Ivan Jacob Agaloos Pesigan 2025-07-22
You can install the CRAN release of semmcci with:
install.packages("semmcci")You can install the development version of semmcci from
GitHub with:
if (!require("remotes")) install.packages("remotes")
remotes::install_github("jeksterslab/semmcci")In the Monte Carlo method, a sampling distribution of parameter estimates is generated from the multivariate normal distribution using the parameter estimates and the sampling variance-covariance matrix. Confidence intervals for defined parameters are generated by obtaining percentiles corresponding to 100(1 - α)% from the generated sampling distribution, where α is the significance level.
Monte Carlo confidence intervals for free and defined parameters in
models fitted in the structural equation modeling package lavaan can
be generated using the semmcci package. The package has three main
functions, namely, MC(), MCMI(), and MCStd(). The output of
lavaan is passed as the first argument to the MC() function or the
MCMI() function to generate Monte Carlo confidence intervals. Monte
Carlo confidence intervals for the standardized estimates can also be
generated by passing the output of the MC() function or the MCMI()
function to the MCStd() function. A description of the package and
code examples are presented in Pesigan and Cheung (2023:
https://doi.org/10.3758/s13428-023-02114-4).
A common application of the Monte Carlo method is to generate confidence
intervals for the indirect effect. In the simple mediation model,
variable X has an effect on variable Y, through a mediating variable
M. This mediating or indirect effect is a product of path coefficients
from the fitted model.
library(semmcci)
library(lavaan)summary(df)
#> X M Y
#> Min. :-3.15272 Min. :-2.770180 Min. :-3.15166
#> 1st Qu.:-0.68013 1st Qu.:-0.617813 1st Qu.:-0.67360
#> Median : 0.02779 Median :-0.008015 Median :-0.03968
#> Mean : 0.00338 Mean : 0.016066 Mean :-0.02433
#> 3rd Qu.: 0.69009 3rd Qu.: 0.668327 3rd Qu.: 0.63753
#> Max. : 2.90756 Max. : 2.712008 Max. : 3.27343
#> NA's :100 NA's :100 NA's :100The indirect effect is defined by the product of the slopes of paths X
to M labeled as a and M to Y labeled as b. In this example, we
are interested in the confidence intervals of indirect defined as the
product of a and b using the := operator in the lavaan model
syntax.
model <- "
Y ~ cp * X + b * M
M ~ a * X
X ~~ X
indirect := a * b
direct := cp
total := cp + (a * b)
"We can now fit the model using the sem() function from lavaan. We
use full-information maximum likelihood to deal with missing values.
fit <- sem(data = df, model = model, missing = "fiml")The fit lavaan object can then be passed to the MC() function to
generate Monte Carlo confidence intervals.
mc <- MC(fit, R = 20000L, alpha = 0.05)
mc
#> Monte Carlo Confidence Intervals
#> est se R 2.5% 97.5%
#> cp 0.2491 0.0311 20000 0.1872 0.3099
#> b 0.4685 0.0321 20000 0.4064 0.5318
#> a 0.5032 0.0274 20000 0.4500 0.5562
#> X~~X 1.0556 0.0496 20000 0.9573 1.1532
#> Y~~Y 0.5830 0.0284 20000 0.5279 0.6393
#> M~~M 0.6790 0.0328 20000 0.6133 0.7427
#> Y~1 -0.0307 0.0258 20000 -0.0817 0.0204
#> M~1 0.0134 0.0278 20000 -0.0410 0.0675
#> X~1 0.0029 0.0337 20000 -0.0634 0.0695
#> indirect 0.2358 0.0206 20000 0.1973 0.2777
#> direct 0.2491 0.0311 20000 0.1872 0.3099
#> total 0.4849 0.0284 20000 0.4294 0.5406The MCMI() function can be used to handle missing values using
multiple imputation. The MCMI() accepts the output of mice::mice(),
Amelia::amelia(), or a list of multiply imputed data sets. In this
example, we impute multivariate missing data under the normal model.
mi <- mice::mice(
df,
method = "norm",
m = 100,
print = FALSE,
seed = 42
)We fit the model using lavaan using the default listwise deletion.
