Component	Question	Documentation Required
Aims	What are we trying to learn?	Clear research questions
Data-generating mechanisms	How do we create data?	Full DGP specification
Estimands	What are we estimating?	Mathematical definition
Methods	What estimators do we compare?	Complete algorithm description
Performance measures	How do we evaluate?	Bias, variance, coverage

Morris et al. Reporting Checklist

□ Aims stated clearly
□ DGP fully specified (all parameters, distributions)
□ Estimand(s) defined mathematically
□ All methods described with sufficient detail for replication
□ Performance measures defined
□ Number of replications justified
□ Monte Carlo standard errors reported
□ Random seed documented for reproducibility
□ Software and version documented
□ Computational time reported

Replication Counts

How Many Replications Are Needed?

Monte Carlo Standard Error (MCSE) formula:

$$\text{MCSE}(\hat{\theta}) = \frac{\hat{\sigma}}{\sqrt{B}}$$

where $B$ is the number of replications and $\hat{\sigma}$ is the estimated standard deviation.

Recommended Replications by Purpose

Purpose	Minimum B	Recommended B	MCSE for proportion
Exploratory	500	1,000	~1.4% at 95% coverage
Publication	1,000	2,000	~1.0% at 95% coverage
Definitive	5,000	10,000	~0.4% at 95% coverage
Precision	10,000+	50,000	~0.2% at 95% coverage

MCSE Calculation

# Calculate Monte Carlo standard errors
calculate_mcse <- function(estimates, coverage_indicators = NULL) {
  B <- length(estimates)

  list(
    # MCSE for mean (bias)
    mcse_mean = sd(estimates) / sqrt(B),

    # MCSE for standard deviation
    mcse_sd = sd(estimates) / sqrt(2 * (B - 1)),

    # MCSE for coverage (proportion)
    mcse_coverage = if (!is.null(coverage_indicators)) {
      p <- mean(coverage_indicators)
      sqrt(p * (1 - p) / B)
    } else NA
  )
}

# Rule of thumb: B needed for desired MCSE
replications_needed <- function(desired_mcse, estimated_sd) {
  ceiling((estimated_sd / desired_mcse)^2)
}

R Code Templates

Complete Simulation Template

# Full simulation study template following Morris et al. guidelines
run_simulation_study <- function(
  n_sims = 2000,
  n_vec = c(200, 500, 1000),
  seed = 42,
  parallel = TRUE,
  n_cores = parallel::detectCores() - 1
) {

  set.seed(seed)

  # Define parameter grid
  params <- expand.grid(
    n = n_vec,
    effect_size = c(0, 0.14, 0.39),
    model_spec = c("correct", "misspecified")
  )

  # Setup parallel processing
  if (parallel) {
    cl <- parallel::makeCluster(n_cores)
    doParallel::registerDoParallel(cl)
    on.exit(parallel::stopCluster(cl))
  }

  # Run simulations
  results <- foreach(
    i = 1:nrow(params),
    .combine = rbind,
    .packages = c("tidyverse", "mediation")
  ) %dopar% {

    scenario <- params[i, ]
    sim_results <- replicate(n_sims, {
      data <- generate_dgp(scenario)
      estimates <- apply_methods(data)
      evaluate_performance(estimates, truth = scenario$effect_size)
    }, simplify = FALSE)

    summarize_scenario(sim_results, scenario)
  }

  # Add MCSE
  results <- add_monte_carlo_errors(results, n_sims)

  results
}

# Summarize with MCSE
add_monte_carlo_errors <- function(results, B) {
  results %>%
    mutate(
      mcse_bias = empirical_se / sqrt(B),
      mcse_coverage = sqrt(coverage * (1 - coverage) / B),
      mcse_rmse = rmse / sqrt(2 * B)
    )
}

Parallel Simulation Template

# Memory-efficient parallel simulation
run_parallel_simulation <- function(scenario, n_sims, n_cores = 4) {
  library(future)
  library(future.apply)

  plan(multisession, workers = n_cores)

  results <- future_replicate(n_sims, {
    data <- generate_dgp(scenario$n, scenario$params)
    est <- estimate_effect(data)
    list(
      estimate = est$point,
      se = est$se,
      covered = abs(est$point - scenario$truth) < 1.96 * est$se
    )
  }, simplify = FALSE)

  plan(sequential)  # Reset

  # Aggregate
  estimates <- sapply(results, `[[`, "estimate")
  ses <- sapply(results, `[[`, "se")
  covered <- sapply(results, `[[`, "covered")

  list(
    bias = mean(estimates) - scenario$truth,
    empirical_se = sd(estimates),
    mean_se = mean(ses),
    coverage = mean(covered),
    mcse_bias = sd(estimates) / sqrt(n_sims),
    mcse_coverage = sqrt(mean(covered) * (1 - mean(covered)) / n_sims)
  )
}

Core Principles (Morris et al., 2019)

ADEMP Framework

Aims: What question does the simulation answer?
Data-generating mechanisms: How is data simulated?
Estimands: What is being estimated?
Methods: What estimators are compared?
Performance measures: How is performance assessed?

