{\boldsymbol{\Sigma}}\)) from the original data set.
- delta_t
Numeric. Time interval (\(\Delta t\)).
- ncores
Positive integer. Number of cores to use. If
ncores = NULL, use a single core. Consider using multiple cores when number of replicationsRis a large value.- tol
Numeric. Smallest possible time interval to allow.
Value
Returns an object
of class ctmedboot which is a list with the following elements:
- call
Function call.
- args
Function arguments.
- fun
Function used ("BootBetaStd").
- output
A list with length of
length(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 bootstrap elements of the standardized matrix of lagged coefficients.
Details
See TotalStd().
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:
BootBeta(),
BootIndirectCentral(),
BootMed(),
BootMedStd(),
BootTotalCentral(),
DeltaBeta(),
DeltaBetaStd(),
DeltaIndirectCentral(),
DeltaMed(),
DeltaMedStd(),
DeltaTotalCentral(),
Direct(),
DirectStd(),
Indirect(),
IndirectCentral(),
IndirectStd(),
MCBeta(),
MCBetaStd(),
MCIndirectCentral(),
MCMed(),
MCMedStd(),
MCPhi(),
MCPhiSigma(),
MCTotalCentral(),
Med(),
MedStd(),
PosteriorBeta(),
PosteriorIndirectCentral(),
PosteriorMed(),
PosteriorTotalCentral(),
Total(),
TotalCentral(),
TotalStd(),
Trajectory()
Examples
# \donttest{
library(bootStateSpace)
# prepare parameters
## number of individuals
n <- 50
## time points
time <- 100
delta_t <- 0.10
## dynamic structure
p <- 3
mu0 <- rep(x = 0, times = p)
sigma0 <- matrix(
data = c(
1.0,
0.2,
0.2,
0.2,
1.0,
0.2,
0.2,
0.2,
1.0
),
nrow = p
)
sigma0_l <- t(chol(sigma0))
mu <- rep(x = 0, times = p)
phi <- matrix(
data = c(
-0.357,
0.771,
-0.450,
0.0,
-0.511,
0.729,
0,
0,
-0.693
),
nrow = p
)
sigma <- matrix(
data = c(
0.24455556,
0.02201587,
-0.05004762,
0.02201587,
0.07067800,
0.01539456,
-0.05004762,
0.01539456,
0.07553061
),
nrow = p
)
sigma_l <- t(chol(sigma))
## measurement model
k <- 3
nu <- rep(x = 0, times = k)
lambda <- diag(k)
theta <- 0.2 * diag(k)
theta_l <- t(chol(theta))
boot <- PBSSMOUFixed(
R = 10L, # use at least 1000 in actual research
path = getwd(),
prefix = "ou",
n = n,
time = time,
delta_t = delta_t,
mu0 = mu0,
sigma0_l = sigma0_l,
mu = mu,
phi = phi,
sigma_l = sigma_l,
nu = nu,
lambda = lambda,
theta_l = theta_l,
ncores = NULL, # consider using multiple cores
seed = 42
)
phi_hat <- phi
colnames(phi_hat) <- rownames(phi_hat) <- c("x", "m", "y")
sigma_hat <- sigma
phi <- extract(object = boot, what = "phi")
sigma <- extract(object = boot, what = "sigma")
# Specific time interval ----------------------------------------------------
BootBetaStd(
phi = phi,
sigma = sigma,
phi_hat = phi_hat,
sigma_hat = sigma_hat,
delta_t = 1
)
#>
#> Total, Direct, and Indirect Effects
