Illustration 2 - Confidence Intervals for the Direct, Indirect, and Total Effects
Ivan Jacob Agaloos Pesigan
Source:vignettes/fig-example-2.Rmd
fig-example-2.RmdThis vignette accompanies Illustration 2. The goal of the
illustration is to calculate confidence intervals for the direct,
indirect, and total effects from the continuous-time vector
autoregressive model drift matrix
and process noise covariance matrix
for a range of time intervals. This example features the
DeltaMed, MCMed, BootMed,
DeltaMedStd, MCMedStd, and
BootMedStd functions from the cTMed
package.
Continuous-Time Vector Autoregressive Model Estimates
The object fit contains the fitted dynr
model. Data is generated using the
manCTMed::IllustrationGenData function. The model was
fitted using the manCTMed::IllustrationFitDynr.
summary(fit)
#> Coefficients:
#> Estimate Std. Error t value ci.lower ci.upper Pr(>|t|)
#> phi_11 -0.1157754 0.0174197 -6.646 -0.1499173 -0.0816334 <2e-16 ***
#> phi_12 0.0203747 0.0550571 0.370 -0.0875352 0.1282846 0.3557
#> phi_13 -0.0070992 0.0535302 -0.133 -0.1120165 0.0978181 0.4472
#> phi_21 -0.1226909 0.0187426 -6.546 -0.1594258 -0.0859560 <2e-16 ***
#> phi_22 -0.8819998 0.0653717 -13.492 -1.0101259 -0.7538737 <2e-16 ***
#> phi_23 0.4737224 0.0594135 7.973 0.3572740 0.5901708 <2e-16 ***
#> phi_31 -0.0877980 0.0178492 -4.919 -0.1227817 -0.0528143 <2e-16 ***
#> phi_32 0.0595814 0.0586861 1.015 -0.0554414 0.1746041 0.1550
#> phi_33 -0.6636176 0.0616495 -10.764 -0.7844485 -0.5427867 <2e-16 ***
#> sigma_11 0.0954899 0.0049239 19.393 0.0858393 0.1051405 <2e-16 ***
#> sigma_12 0.0009624 0.0040987 0.235 -0.0070710 0.0089957 0.4072
#> sigma_13 0.0070297 0.0040069 1.754 -0.0008237 0.0148830 0.0397 *
#> sigma_22 0.1027182 0.0078961 13.009 0.0872422 0.1181942 <2e-16 ***
#> sigma_23 0.0009240 0.0049248 0.188 -0.0087285 0.0105764 0.4256
#> sigma_33 0.0991956 0.0074695 13.280 0.0845558 0.1138355 <2e-16 ***
#> theta_11 0.1016133 0.0014901 68.194 0.0986928 0.1045338 <2e-16 ***
#> theta_22 0.0996380 0.0015668 63.592 0.0965671 0.1027089 <2e-16 ***
#> theta_33 0.0992561 0.0015460 64.202 0.0962260 0.1022862 <2e-16 ***
#> mu0_1 -0.1217526 0.0521444 -2.335 -0.2239538 -0.0195514 0.0098 **
#> mu0_2 0.0160785 0.0279075 0.576 -0.0386192 0.0707762 0.2823
#> mu0_3 0.0204163 0.0288928 0.707 -0.0362125 0.0770451 0.2399
#> sigma0_11 0.3346315 0.0446548 7.494 0.2471097 0.4221533 <2e-16 ***
#> sigma0_12 -0.0644525 0.0177503 -3.631 -0.0992424 -0.0296626 0.0001 ***
#> sigma0_13 -0.0519690 0.0179974 -2.888 -0.0872433 -0.0166946 0.0019 **
#> sigma0_22 0.0691741 0.0127707 5.417 0.0441439 0.0942043 <2e-16 ***
#> sigma0_23 0.0364469 0.0098132 3.714 0.0172134 0.0556805 0.0001 ***
#> sigma0_33 0.0792446 0.0137057 5.782 0.0523819 0.1061072 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> -2 log-likelihood value at convergence = 32662.46
#> AIC = 32716.46
#> BIC = 32919.11Parameter Estimates
We extract parameter estimates from the fit object.