fit <- sem(data = df, model = model)The fit lavaan object and mi object can then be passed to the
MCMI() function to generate Monte Carlo confidence intervals.
mcmi <- MCMI(fit, mi = mi, R = 20000L, alpha = 0.05, seed = 42)
mcmi
#> Monte Carlo Confidence Intervals (Multiple Imputation Estimates)
#> est se R 2.5% 97.5%
#> cp 0.2479 0.0311 20000 0.1874 0.3086
#> b 0.4689 0.0332 20000 0.4041 0.5338
#> a 0.5031 0.0275 20000 0.4499 0.5572
#> X~~X 1.0568 0.0502 20000 0.9587 1.1554
#> Y~~Y 0.5827 0.0289 20000 0.5257 0.6396
#> M~~M 0.6794 0.0328 20000 0.6151 0.7433
#> indirect 0.2359 0.0207 20000 0.1967 0.2775
#> direct 0.2479 0.0311 20000 0.1874 0.3086
#> total 0.4838 0.0284 20000 0.4276 0.5396Standardized Monte Carlo Confidence intervals can be generated by
passing the result of the MC() function or the MCMI() function to
MCStd().
MCStd(mc, alpha = 0.05)
#> Standardized Monte Carlo Confidence Intervals
#> est se R 2.5% 97.5%
#> cp 0.2585 0.0318 20000 0.1951 0.3204
#> b 0.4603 0.0296 20000 0.4017 0.5178
#> a 0.5315 0.0247 20000 0.4824 0.5791
#> X~~X 1.0000 0.0000 20000 1.0000 1.0000
#> Y~~Y 0.5948 0.0260 20000 0.5431 0.6449
#> M~~M 0.7175 0.0262 20000 0.6646 0.7672
#> indirect -0.0310 0.0196 20000 0.2072 0.2838
#> direct 0.0137 0.0318 20000 0.1951 0.3204
#> total 0.0028 0.0257 20000 0.4508 0.5515MCStd(mcmi, alpha = 0.05)
#> Standardized Monte Carlo Confidence Intervals
#> est se R 2.5% 97.5%
#> cp 0.2626 0.0320 20000 0.1953 0.3195
#> b 0.4461 0.0304 20000 0.4004 0.5190
#> a 0.5187 0.0250 20000 0.4819 0.5799
#> X~~X 1.0000 0.0000 20000 1.0000 1.0000
#> Y~~Y 0.6105 0.0265 20000 0.5419 0.6460
#> M~~M 0.7310 0.0265 20000 0.6637 0.7678
#> indirect 0.2314 0.0196 20000 0.2071 0.2836
#> direct 0.2626 0.0320 20000 0.1953 0.3195
#> total 0.4940 0.0260 20000 0.4503 0.5519See GitHub Pages for package documentation.
To cite semmcci in publications, please cite Pesigan & Cheung (2023).
MacKinnon, D. P., Lockwood, C. M., & Williams, J. (2004). Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate Behavioral Research, 39(1), 99–128. https://doi.org/10.1207/s15327906mbr3901_4
Pesigan, I. J. A., & Cheung, S. F. (2023). Monte Carlo confidence intervals for the indirect effect with missing data. Behavior Research Methods, 56(3), 1678–1696. https://doi.org/10.3758/s13428-023-02114-4
Preacher, K. J., & Selig, J. P. (2012). Advantages of Monte Carlo confidence intervals for indirect effects. Communication Methods and Measures, 6(2), 77–98. https://doi.org/10.1080/19312458.2012.679848
Tofighi, D., & Kelley, K. (2019). Indirect effects in sequential mediation models: Evaluating methods for hypothesis testing and confidence interval formation. Multivariate Behavioral Research, 55(2), 188–210. https://doi.org/10.1080/00273171.2019.1618545
Tofighi, D., & MacKinnon, D. P. (2015). Monte Carlo confidence intervals for complex functions of indirect effects. Structural Equation Modeling: A Multidisciplinary Journal, 23(2), 194–205. https://doi.org/10.1080/10705511.2015.1057284