#> type = pc
#> $`1`
#> interval est se R 2.5% 97.5%
#> from x to x 1 0.6998 0.0462 10 0.6266 0.7449
#> from x to m 1 0.3888 0.0195 10 0.3519 0.4120
#> from x to y 1 -0.1069 0.0146 10 -0.1388 -0.0966
#> from m to x 1 0.0000 0.0607 10 -0.0633 0.1097
#> from m to m 1 0.5999 0.0277 10 0.5667 0.6537
#> from m to y 1 0.5494 0.0196 10 0.5280 0.5823
#> from y to x 1 0.0000 0.0379 10 -0.0687 0.0476
#> from y to m 1 0.0000 0.0200 10 -0.0264 0.0323
#> from y to y 1 0.5001 0.0157 10 0.4886 0.5323
#>
# Range of time intervals ---------------------------------------------------
boot <- BootBetaStd(
phi = phi,
sigma = sigma,
phi_hat = phi_hat,
sigma_hat = sigma_hat,
delta_t = 1:5
)
plot(boot)
plot(boot, type = "bc") # bias-corrected
# Methods -------------------------------------------------------------------
# BootBetaStd has a number of methods including
# print, summary, confint, and plot
print(boot)
#>
#> Total, Direct, and Indirect Effects
#> type = pc
#> $`1`
#> interval est se R 2.5% 97.5%
#> from x to x 1 0.6998 0.0462 10 0.6266 0.7449
#> from x to m 1 0.3888 0.0195 10 0.3519 0.4120
#> from x to y 1 -0.1069 0.0146 10 -0.1388 -0.0966
#> from m to x 1 0.0000 0.0607 10 -0.0633 0.1097
#> from m to m 1 0.5999 0.0277 10 0.5667 0.6537
#> from m to y 1 0.5494 0.0196 10 0.5280 0.5823
#> from y to x 1 0.0000 0.0379 10 -0.0687 0.0476
#> from y to m 1 0.0000 0.0200 10 -0.0264 0.0323
#> from y to y 1 0.5001 0.0157 10 0.4886 0.5323
#>
#> $`2`
#> interval est se R 2.5% 97.5%
#> from x to x 2 0.4897 0.0483 10 0.4098 0.5591
#> from x to m 2 0.5053 0.0258 10 0.4619 0.5289
#> from x to y 2 0.0854 0.0170 10 0.0435 0.0934
#> from m to x 2 0.0000 0.0632 10 -0.0603 0.1075
#> from m to m 2 0.3599 0.0460 10 0.3184 0.4541
#> from m to y 2 0.6044 0.0217 10 0.5844 0.6503
#> from y to x 2 0.0000 0.0465 10 -0.0841 0.0600
#> from y to m 2 0.0000 0.0322 10 -0.0508 0.0405
#> from y to y 2 0.2501 0.0190 10 0.2345 0.2841
#>
#> $`3`
#> interval est se R 2.5% 97.5%
#> from x to x 3 0.3427 0.0463 10 0.2751 0.4257
#> from x to m 3 0.4936 0.0310 10 0.4390 0.5293
#> from x to y 3 0.2680 0.0175 10 0.2321 0.2816
#> from m to x 3 0.0000 0.0518 10 -0.0592 0.0783
#> from m to m 3 0.2159 0.0528 10 0.1761 0.3210
#> from m to y 3 0.4999 0.0314 10 0.4755 0.5617
#> from y to x 3 0.0000 0.0435 10 -0.0800 0.0559
#> from y to m 3 0.0000 0.0395 10 -0.0690 0.0401
#> from y to y 3 0.1251 0.0217 10 0.0991 0.1646
#>
#> $`4`
#> interval est se R 2.5% 97.5%
#> from x to x 4 0.2398 0.0451 10 0.1818 0.3288
#> from x to m 4 0.4293 0.0370 10 0.3675 0.4865
#> from x to y 4 0.3686 0.0193 10 0.3339 0.3881
#> from m to x 4 0.0000 0.0407 10 -0.0597 0.0535
#> from m to m 4 0.1295 0.0508 10 0.0907 0.2251
