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Benchmark: Comparing the Monte Carlo Method with Nonparametric Bootstrapping (FIML)

Ivan Jacob Agaloos Pesigan

2025-10-19

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

We compare the Monte Carlo (MC) method with nonparametric bootstrapping (NB) using the simple mediation model with missing data using full-information maximum likelihood. 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)

# Create data set with missing values.

miss <- sample(1:dim(df)[1], 300)
df[miss[1:100], "X"] <- NA
df[miss[101:200], "M"] <- NA
df[miss[201:300], "Y"] <- NA

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. We are using missing = "fiml" to handle missing data in 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       0.2419 0.0332 100 0.1792 0.3070
#> b        0.5166 0.0308 100 0.4580 0.5785
#> a        0.4989 0.0319 100 0.4448 0.5615
#> X~~X     1.0951 0.0621 100 0.9856 1.2026
#> Y~~Y     0.5796 0.0307 100 0.5257 0.6413
#> M~~M     0.8045 0.0464 100 0.7325 0.9106
#> indirect 0.2577 0.0210 100 0.2234 0.3031
#> direct   0.2419 0.0332 100 0.1792 0.3070
#> total    0.4996 0.0322 100 0.4550 0.5681

Nonparametric Bootstrap Confidence Intervals

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

parameterEstimates(
  sem(
    data = df,
    model = model,
    missing = "fiml",
    se = "bootstrap",
    bootstrap = 100L
  )
)
#>         lhs op      rhs    label    est    se      z pvalue ci.lower ci.upper
#> 1         Y  ~        X       cp  0.234 0.030  7.721  0.000    0.169    0.287
#> 2         Y  ~        M        b  0.511 0.035 14.704  0.000    0.442    0.585
#> 3         M  ~        X        a  0.481 0.028 17.117  0.000    0.425    0.532
#> 4         X ~~        X           1.059 0.049 21.539  0.000    0.979    1.148
#> 5         Y ~~        Y           0.554 0.029 19.264  0.000    0.490    0.607
#> 6         M ~~        M           0.756 0.032 23.389  0.000    0.693    0.820
#> 7         Y ~1                   -0.013 0.027 -0.473  0.636   -0.065    0.056
#> 8         M ~1                   -0.022 0.030 -0.744  0.457   -0.077    0.044
#> 9         X ~1                    0.002 0.036  0.069  0.945   -0.072    0.074
#> 10 indirect :=      a*b indirect  0.246 0.021 11.534  0.000    0.202    0.286
#> 11   direct :=       cp   direct  0.234 0.030  7.721  0.000    0.169    0.287
#> 12    total := cp+(a*b)    total  0.479 0.030 16.081  0.000    0.417    0.547

Benchmark

Arguments

Variables Values Notes
R 1000 Number of Monte Carlo replications.
B 1000 Number of bootstrap samples. 
benchmark_fiml_01 <- microbenchmark(
  MC = {
    fit <- sem(
      data = df,
      model = model,
      missing = "fiml"
    )
    MC(
      fit,
      R = R,
      decomposition = "chol",
      pd = FALSE
    )
  },
  NB = sem(
    data = df,
    model = model,
    missing = "fiml",
    se = "bootstrap",
    bootstrap = B
  ),
  times = 10
)

Summary of Benchmark Results 

summary(benchmark_fiml_01, unit = "ms")
#>   expr        min         lq       mean     median         uq        max neval
#> 1   MC   121.7111   122.5368   124.5841   125.1569   125.8596   127.6093    10
#> 2   NB 27922.9926 27984.3329 28009.1665 27992.8715 28057.7444 28133.4957    10

Summary of Benchmark Results Relative to the Faster Method 

summary(benchmark_fiml_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 229.4202 228.3749 224.8214 223.6622 222.9289 220.4658    10

Plot

Benchmark - Monte Carlo Method with Precalculated Estimates 

fit <- sem(
  data = df,
  model = model,
  missing = "fiml"
)
benchmark_fiml_02 <- microbenchmark(
  MC = MC(
    fit,
    R = R,
    decomposition = "chol",
    pd = FALSE
  ),
  NB = sem(
    data = df,
    model = model,
    missing = "fiml",
    se = "bootstrap",
    bootstrap = B
  ),
  times = 10
)

Summary of Benchmark Results 

summary(benchmark_fiml_02, unit = "ms")
#>   expr         min          lq        mean      median          uq         max
#> 1   MC    21.87685    22.12413    24.21625    23.49522    26.27235    27.63496
#> 2   NB 27614.25595 27851.34753 27899.64184 27950.19922 28000.03878 28058.31435
#>   neval
#> 1    10
#> 2    10

Summary of Benchmark Results Relative to the Faster Method 

summary(benchmark_fiml_02, unit = "relative")
#>   expr      min       lq     mean   median       uq     max neval
#> 1   MC    1.000    1.000    1.000    1.000    1.000    1.00    10
#> 2   NB 1262.259 1258.867 1152.104 1189.612 1065.761 1015.32    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