Example 2: The Serial Mediation Model

In this example, the Monte Carlo method is used to generate confidence intervals for the indirect effects in a serial mediation model with two mediators where X is the predictor, M1 is the first mediator, M2 is the second mediator, and Y is the dependent variable.

$The Serial Mediation Model$

The Serial Mediation Model

library(semmcci)
library(lavaan)

Data

summary(df)
#>        X                  M1                 M2                 Y           
#>  Min.   :-3.37174   Min.   :-3.22690   Min.   :-4.33590   Min.   :-4.29020  
#>  1st Qu.:-0.67546   1st Qu.:-0.73709   1st Qu.:-0.82188   1st Qu.:-0.86035  
#>  Median :-0.01313   Median :-0.01651   Median :-0.03903   Median :-0.02704  
#>  Mean   :-0.02582   Mean   :-0.01823   Mean   :-0.01620   Mean   :-0.03338  
#>  3rd Qu.: 0.66401   3rd Qu.: 0.72825   3rd Qu.: 0.80016   3rd Qu.: 0.81721  
#>  Max.   : 3.49530   Max.   : 3.69001   Max.   : 3.65147   Max.   : 4.05239
colMeans(df)
#>           X          M1          M2           Y 
#> -0.02582443 -0.01823021 -0.01619576 -0.03337865
var(df)
#>            X        M1        M2         Y
#> X  1.0050488 0.5123920 0.3848638 0.3333458
#> M1 0.5123920 1.2334461 0.6645408 0.5108946
#> M2 0.3848638 0.6645408 1.4321822 0.8012638
#> Y  0.3333458 0.5108946 0.8012638 1.4504417

Model Specification

We can define several indirect effects in this example:

$X \rightarrow M1 \rightarrow M2 \rightarrow Y$
$X \rightarrow M1 \rightarrow M2$
$X \rightarrow M1 \rightarrow Y$
$M1 \rightarrow M2 \rightarrow Y$

These indirect effects are defined using the := operator in the lavaan model syntax.

model <- "
  Y ~ cp * X + b1 * M1 + b2 * M2
  M2 ~ a2 * X + k * M1
  M1 ~ a1 * X
  # X -> M1 -> M2 -> Y
  a1kb2 := a1 * k * b2
  # X -> M1 -> M2
  a1k := a1 * k
  # X -> M1 -> Y
  a2b2 := a2 * b2
  # M1 -> M2 -> Y
  kb2 := k * b2
"

Model Fitting

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

Monte Carlo Confidence Intervals

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

MC(fit, R = 20000L, alpha = 0.05)
#> Monte Carlo Confidence Intervals
#>           est     se     R   2.5%  97.5%
#> cp     0.0868 0.0355 20000 0.0169 0.1554
#> b1     0.1190 0.0350 20000 0.0513 0.1880
#> b2     0.4809 0.0305 20000 0.4215 0.5404
#> a2     0.1373 0.0367 20000 0.0657 0.2098
#> k      0.4817 0.0328 20000 0.4175 0.5463
#> a1     0.5098 0.0310 20000 0.4483 0.5700
#> Y~~Y   0.9744 0.0439 20000 0.8885 1.0603
#> M2~~M2 1.0581 0.0477 20000 0.9641 1.1510
#> M1~~M1 0.9712 0.0431 20000 0.8862 1.0553
#> X~~X   1.0040 0.0000 20000 1.0040 1.0040
#> a1kb2  0.1181 0.0132 20000 0.0936 0.1451
#> a1k    0.2456 0.0224 20000 0.2034 0.2909
#> a2b2   0.0660 0.0181 20000 0.0310 0.1027
#> kb2    0.2317 0.0217 20000 0.1909 0.2754

Standardized Monte Carlo Confidence Intervals

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

Note: We recommend setting fixed.x = FALSE when generating standardized estimates and confidence intervals to model the variances and covariances of the exogenous observed variables if they are assumed to be random. If fixed.x = TRUE, which is the default setting in lavaan, MC() will fix the variances and the covariances of the exogenous observed variables to the sample values.

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

MCStd(unstd, alpha = 0.05)
#> Standardized Monte Carlo Confidence Intervals
#>           est     se     R   2.5%  97.5%
#> cp     0.0723 0.0291 20000 0.0147 0.1291
#> b1     0.1098 0.0317 20000 0.0485 0.1723
#> b2     0.4779 0.0273 20000 0.4234 0.5309
#> a2     0.1151 0.0307 20000 0.0546 0.1736
#> k      0.4470 0.0281 20000 0.3911 0.5016
#> a1     0.4602 0.0249 20000 0.4108 0.5084
#> Y~~Y   0.6725 0.0242 20000 0.6230 0.7175
#> M2~~M2 0.7396 0.0237 20000 0.6920 0.7846
#> M1~~M1 0.7882 0.0229 20000 0.7415 0.8312
#> X~~X   1.0000 0.0000 20000 1.0000 1.0000
#> a1kb2  0.0983 0.0104 20000 0.0787 0.1194
#> a1k    0.2057 0.0175 20000 0.1717 0.2405
#> a2b2   0.0550 0.0150 20000 0.0259 0.0845
#> kb2    0.2136 0.0187 20000 0.1777 0.2509

References

Pesigan, I. J. A., & Cheung, S. F. (2024). 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