Data-Generating Process Design

Standard Mediation DGP

generate_mediation_data <- function(n, params) {
  # Confounders
  X <- rnorm(n)

  # Treatment (binary)
  ps <- plogis(params$gamma0 + params$gamma1 * X)
  A <- rbinom(n, 1, ps)

  # Mediator
  M <- params$alpha0 + params$alpha1 * A + params$alpha2 * X +
       rnorm(n, sd = params$sigma_m)

  # Outcome
  Y <- params$beta0 + params$beta1 * A + params$beta2 * M +
       params$beta3 * X + params$beta4 * A * M +
       rnorm(n, sd = params$sigma_y)

  data.frame(Y = Y, A = A, M = M, X = X)
}

DGP Variations to Consider

Linearity: Linear vs nonlinear relationships
Model specification: Correct vs misspecified
Error structure: Homoscedastic vs heteroscedastic
Interaction: No interaction vs A×M interaction
Confounding: Measured vs unmeasured
Treatment: Binary vs continuous
Mediator: Continuous vs binary vs count

Parameter Grid Design

Sample Size Selection

Size	Label	Purpose
100-200	Small	Stress test
500	Medium	Typical study
1000-2000	Large	Asymptotic behavior
5000+	Very large	Efficiency comparison

Effect Size Selection

Effect	Interpretation
0	Null (Type I error)
0.1	Small
0.3	Medium
0.5	Large

Recommended Grid Structure

params <- expand.grid(
  n = c(200, 500, 1000, 2000),
  effect = c(0, 0.14, 0.39, 0.59),  # Small/medium/large per Cohen
  confounding = c(0, 0.3, 0.6),
  misspecification = c(FALSE, TRUE)
)

Performance Metrics

Primary Metrics

Metric	Formula	Target	MCSE Formula
Bias	$\bar{\hat\psi} - \psi_0$	≈ 0	$\sqrt{\text{Var}(\hat\psi)/n_{sim}}$
Empirical SE	$\text{SD}(\hat\psi)$	—	Complex
Average SE	$\bar{\widehat{SE}}$	≈ Emp SE	$\text{SD}(\widehat{SE})/\sqrt{n_{sim}}$
Coverage	$\frac{1}{n_{sim}}\sum I(\psi_0 \in CI)$	≈ 0.95	$\sqrt{p(1-p)/n_{sim}}$
MSE	$\text{Bias}^2 + \text{Var}$	Minimize	—
Power	% rejecting $H_0$	Context-dependent	$\sqrt{p(1-p)/n_{sim}}$

Relative Metrics (for method comparison)

Relative bias: $\text{Bias}/\psi_0$ (when $\psi_0 \neq 0$)
Relative efficiency: $\text{Var}(\hat\psi_1)/\text{Var}(\hat\psi_2)$
Relative MSE: $\text{MSE}_1/\text{MSE}_2$

Replication Guidelines

Minimum Replications

Metric	Minimum	Recommended
Bias	1000	2000
Coverage	2000	5000
Power	1000	2000

Monte Carlo Standard Error

Always report MCSE for key metrics:

Coverage MCSE at 95%: $\sqrt{0.95 \times 0.05 / n_{sim}} \approx 0.007$ for $n_{sim}=1000$
Need ~2500 reps for MCSE < 0.005

R Implementation Template

#' Run simulation study
#' @param scenario Parameter list for this scenario
#' @param n_rep Number of replications
#' @param seed Random seed
run_simulation <- function(scenario, n_rep = 2000, seed = 42) {
  set.seed(seed)

  results <- future_map(1:n_rep, function(i) {
    # Generate data
    data <- generate_data(scenario$n, scenario$params)

    # Fit methods
    fit1 <- method1(data)
    fit2 <- method2(data)

    # Extract estimates
    tibble(
      rep = i,
      method = c("method1", "method2"),
      estimate = c(fit1$est, fit2$est),
      se = c(fit1$se, fit2$se),
      ci_lower = estimate - 1.96 * se,
      ci_upper = estimate + 1.96 * se
    )
  }, .options = furrr_options(seed = TRUE)) %>%
    bind_rows()

  # Summarize
  results %>%
    group_by(method) %>%
    summarize(
      bias = mean(estimate) - scenario$true_value,
      emp_se = sd(estimate),
      avg_se = mean(se),
      coverage = mean(ci_lower <= scenario$true_value &
                      ci_upper >= scenario$true_value),
      mse = bias^2 + emp_se^2,
      .groups = "drop"
    )
}

Results Presentation

Standard Table Format

Table X: Simulation Results (n_rep = 2000)

                    Method 1                    Method 2
n       Bias   SE    Cov   MSE      Bias   SE    Cov   MSE
-----------------------------------------------------------
200     0.02   0.15  0.94  0.023    0.01   0.12  0.95  0.014
500     0.01   0.09  0.95  0.008    0.00   0.08  0.95  0.006
1000    0.00   0.06  0.95  0.004    0.00   0.05  0.95  0.003

Note: Cov = 95% CI coverage. MCSE for coverage ≈ 0.005.

Visualization Guidelines

Use faceted plots for multiple scenarios
Show confidence bands for metrics
Compare methods side-by-side
Log scale for MSE if range is large

Checkpoints and Reproducibility

Checkpointing Strategy

# Save results incrementally
if (i %% 100 == 0) {
  saveRDS(results_so_far,
          file = sprintf("checkpoint_%s_rep%d.rds", scenario_id, i))
}

Reproducibility Requirements

Set seed explicitly
Record package versions (sessionInfo())
Use furrr_options(seed = TRUE) for parallel
Save full results, not just summaries
Document any manual interventions

Common Pitfalls

Design Pitfalls

Too few replications for coverage assessment
Unrealistic parameter combinations
Missing null scenario (effect = 0)
No misspecification scenarios

Implementation Pitfalls

Not setting seeds properly in parallel
Ignoring convergence failures
Not checking for numerical issues
Insufficient burn-in for MCMC methods

Reporting Pitfalls

Missing MCSE
Not reporting convergence failures
Cherry-picking scenarios
Inadequate description of DGP

Key References

Morris et al 2019
Burton et al
White