# drift matrix
phi <- matrix(
data = coef(fit)[
c(
"phi_11", "phi_21", "phi_31",
"phi_12", "phi_22", "phi_32",
"phi_13", "phi_23", "phi_33"
)
],
nrow = 3
)
# column names and row names are needed to define the indirect effects
colnames(phi) <- rownames(phi) <- c(
"conflict",
"knowledge",
"competence"
)
# process noise covariance matrix
sigma <- matrix(
data = coef(fit)[
c(
"sigma_11", "sigma_12", "sigma_13",
"sigma_12", "sigma_22", "sigma_23",
"sigma_13", "sigma_23", "sigma_33"
)
],
nrow = 3
)
# measurement error covariance matrix
theta <- matrix(
data = coef(fit)[
c(
"theta_11", 0, 0,
0, "theta_22", 0,
0, 0, "theta_33"
)
],
nrow = 3
)
# initial condition
mu0 <- coef(fit)[
c(
"mu0_1", "mu0_2", "mu0_3"
)
]
sigma0 <- matrix(
data = coef(fit)[
c(
"sigma0_11", "sigma0_12", "sigma0_13",
"sigma0_12", "sigma0_22", "sigma0_23",
"sigma0_13", "sigma0_23", "sigma0_33"
)
],
nrow = 3
)Sampling Variance-Covariance Matrix of Parameter Estimates
# combining the drift matrix and the process noise covariance matrix
phi_sigma_names <- c(
"phi_11", "phi_21", "phi_31",
"phi_12", "phi_22", "phi_32",
"phi_13", "phi_23", "phi_33",
"sigma_11", "sigma_12", "sigma_13",
"sigma_22", "sigma_23",
"sigma_33"
)
vcov_theta <- vcov(fit)[phi_sigma_names, phi_sigma_names]
phi_names <- c(
"phi_11", "phi_21", "phi_31",
"phi_12", "phi_22", "phi_32",
"phi_13", "phi_23", "phi_33"
)
vcov_phi_vec <- vcov_theta[phi_names, phi_names]Delta Method Confidence Intervals For The Direct, Indirect, and Total Effects
Using the DeltaMed function from the cTMed
package, confidence intervals for the direct, indirect, and total
effects for a time interval of one, two, and three are given below.
delta <- DeltaMed(
phi = phi,
vcov_phi_vec = vcov_phi_vec,
delta_t = c(1, 2, 3),
from = "conflict",
to = "competence",
med = "knowledge",
ncores = parallel::detectCores()
)
summary(delta)
#> effect interval est se z p
#> 1 total 1 -0.062555476 0.011487680 -5.445440 5.167744e-08
#> 2 direct 1 -0.060214915 0.012242403 -4.918553 8.718614e-07
#> 3 indirect 1 -0.002340561 0.002308034 -1.014093 3.105384e-01
#> 4 total 2 -0.090728400 0.015648577 -5.797869 6.716285e-09
#> 5 direct 2 -0.084667452 0.017491657 -4.840448 1.295468e-06
#> 6 indirect 2 -0.006060948 0.005969252 -1.015361 3.099336e-01
#> 7 total 3 -0.100363120 0.016803252 -5.972839 2.331598e-09
#> 8 direct 3 -0.091430222 0.019347394 -4.725713 2.293097e-06
#> 9 indirect 3 -0.008932897 0.008860944 -1.008120 3.133968e-01
#> 2.5% 97.5%
#> 1 -0.085070916 -0.040040036
#> 2 -0.084209585 -0.036220245
#> 3 -0.006864224 0.002183102
#> 4 -0.121399047 -0.060057753
#> 5 -0.118950469 -0.050384435
#> 6 -0.017760467 0.005638571
#> 7 -0.133296888 -0.067429351
#> 8 -0.129350417 -0.053510028
#> 9 -0.026300029 0.008434235Monte Carlo Method Confidence Intervals For The Direct, Indirect, and Total Effects
Using the MCMed function from the cTMed
package, confidence intervals for the direct, indirect, and total
effects for a time interval of one, two, and three are given below.
mc <- MCMed(
phi = phi,
vcov_phi_vec = vcov_phi_vec,
delta_t = c(1, 2, 3),
from = "conflict",
to = "competence",
med = "knowledge",
ncores = parallel::detectCores(),
R = 20000L,
seed = 42
)
summary(mc)
#> effect interval est se R 2.5% 97.5%
#> 1 total 1 -0.062555476 0.011439708 20000 -0.084799896 -0.039979723
#> 2 direct 1 -0.060214915 0.012208668 20000 -0.084194613 -0.036335836
#> 3 indirect 1 -0.002340561 0.002348285 20000 -0.006960547 0.002333871
#> 4 total 2 -0.090728400 0.015621708 20000 -0.121252097 -0.059993004
#> 5 direct 2 -0.084667452 0.017502554 20000 -0.119820771 -0.050722636
#> 6 indirect 2 -0.006060948 0.006078536 20000 -0.017667159 0.006440646
#> 7 total 3 -0.100363120 0.016825327 20000 -0.133440741 -0.067395319
#> 8 direct 3 -0.091430222 0.019442548 20000 -0.130989746 -0.054463617
#> 9 indirect 3 -0.008932897 0.009051204 20000 -0.025860319 0.009937107Parametric Bootstrap Method Confidence Intervals For The Direct, Indirect, and Total Effects
The parametric bootstrap approach involves generating data from the
parametric model and fitting the model on the generated data multiple
times. The data generation and model fitting is hadled by the
bootStateSpace package. The estimated parameters are passed
as arguments to the PBSSMOUFixed function from the
bootStateSpace package, which generates a parametric
bootstrap sampling distribution of the parameter estimates. The argument
R specifies the number of bootstrap replications. The
generated data and model estimates are be stored in path
using the specified prefix for the file names. The
ncores = parallel::detectCores() argument instructs the
function to use all available CPU cores in the system.
