AIPE Sample Size Planning for a Fixed Effect in a Two-Level Mixed-Effects Model

Computes the minimum number of clusters (level-2 units) needed so that the confidence interval on a level-1 fixed-effect slope has expected full width no larger than \(\omega\) (Kelley, 2007; Raudenbush & Liu, 2001; Snijders & Bosker, 2012). The function inverts the closed-form approximation for the variance of a fixed-effect slope in a balanced two-level random-intercept model.

Usage

ss_aipe_mixed_effects(
  sigma2_y,
  sigma2_x,
  icc,
  width,
  cluster_size = 20L,
  conf_level = 0.95
)

Arguments

sigma2_y

Total variance of the outcome variable.

sigma2_x

Variance of the level-1 predictor (covariate).

icc

Intraclass correlation of the outcome.

width

Target full CI width on the slope.

cluster_size

Per-cluster sample size (number of level-1 units per level-2 unit). Default 20L.

conf_level

Confidence level. Default 0.95.

Value

A data.frame with rows for the recommended number of clusters necessary_n_clusters, the implied total sample size total_N (necessary_n_clusters * cluster_size), the target width, the intraclass correlation icc, the cluster_size, and the resulting ci_width_expected (the expected full CI width at the recommended size).

Details

Variance of the slope. For a level-1 predictor centered within cluster, the asymptotic variance of \(\hat\beta\) is approximately $$\mathrm{Var}(\hat\beta) \;\approx\; \frac{\sigma^2_y (1 - \rho_I)}{N \sigma^2_x},$$ where \(N = n_{\mathrm{clusters}} \cdot m\) is the total number of level-1 units, \(m\) is the cluster size, and \(\rho_I\) the intraclass correlation. Because within-cluster centering removes the cluster-level variation from the predictor, the design effect \(1 + (m - 1) \rho_I\) that inflates the variance of a cluster-level estimand does not appear here; clustering enters only through the residual variance \(\sigma^2_y (1 - \rho_I)\). The function inverts this expression for \(N\). No anticipated slope value is needed: \(\beta\) does not appear in the variance, so the recommended number of clusters is the same whatever the slope.

Scope. Planning is for the most common single-level covariate case (random intercept, fixed slope, level-1 predictor centered within cluster). For cross-level interactions or random slopes, the variance formula changes and a Monte Carlo planner should be used instead (Schoemann, Boulton, & Short, 2017).

References

Kelley, K. (2007). Confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20(8), 1–24. doi:10.18637/jss.v020.i08

Raudenbush, S. W., & Liu, X.-F. (2001). Effects of study duration, frequency of observation, and sample size on power in studies of group differences in polynomial change. Psychological Methods, 6(4), 387–401. doi:10.1037/1082-989X.6.4.387

Schoemann, A. M., Boulton, A. J., & Short, S. D. (2017). Determining power and sample size for simple and complex mediation models. Social Psychological and Personality Science, 8(4), 379–386. doi:10.1177/1948550617715068

Snijders, T. A. B., & Bosker, R. J. (2012). Multilevel analysis: An introduction to basic and advanced multilevel modeling (2nd ed.). Sage.

See also

ss_power_mixed_effects, var_icc, ss_aipe_icc

design_consequences for what a chosen design delivers: power, the Type S (sign) and Type M (exaggeration) errors of the significance filter, and the expected confidence interval width.

Other sample size for power: power_fisher_exact(), ss_power_R2(), ss_power_R2_sensitivity(), ss_power_c(), ss_power_c_ancova(), ss_power_composite_ancova(), ss_power_composite_ancova_2group(), ss_power_composite_anova(), ss_power_composite_factorial_ancova(), ss_power_composite_factorial_ancova_het(), ss_power_composite_factorial_anova(), ss_power_composite_sem(), ss_power_contrast(), ss_power_equivalence_c(), ss_power_factorial_ancova(), ss_power_factorial_anova(), ss_power_indirect_effect(), ss_power_mixed_effects(), ss_power_one_way_anova(), ss_power_pcm(), ss_power_r(), ss_power_rc(), ss_power_reg_coef(), ss_power_reg_coef_sensitivity(), ss_power_rm_anova(), ss_power_sc(), ss_power_sem(), ss_power_smd(), ss_power_split_plot_anova()

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

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Plan a two-level study with cluster size 20, ICC = 0.10,
#        sigma_y = 1, sigma_x = 1, target CI width = 0.20:
ss_aipe_mixed_effects(sigma2_y = 1, sigma2_x = 1, icc = 0.10,
                           width = 0.20, cluster_size = 20)
#>  term                 value
#>  necessary_n_clusters 18   
#>  total_N              360  
#>  width                0.2  
#>  icc                  0.1  
#>  cluster_size         20   
#>  ci_width_expected    0.196
#> 
#> Confidence level: 95%

# 2. The same study with stronger clustering (ICC = 0.20):
ss_aipe_mixed_effects(sigma2_y = 1, sigma2_x = 1, icc = 0.20,
                           width = 0.20, cluster_size = 20)
#>  term                 value
#>  necessary_n_clusters 16   
#>  total_N              320  
#>  width                0.2  
#>  icc                  0.2  
#>  cluster_size         20   
#>  ci_width_expected    0.196
#> 
#> Confidence level: 95%