#> from m to y 4 0.3686 0.0420 10 0.3379 0.4582
#> from y to x 4 0.0000 0.0366 10 -0.0691 0.0460
#> from y to m 4 0.0000 0.0417 10 -0.0760 0.0425
#> from y to y 4 0.0625 0.0266 10 0.0262 0.1026
#>
#> $`5`
#> interval est se R 2.5% 97.5%
#> from x to x 5 0.1678 0.0436 10 0.1193 0.2574
#> from x to m 5 0.3508 0.0418 10 0.2884 0.4258
#> from x to y 5 0.3946 0.0256 10 0.3507 0.4268
#> from m to x 5 0.0000 0.0327 10 -0.0543 0.0472
#> from m to m 5 0.0777 0.0448 10 0.0312 0.1541
#> from m to y 5 0.2555 0.0473 10 0.2231 0.3556
#> from y to x 5 0.0000 0.0292 10 -0.0566 0.0356
#> from y to m 5 0.0000 0.0400 10 -0.0749 0.0424
#> from y to y 5 0.0313 0.0316 10 -0.0151 0.0702
#>
summary(boot)
#> effect interval est se R 2.5% 97.5%
#> 1 from x to x 1 0.69977250 0.04619116 10 0.62664729 0.74488459
#> 2 from x to m 1 0.38882458 0.01953203 10 0.35192261 0.41196870
#> 3 from x to y 1 -0.10689374 0.01457135 10 -0.13880836 -0.09659594
#> 4 from m to x 1 0.00000000 0.06073486 10 -0.06326864 0.10973836
#> 5 from m to m 1 0.59989538 0.02765975 10 0.56673563 0.65366698
#> 6 from m to y 1 0.54943087 0.01959635 10 0.52803581 0.58233075
#> 7 from y to x 1 0.00000000 0.03787369 10 -0.06873167 0.04761120
#> 8 from y to m 1 0.00000000 0.02001760 10 -0.02637969 0.03232691
#> 9 from y to y 1 0.50007360 0.01565101 10 0.48863283 0.53227520
#> 10 from x to x 2 0.48968155 0.04829661 10 0.40983641 0.55913746
#> 11 from x to m 2 0.50534282 0.02575876 10 0.46193377 0.52888561
#> 12 from x to y 2 0.08537619 0.01700777 10 0.04354753 0.09341638
#> 13 from m to x 2 0.00000000 0.06319917 10 -0.06032332 0.10754010
#> 14 from m to m 2 0.35987447 0.04599035 10 0.31838727 0.45406311
#> 15 from m to y 2 0.60435691 0.02171160 10 0.58439277 0.65028161
#> 16 from y to x 2 0.00000000 0.04647067 10 -0.08413191 0.05999321
#> 17 from y to m 2 0.00000000 0.03217802 10 -0.05083872 0.04051302
#> 18 from y to y 2 0.25007360 0.01902263 10 0.23452458 0.28407526
#> 19 from x to x 3 0.34266568 0.04625981 10 0.27505399 0.42572708
#> 20 from x to m 3 0.49355305 0.03103092 10 0.43895163 0.52930052
#> 21 from x to y 3 0.26800143 0.01745794 10 0.23207899 0.28160738
#> 22 from m to x 3 0.00000000 0.05182092 10 -0.05918335 0.07830911
#> 23 from m to m 3 0.21588703 0.05277385 10 0.17613806 0.32096736
#> 24 from m to y 3 0.49994907 0.03139222 10 0.47550794 0.56172186
#> 25 from y to x 3 0.00000000 0.04348230 10 -0.07996316 0.05587709
#> 26 from y to m 3 0.00000000 0.03946202 10 -0.06895053 0.04010735
#> 27 from y to y 3 0.12505520 0.02167083 10 0.09906528 0.16456126
#> 28 from x to x 4 0.23978802 0.04513964 10 0.18178975 0.32883844
#> 29 from x to m 4 0.42931704 0.03695913 10 0.36749533 0.48650604
