Skip to contents

Benchmark: Comparing the Monte Carlo Method with Nonparametric Bootstrapping

Ivan Jacob Agaloos Pesigan

2024-04-14

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 0.9751 1.1296
#> Y~~Y     0.5462 0.0231 100 0.5064 0.5959
#> M~~M     0.7527 0.0337 100 0.7024 0.8208
#> 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 12.381      0    0.209    0.289
#> 8   direct :=       cp   direct 0.233 0.025  9.348      0    0.183    0.278
#> 9    total := cp+(a*b)    total 0.478 0.027 17.876      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   50.183   63.3979    76.76355    71.62314    94.36132   112.1057    10
#> 2   NB 9226.528 9864.4038 10542.68944 10366.60788 11702.20369 11970.5139    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 183.8576 155.5951 137.3398 144.7383 124.0148 106.7788    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    11.21167    12.33705    18.12138    16.59762    21.01478    29.89916
#> 2   NB 10655.40123 11177.79404 11659.56539 11471.02657 12086.97888 13104.42548
#>   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   1.000   1.0000   1.0000    10
#> 2   NB 950.3846 906.0343 643.4149 691.125 575.1655 438.2874    10

Plot

References

Pesigan, I. J. A., & Cheung, S. F. (2023). Monte Carlo confidence intervals for the indirect effect with missing data. Behavior Research Methods. https://doi.org/10.3758/s13428-023-02114-4