Monte Carlo Confidence Intervals in Structural Equation Modeling • semmcci

Ivan Jacob Agaloos Pesigan 2023-01-25

Installation

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")

Description

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 two main functions, namely, MC() and MCStd(). The output of lavaan is passed as the first argument to the MC() 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 to the MCStd() function.

Example

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)

Data

n <- 1000
a <- 0.50
b <- 0.50
cp <- 0.25
s2_em <- 1 - a^2
s2_ey <- 1 - cp^2 - a^2 * b^2 - b^2 * s2_em - 2 * cp * a * b
em <- rnorm(n = n, mean = 0, sd = sqrt(s2_em))
ey <- rnorm(n = n, mean = 0, sd = sqrt(s2_ey))
X <- rnorm(n = n)
M <- a * X + em
Y <- cp * X + b * M + ey
df <- data.frame(X, M, Y)

Model Specification

The 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
  indirect := a * b
  direct := cp
  total := cp + (a * b)
"

Model Fitting

We can now fit the model using the sem() function from lavaan.

fit <- sem(data = df, model = model)

Monte Carlo Confidence Intervals

The fit lavaan object can then be passed to the MC() function to generate Monte Carlo confidence intervals.

MC(fit, R = 20000L, alpha = c(0.001, 0.01, 0.05))
#> Monte Carlo Confidence Intervals
#>             est     se     R  0.05%   0.5%   2.5%  97.5%  99.5% 99.95%
#> cp       0.2333 0.0264 20000 0.1465 0.1651 0.1817 0.2846 0.3012 0.3165
#> b        0.5082 0.0272 20000 0.4161 0.4385 0.4551 0.5611 0.5786 0.5988
#> a        0.4820 0.0264 20000 0.3997 0.4144 0.4302 0.5340 0.5499 0.5680
#> Y~~Y     0.5462 0.0244 20000 0.4660 0.4832 0.4979 0.5944 0.6095 0.6263
#> M~~M     0.7527 0.0339 20000 0.6434 0.6667 0.6867 0.8196 0.8399 0.8612
#> indirect 0.2449 0.0187 20000 0.1881 0.1983 0.2093 0.2825 0.2952 0.3081
#> direct   0.2333 0.0264 20000 0.1465 0.1651 0.1817 0.2846 0.3012 0.3165
#> total    0.4782 0.0267 20000 0.3920 0.4091 0.4257 0.5303 0.5459 0.5647

Standardized Monte Carlo Confidence Intervals

Standardized Monte Carlo Confidence intervals can be generated by passing the result of the MC() function to MCStd().

Note: We recommend setting fixed.x = FALSE when generating standardized estimates and confidence intervals to model the variances and covariances of the predictors if they are assumed to be random.

fit <- sem(data = df, model = model, fixed.x = FALSE)
unstd <- MC(fit, R = 20000L, alpha = c(0.001, 0.01, 0.05))

MCStd(unstd)
#> Standardized Monte Carlo Confidence Intervals
#>             est     se     R  0.05%   0.5%   2.5%  97.5%  99.5% 99.95%
#> cp       0.2422 0.0267 20000 0.1513 0.1722 0.1894 0.2939 0.3111 0.3284
#> b        0.5123 0.0247 20000 0.4319 0.4488 0.4636 0.5604 0.5764 0.5932
#> a        0.4963 0.0240 20000 0.4144 0.4323 0.4478 0.5414 0.5557 0.5709
#> Y~~Y     0.5558 0.0236 20000 0.4785 0.4948 0.5087 0.6017 0.6156 0.6320
#> M~~M     0.7537 0.0237 20000 0.6741 0.6913 0.7069 0.7995 0.8132 0.8282
#> X~~X     1.0000 0.0000 20000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
#> indirect 0.2542 0.0176 20000 0.1967 0.2093 0.2200 0.2890 0.3008 0.3123
#> direct   0.2422 0.0267 20000 0.1513 0.1722 0.1894 0.2939 0.3111 0.3284
#> total    0.4964 0.0240 20000 0.4177 0.4327 0.4478 0.5417 0.5566 0.5738

Documentation

See GitHub Pages for package documentation.