NOTE: Fitting the CT-VAR model multiple times is computationally intensive.
R <- 1000L
path <- root$find_file(
".setup",
"data-raw"
)
prefix <- "illustration_pb"
phi_hat <- phi
sigma_hat <- sigma
boot <- PBSSMOUFixed(
R = R,
path = path,
prefix = prefix,
n = 133,
time = 101,
delta_t = 0.10,
mu0 = mu0,
sigma0_l = t(chol(sigma0)),
mu = c(0, 0, 0),
phi = phi,
sigma_l = t(chol(sigma)),
nu = c(0, 0, 0),
lambda = diag(3),
theta_l = t(chol(theta)),
mu0_fixed = FALSE,
sigma0_fixed = FALSE,
ncores = parallel::detectCores(),
seed = 42,
clean = TRUE
)Using the BootMed function from the cTMed
package, confidence intervals for the direct, indirect, and total
effects for a time interval of one, two, and three are given below.
ci <- BootMed(
phi = extract(object = boot, what = "phi"), # extracts the bootstrap drift matrices
phi_hat = phi_hat,
delta_t = c(1, 2, 3),
from = "conflict",
to = "competence",
med = "knowledge",
ncores = parallel::detectCores()
)
summary(ci)
#> effect interval est se R 2.5% 97.5%
#> 1 total 1 -0.062555476 0.011895369 1000 -0.085729558 -0.040227248
#> 2 direct 1 -0.060214915 0.012644604 1000 -0.085588908 -0.035900504
#> 3 indirect 1 -0.002340561 0.002290933 1000 -0.007052772 0.001949044
#> 4 total 2 -0.090728400 0.016147361 1000 -0.121302346 -0.060151196
#> 5 direct 2 -0.084667452 0.017992384 1000 -0.120845704 -0.050280112
#> 6 indirect 2 -0.006060948 0.005898012 1000 -0.017854673 0.005354471
#> 7 total 3 -0.100363120 0.017285994 1000 -0.133607192 -0.067431119
#> 8 direct 3 -0.091430222 0.019883297 1000 -0.131147843 -0.052983876
#> 9 indirect 3 -0.008932897 0.008742578 1000 -0.026055200 0.008494383
summary(ci, type = "bc") # bias-corrected
#> effect interval est se R 2.5% 97.5%
#> 1 total 1 -0.062555476 0.011895369 1000 -0.085633650 -0.040102172
#> 2 direct 1 -0.060214915 0.012644604 1000 -0.085457442 -0.035826089
#> 3 indirect 1 -0.002340561 0.002290933 1000 -0.007326832 0.001773306
#> 4 total 2 -0.090728400 0.016147361 1000 -0.122605069 -0.060743351
#> 5 direct 2 -0.084667452 0.017992384 1000 -0.120949487 -0.050297783
#> 6 indirect 2 -0.006060948 0.005898012 1000 -0.018334252 0.004930768
#> 7 total 3 -0.100363120 0.017285994 1000 -0.134702537 -0.067590637
#> 8 direct 3 -0.091430222 0.019883297 1000 -0.132197696 -0.053605916
#> 9 indirect 3 -0.008932897 0.008742578 1000 -0.026669169 0.007904886Delta Method Confidence Intervals For The Standardized Direct, Indirect, and Total Effects
Using the DeltaMedStd function from the
cTMed package, confidence intervals for the standardized
direct, indirect, and total effects for a time interval of one, two, and
three are given below.
