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Benchmark: Comparing the Monte Carlo Method with Nonparametric Bootstrapping

Ivan Jacob Agaloos Pesigan

2026-03-01

Source: vignettes/benchmark-complete.Rmd
benchmark-complete.Rmd

We compare the Monte Carlo (MC) method with nonparametric bootstrapping (NB) using the simple mediation model with complete data. One advantage of MC over NB is speed. This is because the model is only fitted once in MC whereas it is fitted many times in NB.

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
  X ~~ 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 from semmcci to generate Monte Carlo confidence intervals.

MC(fit, R = 100L, alpha = 0.05)
#> Monte Carlo Confidence Intervals
#>             est     se   R   2.5%  97.5%
#> cp       0.2333 0.0296 100 0.1806 0.2903
#> b        0.5082 0.0279 100 0.4555 0.5527
#> a        0.4820 0.0280 100 0.4220 0.5301
#> X~~X     1.0590 0.0426 100 0.9883 1.1428
#> Y~~Y     0.5462 0.0231 100 0.5064 0.5959
#> M~~M     0.7527 0.0337 100 0.6846 0.8029
#> indirect 0.2449 0.0179 100 0.2058 0.2738
#> direct   0.2333 0.0296 100 0.1806 0.2903
#> total    0.4782 0.0295 100 0.4162 0.5283

Nonparametric Bootstrap Confidence Intervals

Nonparametric bootstrap confidence intervals can be generated in lavaan using the following.

parameterEstimates(
  sem(
    data = df,
    model = model,
    se = "bootstrap",
    bootstrap = 100L
  )
)
#>        lhs op      rhs    label   est    se      z pvalue ci.lower ci.upper
#> 1        Y  ~        X       cp 0.233 0.025  9.395      0    0.183    0.278
#> 2        Y  ~        M        b 0.508 0.028 18.057      0    0.454    0.568
#> 3        M  ~        X        a 0.482 0.026 18.550      0    0.433    0.535
#> 4        X ~~        X          1.059 0.046 23.224      0    0.969    1.161
#> 5        Y ~~        Y          0.546 0.023 23.640      0    0.508    0.593
#> 6        M ~~        M          0.753 0.033 23.131      0    0.692    0.814
#> 7 indirect :=      a*b indirect 0.245 0.020 12.443      0    0.209    0.289
#> 8   direct :=       cp   direct 0.233 0.025  9.395      0    0.183    0.278
#> 9    total := cp+(a*b)    total 0.478 0.027 17.966      0    0.418    0.518

Benchmark

Arguments

Variables Values Notes
R 1000 Number of Monte Carlo replications.
B 1000 Number of bootstrap samples. 
benchmark_complete_01 <- microbenchmark(
  MC = {
    fit <- sem(
      data = df,
      model = model
    )
    MC(
      fit,
      R = R,
      decomposition = "chol",
      pd = FALSE
    )
  },
  NB = sem(
    data = df,
    model = model,
    se = "bootstrap",
    bootstrap = B
  ),
  times = 10
)

Summary of Benchmark Results 

summary(benchmark_complete_01, unit = "ms")
#>   expr        min        lq       mean     median         uq        max neval
#> 1   MC   156.2475   163.342   178.0618   174.2547   177.4388   236.8905    10
#> 2   NB 28208.6459 29647.342 30674.5847 31136.5817 31477.8059 31873.9380    10

Summary of Benchmark Results Relative to the Faster Method 

summary(benchmark_complete_01, unit = "relative")
#>   expr      min       lq     mean   median       uq      max neval
#> 1   MC   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000    10
#> 2   NB 180.5382 181.5047 172.2693 178.6843 177.4009 134.5514    10

Plot

Benchmark - Monte Carlo Method with Precalculated Estimates 

fit <- sem(
  data = df,
  model = model
)
benchmark_complete_02 <- microbenchmark(
  MC = MC(
    fit,
    R = R,
    decomposition = "chol",
    pd = FALSE
  ),
  NB = sem(
    data = df,
    model = model,
    se = "bootstrap",
    bootstrap = B
  ),
  times = 10
)

Summary of Benchmark Results 

summary(benchmark_complete_02, unit = "ms")
#>   expr         min          lq        mean      median          uq         max
#> 1   MC    47.95931    50.42981    54.50973    55.05257    57.38705    62.35807
#> 2   NB 25714.54010 26012.19289 27636.27829 27725.97381 29009.61288 29622.84813
#>   neval
#> 1    10
#> 2    10

Summary of Benchmark Results Relative to the Faster Method 

summary(benchmark_complete_02, unit = "relative")
#>   expr      min       lq     mean   median      uq      max neval
#> 1   MC   1.0000   1.0000   1.0000   1.0000   1.000   1.0000    10
#> 2   NB 536.1741 515.8099 506.9971 503.6273 505.508 475.0443    10

Plot

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