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

Ivan Jacob Agaloos Pesigan

2026-03-01

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 0.9875 1.2045
#> Y~~Y     0.5796 0.0307 100 0.5179 0.6336
#> 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   260.1772   260.9986   261.2316   261.2154   261.5805   261.8205    10
#> 2   NB 60196.0501 60273.7339 60304.4039 60293.9547 60347.5345 60434.7181    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   1.000    10
#> 2   NB 231.3656 230.9351 230.8465 230.8208 230.7035 230.825    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    58.32543    58.44053    60.16606    59.65633    60.9178    63.67535
#> 2   NB 60136.15726 60200.18985 60245.00599 60241.29127 60281.3703 60384.01692
#>   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    1.00    1.000    1.000   1.0000   1.0000    10
#> 2   NB 1031.045 1030.11 1001.312 1009.806 989.5526 948.3108    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