#> 30 from x to y 4 0.36856490 0.01930720 10 0.33393489 0.38807815
#> 31 from m to x 4 0.00000000 0.04073717 10 -0.05974970 0.05349997
#> 32 from m to m 4 0.12950963 0.05084726 10 0.09070625 0.22505825
#> 33 from m to y 4 0.36862633 0.04200137 10 0.33793532 0.45816980
#> 34 from y to x 4 0.00000000 0.03662452 10 -0.06907513 0.04599297
#> 35 from y to m 4 0.00000000 0.04169749 10 -0.07602734 0.04248806
#> 36 from y to y 4 0.06253681 0.02659055 10 0.02623799 0.10262406
#> 37 from x to x 5 0.16779706 0.04357747 10 0.11929313 0.25743403
#> 38 from x to m 5 0.35078078 0.04177677 10 0.28840681 0.42580094
#> 39 from x to y 5 0.39455777 0.02561383 10 0.35066698 0.42679774
#> 40 from m to x 5 0.00000000 0.03265509 10 -0.05429136 0.04717457
#> 41 from m to m 5 0.07769223 0.04484573 10 0.03122991 0.15411817
#> 42 from m to y 5 0.25549689 0.04730229 10 0.22314814 0.35557279
#> 43 from y to x 5 0.00000000 0.02922052 10 -0.05661104 0.03561862
#> 44 from y to m 5 0.00000000 0.03997550 10 -0.07487555 0.04241948
#> 45 from y to y 5 0.03127301 0.03156738 10 -0.01509265 0.07019346
confint(boot, level = 0.95)
#> effect interval 2.5 % 97.5 %
#> 1 from x to x 1 0.62664729 0.74488459
#> 2 from x to m 1 0.35192261 0.41196870
#> 3 from x to y 1 -0.13880836 -0.09659594
#> 4 from x to x 2 0.40983641 0.55913746
#> 5 from x to m 2 0.46193377 0.52888561
#> 6 from x to y 2 0.04354753 0.09341638
#> 7 from x to x 3 0.27505399 0.42572708
#> 8 from x to m 3 0.43895163 0.52930052
#> 9 from x to y 3 0.23207899 0.28160738
#> 10 from x to x 4 0.18178975 0.32883844
#> 11 from x to m 4 0.36749533 0.48650604
#> 12 from x to y 4 0.33393489 0.38807815
#> 13 from x to x 5 0.11929313 0.25743403
#> 14 from x to m 5 0.28840681 0.42580094
#> 15 from x to y 5 0.35066698 0.42679774
print(boot, type = "bc") # bias-corrected
#>
#> Total, Direct, and Indirect Effects
#> type = bc
#> $`1`
#> interval est se R 2.5% 97.5%
#> from x to x 1 0.6998 0.0462 10 0.6247 0.7391
#> from x to m 1 0.3888 0.0195 10 0.3620 0.4135
#> from x to y 1 -0.1069 0.0146 10 -0.1156 -0.0966
#> from m to x 1 0.0000 0.0607 10 -0.0633 0.1097
#> from m to m 1 0.5999 0.0277 10 0.5653 0.6134
#> from m to y 1 0.5494 0.0196 10 0.5280 0.5823
#> from y to x 1 0.0000 0.0379 10 -0.0687 0.0476
#> from y to m 1 0.0000 0.0200 10 -0.0243 0.0355
#> from y to y 1 0.5001 0.0157 10 0.4886 0.5323
#>
#> $`2`
#> interval est se R 2.5% 97.5%
#> from x to x 2 0.4897 0.0483 10 0.3995 0.5203
#> from x to m 2 0.5053 0.0258 10 0.4619 0.5289
#> from x to y 2 0.0854 0.0170 10 0.0711 0.0945
#> from m to x 2 0.0000 0.0632 10 -0.0598 0.1086
#> from m to m 2 0.3599 0.0460 10 0.3177 0.4386
#> from m to y 2 0.6044 0.0217 10 0.5804 0.6336
