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

Ivan Jacob Agaloos Pesigan

2025-07-22

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 11.476  0.000    0.202    0.286
#> 11   direct :=       cp   direct  0.234 0.030  7.682  0.000    0.169    0.287
#> 12    total := cp+(a*b)    total  0.479 0.030 16.001  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   106.5657   107.333   109.2974   108.5297   111.7913   113.4185    10
#> 2   NB 46897.5716 47470.718 47936.1980 47855.8885 48482.9339 48959.5930    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   1.000   1.0000   1.0000   1.0000   1.0000    10
#> 2   NB 440.0815 442.275 438.5848 440.9474 433.6916 431.6722    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    19.12036    20.06588    21.29987    20.41911    23.00196    24.2131
#> 2   NB 47174.69427 47190.17818 47779.39203 47463.06558 48740.24230 48910.4056
#>   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    1.00    1.000    10
#> 2   NB 2467.249 2351.762 2243.178 2324.444 2118.96 2019.998    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