delta <- DeltaMedStd(
phi = phi,
sigma = sigma,
vcov_theta = vcov_theta,
delta_t = c(1, 2, 3),
from = "conflict",
to = "competence",
med = "knowledge",
ncores = parallel::detectCores()
)
summary(delta)
#> effect interval est se z p 2.5%
#> 1 total 1 -0.137467669 0.025247997 -5.444696 5.189394e-08 -0.18695283
#> 2 direct 1 -0.132324211 0.026919970 -4.915466 8.857130e-07 -0.18508638
#> 3 indirect 1 -0.005143458 0.005069136 -1.014662 3.102671e-01 -0.01507878
#> 4 total 2 -0.199378575 0.034680527 -5.749006 8.976937e-09 -0.26735116
#> 5 direct 2 -0.186059447 0.038717618 -4.805550 1.543267e-06 -0.26194458
#> 6 indirect 2 -0.013319129 0.013110257 -1.015932 3.096618e-01 -0.03901476
#> 7 total 3 -0.220551181 0.037701725 -5.849896 4.918794e-09 -0.29444520
#> 8 direct 3 -0.200920852 0.043175066 -4.653632 3.261393e-06 -0.28554243
#> 9 indirect 3 -0.019630329 0.019462994 -1.008598 3.131677e-01 -0.05777710
#> 97.5%
#> 1 -0.087982504
#> 2 -0.079562039
#> 3 0.004791866
#> 4 -0.131405991
#> 5 -0.110174311
#> 6 0.012376504
#> 7 -0.146657159
#> 8 -0.116299277
#> 9 0.018516438Monte Carlo Method Confidence Intervals For The Standardized Direct, Indirect, and Total Effects
Using the MCMedStd function from the cTMed
package, confidence intervals for the standardized direct, indirect, and
total effects for a time interval of one, two, and three are given
below.
mc <- MCMedStd(
phi = phi,
sigma = sigma,
vcov_theta = vcov_theta,
delta_t = c(1, 2, 3),
from = "conflict",
to = "competence",
med = "knowledge",
ncores = parallel::detectCores(),
R = 20000L,
seed = 42
)
summary(mc)
#> effect interval est se R 2.5% 97.5%
#> 1 total 1 -0.137467669 0.025504385 20000 -0.18798020 -0.088043163
#> 2 direct 1 -0.132324211 0.027233030 20000 -0.18660861 -0.079607848
#> 3 indirect 1 -0.005143458 0.005119941 20000 -0.01518742 0.005082213
#> 4 total 2 -0.199378575 0.035103138 20000 -0.26963838 -0.131719130
#> 5 direct 2 -0.186059447 0.039305150 20000 -0.26595529 -0.111519619
#> 6 indirect 2 -0.013319129 0.013255409 20000 -0.03850939 0.013835196
#> 7 total 3 -0.220551181 0.038248518 20000 -0.29904341 -0.147739964
#> 8 direct 3 -0.200920852 0.044020957 20000 -0.29283279 -0.119764424
#> 9 indirect 3 -0.019630329 0.019742787 20000 -0.05630960 0.021700396Parametric Bootstrap Method Confidence Intervals For The Standardized Direct, Indirect, and Total Effects
Using the BootMedStd function from the
cTMed package, confidence intervals for the standardized
direct, indirect, and total effects for a time interval of one, two, and
three are given below.
pb <- BootMedStd(
phi = extract(object = boot, what = "phi"), # extracts the bootstrap drift matrices
sigma = extract(object = boot, what = "sigma"), # extracts the bootstrap process noise covariance matrices
phi_hat = phi_hat,
sigma_hat = sigma_hat,
delta_t = c(1, 2, 3),
from = "conflict",
to = "competence",
med = "knowledge",
ncores = parallel::detectCores()
)
summary(pb)
#> effect interval est se R 2.5% 97.5%
#> 1 total 1 -0.137467669 0.026253764 1000 -0.18851716 -0.08897451
#> 2 direct 1 -0.132324211 0.027932919 1000 -0.18663188 -0.07948977
#> 3 indirect 1 -0.005143458 0.005055074 1000 -0.01576038 0.00425024
#> 4 total 2 -0.199378575 0.035966425 1000 -0.26884004 -0.13182073
#> 5 direct 2 -0.186059447 0.040041763 1000 -0.26340729 -0.11036386
#> 6 indirect 2 -0.013319129 0.013020860 1000 -0.03934067 0.01176595
#> 7 total 3 -0.220551181 0.039021505 1000 -0.29642911 -0.14852508
#> 8 direct 3 -0.200920852 0.044639569 1000 -0.28761730 -0.11848043
#> 9 indirect 3 -0.019630329 0.019311891 1000 -0.05705291 0.01906322
summary(pb, type = "bc") # bias-corrected
#> effect interval est se R 2.5% 97.5%
#> 1 total 1 -0.137467669 0.026253764 1000 -0.18883080 -0.089561065
#> 2 direct 1 -0.132324211 0.027932919 1000 -0.18576191 -0.078536696
#> 3 indirect 1 -0.005143458 0.005055074 1000 -0.01615588 0.003820513
#> 4 total 2 -0.199378575 0.035966425 1000 -0.27257598 -0.133985668
#> 5 direct 2 -0.186059447 0.040041763 1000 -0.26264001 -0.109715342
#> 6 indirect 2 -0.013319129 0.013020860 1000 -0.04093186 0.010552441
#> 7 total 3 -0.220551181 0.039021505 1000 -0.29872972 -0.152451925
#> 8 direct 3 -0.200920852 0.044639569 1000 -0.28483004 -0.117621516
#> 9 indirect 3 -0.019630329 0.019311891 1000 -0.05923582 0.016686814