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

Ivan Jacob Agaloos Pesigan

2024-10-22

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 0.4363 0.5530
#> 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 0.0241 100 0.2154 0.3031
#> direct   0.2419 0.0332 100 0.1792 0.3070
#> total    0.4996 0.0342 100 0.4333 0.5626

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   163.9536   168.6076   203.8065   198.3323   238.4714   260.1297    10
#> 2   NB 54766.8708 55844.2892 65956.6550 59521.2283 72928.0038 93144.6298    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 334.0388 331.2086 323.6239 300.1086 305.8145 358.0699    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    24.81573    24.96414    26.49458    26.32657    28.03841    28.66643
#> 2   NB 54559.68598 54682.82524 54724.62625 54715.37411 54764.59709 54907.04943
#>   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    1.000    10
#> 2   NB 2198.593 2190.455 2065.503 2078.333 1953.199 1915.378    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