#> from y to x 2 0.0000 0.0465 10 -0.0841 0.0600
#> from y to m 2 0.0000 0.0322 10 -0.0508 0.0405
#> from y to y 2 0.2501 0.0190 10 0.2339 0.2829
#>
#> $`3`
#> interval est se R 2.5% 97.5%
#> from x to x 3 0.3427 0.0463 10 0.2607 0.3854
#> from x to m 3 0.4936 0.0310 10 0.4303 0.5197
#> from x to y 3 0.2680 0.0175 10 0.2652 0.2856
#> from m to x 3 0.0000 0.0518 10 -0.0557 0.0783
#> from m to m 3 0.2159 0.0528 10 0.1761 0.3210
#> from m to y 3 0.4999 0.0314 10 0.4751 0.5440
#> from y to x 3 0.0000 0.0435 10 -0.0800 0.0559
#> from y to m 3 0.0000 0.0395 10 -0.0690 0.0401
#> from y to y 3 0.1251 0.0217 10 0.0990 0.1432
#>
#> $`4`
#> interval est se R 2.5% 97.5%
#> from x to x 4 0.2398 0.0451 10 0.1710 0.2551
#> from x to m 4 0.4293 0.0370 10 0.3537 0.4512
#> from x to y 4 0.3686 0.0193 10 0.3357 0.3882
#> from m to x 4 0.0000 0.0407 10 -0.0494 0.0537
#> from m to m 4 0.1295 0.0508 10 0.0907 0.2251
#> from m to y 4 0.3686 0.0420 10 0.3361 0.4247
#> from y to x 4 0.0000 0.0366 10 -0.0691 0.0460
#> from y to m 4 0.0000 0.0417 10 -0.0760 0.0425
#> from y to y 4 0.0625 0.0266 10 0.0246 0.0917
#>
#> $`5`
#> interval est se R 2.5% 97.5%
#> from x to x 5 0.1678 0.0436 10 0.1135 0.2148
#> from x to m 5 0.3508 0.0418 10 0.2717 0.3772
#> from x to y 5 0.3946 0.0256 10 0.3440 0.4258
#> from m to x 5 0.0000 0.0327 10 -0.0543 0.0472
#> from m to m 5 0.0777 0.0448 10 0.0412 0.1572
#> from m to y 5 0.2555 0.0473 10 0.2231 0.3556
#> from y to x 5 0.0000 0.0292 10 -0.0566 0.0356
#> from y to m 5 0.0000 0.0400 10 -0.0749 0.0424
#> from y to y 5 0.0313 0.0316 10 -0.0152 0.0641
#>
summary(boot, type = "bc")
#> effect interval est se R 2.5% 97.5%
#> 1 from x to x 1 0.69977250 0.04619116 10 0.62468140 0.73907938
#> 2 from x to m 1 0.38882458 0.01953203 10 0.36196513 0.41348561
#> 3 from x to y 1 -0.10689374 0.01457135 10 -0.11556784 -0.09656558
#> 4 from m to x 1 0.00000000 0.06073486 10 -0.06326864 0.10973836
#> 5 from m to m 1 0.59989538 0.02765975 10 0.56533542 0.61343471
#> 6 from m to y 1 0.54943087 0.01959635 10 0.52803581 0.58233075
#> 7 from y to x 1 0.00000000 0.03787369 10 -0.06873167 0.04761120
#> 8 from y to m 1 0.00000000 0.02001760 10 -0.02431892 0.03549856
#> 9 from y to y 1 0.50007360 0.01565101 10 0.48863283 0.53227520
#> 10 from x to x 2 0.48968155 0.04829661 10 0.39951058 0.52028969
#> 11 from x to m 2 0.50534282 0.02575876 10 0.46193377 0.52888561
#> 12 from x to y 2 0.08537619 0.01700777 10 0.07107420 0.09450813
#> 13 from m to x 2 0.00000000 0.06319917 10 -0.05983395 0.10862931
#> 14 from m to m 2 0.35987447 0.04599035 10 0.31766333 0.43855897
#> 15 from m to y 2 0.60435691 0.02171160 10 0.58044349 0.63358363
