Sensitivity Analysis for Sample Size Planning From the AIPE Perspective for a Mixed-Effects Fixed Effect

Quantifies how much misspecification of the variance components ($\sigma^2_Y$, $\sigma^2_X$, and the intraclass correlation $\mathrm{icc}$) distorts an AIPE-based sample size plan for a cluster-level fixed effect under a two-level random-intercept model. On each replication the function simulates K clusters of cluster_size observations each from $$Y_{ki} = \beta\,X_k + u_k + \epsilon_{ki},$$ with $X_k \sim N(0, \sigma^2_X)$, $u_k \sim N(0, \mathrm{icc}\cdot\sigma^2_Y)$, and $\epsilon_{ki} \sim N(0, (1 - \mathrm{icc})\sigma^2_Y)$. The model is then refit by either lme4::lmer (if available) or by GLS-by-cluster aggregation, and a Wald CI on the fixed effect is recorded.

Usage

ss_aipe_mixed_effects_sensitivity(
  true_sigma2_y = NULL,
  true_sigma2_x = NULL,
  true_icc = NULL,
  true_beta = 0,
  estimated_sigma2_y = NULL,
  estimated_sigma2_x = NULL,
  estimated_icc = NULL,
  width,
  cluster_size = 20L,
  specified_K = NULL,
  conf_level = 0.95,
  G = 1000,
  print_iter = FALSE,
  save = FALSE,
  filename = "ss_aipe_mixed_effects_sensitivity_result.csv"
)

Arguments

true_sigma2_y: Population total variance of Y.
true_sigma2_x: Population variance of the cluster-level predictor.
true_icc: Population intraclass correlation (between-cluster share of total variance).
true_beta: Population fixed-effect slope (default 0).
estimated_sigma2_y, estimated_sigma2_x, estimated_icc: Planning values passed to ss_aipe_mixed_effects; supply all three or specified_K but not both.
width: Desired full width of the CI on the fixed effect.
cluster_size: Number of observations per cluster (assumed balanced).
specified_K: Number of clusters to evaluate.
conf_level: Confidence level (default 0.95).
G: Number of Monte Carlo replications.
print_iter: Logical.
save: Logical. Save per-replication CSV.
filename: Path used when save = TRUE.

Value

A data.frame with rows for mean / median / SD of the realized fixed-effect estimate and CI width, the proportion of intervals at or below width, tail-specific and overall non-coverage of true_beta, and the input echoes.

References

McNeish, D., & Kelley, K. (2019). Fixed effects versus mixed effects models for clustered data: Reviewing the approaches, disentangling the differences, and making recommendations. Psychological Methods, 24, 20–35. doi:10.1037/met0000182

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapters 15 and 16 on mixed-effects models.)

Other mixed models: R2_mixed_effects(), R2_mixed_effects_decomposition(), icc_lmer(), manova_split_plot(), mixed_anova(), ss_aipe_mixed_effects(), ss_power_mixed_effects(), ss_power_split_plot_anova()

Author

Ken Kelley kkelley@nd.edu

Examples

# Monte Carlo sensitivity check, reduced sizes for a fast example.
set.seed(113)
ss_aipe_mixed_effects_sensitivity(
  true_sigma2_y = 1, true_sigma2_x = 1, true_icc = 0.10,
  true_beta = 0.30,
  specified_K = 25, cluster_size = 10,
  width = 0.40,
  G = 25, print_iter = FALSE
)
#>  term               value 
#>  mean_beta          0.325 
#>  median_beta        0.322 
#>  sd_beta            0.108 
#>  mean_ci_width      0.384 
#>  median_ci_width    0.383 
#>  sd_ci_width        0.0694
#>  pct_ci_less_w      0.64  
#>  pct_ci_miss_low    0     
#>  pct_ci_miss_high   0.08  
#>  total_type_I_error 0.08  
#>  n_clusters         25    
#>  cluster_size       10    
#>  total_N            250   
#>  true_sigma2_y      1     
#>  true_sigma2_x      1     
#>  true_icc           0.1   
#>  true_beta          0.3   
#>  estimated_sigma2_y <NA>  
#>  estimated_sigma2_x <NA>  
#>  estimated_icc      <NA>  
#>  width              0.4   
#>  conf_level         0.95  
#> 
#> Confidence level: 95%