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Benchmark: Comparing the Monte Carlo Method with Nonparametric Bootstrapping

Ivan Jacob Agaloos Pesigan

2024-10-22

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 12.381      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   120.6836   186.192   203.7261   215.5088   230.7458   257.3183    10
#> 2   NB 29515.6644 45558.542 56084.4064 50152.7194 70779.4403 76304.1979    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 244.5707 244.6858 275.2932 232.7178 306.7421 296.5363    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    26.92053    31.71674    44.28055    47.78796    50.6716    63.41477
#> 2   NB 29200.31081 41337.92447 44233.31711 46074.02164 49510.4200 52418.48387
#>   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    1.000    1.000   1.0000   1.0000   1.0000   1.0000    10
#> 2   NB 1084.685 1303.347 998.9334 964.1346 977.0842 826.5974    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, 56(3), 1678–1696. https://doi.org/10.3758/s13428-023-02114-4