#> 16 from y to x 2 0.00000000 0.04647067 10 -0.08413191 0.05999321
#> 17 from y to m 2 0.00000000 0.03217802 10 -0.05083872 0.04051302
#> 18 from y to y 2 0.25007360 0.01902263 10 0.23392510 0.28285536
#> 19 from x to x 3 0.34266568 0.04625981 10 0.26069578 0.38535309
#> 20 from x to m 3 0.49355305 0.03103092 10 0.43031824 0.51967621
#> 21 from x to y 3 0.26800143 0.01745794 10 0.26519759 0.28556433
#> 22 from m to x 3 0.00000000 0.05182092 10 -0.05567425 0.07831864
#> 23 from m to m 3 0.21588703 0.05277385 10 0.17613806 0.32096736
#> 24 from m to y 3 0.49994907 0.03139222 10 0.47505535 0.54396596
#> 25 from y to x 3 0.00000000 0.04348230 10 -0.07996316 0.05587709
#> 26 from y to m 3 0.00000000 0.03946202 10 -0.06895053 0.04010735
#> 27 from y to y 3 0.12505520 0.02167083 10 0.09901812 0.14317853
#> 28 from x to x 4 0.23978802 0.04513964 10 0.17104675 0.25509649
#> 29 from x to m 4 0.42931704 0.03695913 10 0.35373868 0.45122545
#> 30 from x to y 4 0.36856490 0.01930720 10 0.33570846 0.38818798
#> 31 from m to x 4 0.00000000 0.04073717 10 -0.04944213 0.05367381
#> 32 from m to m 4 0.12950963 0.05084726 10 0.09070625 0.22505825
#> 33 from m to y 4 0.36862633 0.04200137 10 0.33610824 0.42472686
#> 34 from y to x 4 0.00000000 0.03662452 10 -0.06907513 0.04599297
#> 35 from y to m 4 0.00000000 0.04169749 10 -0.07602734 0.04248806
#> 36 from y to y 4 0.06253681 0.02659055 10 0.02457317 0.09166099
#> 37 from x to x 5 0.16779706 0.04357747 10 0.11350347 0.21483345
#> 38 from x to m 5 0.35078078 0.04177677 10 0.27173887 0.37722926
#> 39 from x to y 5 0.39455777 0.02561383 10 0.34399803 0.42575955
#> 40 from m to x 5 0.00000000 0.03265509 10 -0.05429136 0.04717457
#> 41 from m to m 5 0.07769223 0.04484573 10 0.04123415 0.15717773
#> 42 from m to y 5 0.25549689 0.04730229 10 0.22314814 0.35557279
#> 43 from y to x 5 0.00000000 0.02922052 10 -0.05661104 0.03561862
#> 44 from y to m 5 0.00000000 0.03997550 10 -0.07487555 0.04241948
#> 45 from y to y 5 0.03127301 0.03156738 10 -0.01516324 0.06407115
confint(boot, level = 0.95, type = "bc")
#> effect interval 2.5 % 97.5 %
#> 1 from x to x 1 0.6246814 0.73907938
#> 2 from x to m 1 0.3619651 0.41348561
#> 3 from x to y 1 -0.1155678 -0.09656558
#> 4 from x to x 2 0.3995106 0.52028969
#> 5 from x to m 2 0.4619338 0.52888561
#> 6 from x to y 2 0.0710742 0.09450813
#> 7 from x to x 3 0.2606958 0.38535309
#> 8 from x to m 3 0.4303182 0.51967621
#> 9 from x to y 3 0.2651976 0.28556433
#> 10 from x to x 4 0.1710467 0.25509649
#> 11 from x to m 4 0.3537387 0.45122545
#> 12 from x to y 4 0.3357085 0.38818798
#> 13 from x to x 5 0.1135035 0.21483345
#> 14 from x to m 5 0.2717389 0.37722926
#> 15 from x to y 5 0.3439980 0.42575955
# }