Sample Size or Power for a Treatment Effect in a Two-Level Mixed-Effects Model

Determine the necessary number of level-2 units per arm to achieve a desired level of statistical power for a treatment-versus-control comparison in a two-level mixed-effects model with a random intercept (e.g., individuals nested within clusters in a cluster-randomized trial, or repeated measurements nested within subjects in a person-randomized longitudinal study). Alternatively, given a number of level-2 units per arm, return the realized statistical power.

Usage

ss_power_mixed_effects(
  d,
  n,
  rho,
  J = NULL,
  desired_power = 0.85,
  alpha_level = 0.05,
  directional = FALSE
)

Arguments

d: Standardized treatment effect, defined as the population mean difference divided by the population standard deviation of the level-1 outcome
n: Number of level-1 units per level-2 unit (e.g., individuals per cluster, or measurements per subject); assumed equal across level-2 units
rho: Intra-class correlation (the proportion of total outcome variance attributable to differences between level-2 units); must be in [0, 1)
J: Number of level-2 units per arm (i.e., J treatment clusters and J control clusters); if specified, returns the realized power
desired_power: Desired statistical power (default 0.85)
alpha_level: Type I error rate (default 0.05)
directional: Logical: TRUE for a one-sided test (in the same sign as d), FALSE (default) for a two-sided test

Value

A data.frame with rows for necessary_J_per_arm (or specified_J_per_arm), total_N, noncentrality, and actual_power. The result carries the dmar_ss_power class, so tidy and glance summarize it in broom convention (the reported size is the number of clusters per arm).

Details

This function computes power for the fixed treatment effect at the higher level of a two-level mixed-effects model with random intercept, $$y_{ij} = \beta_0 + \beta_1 T_j + u_j + \epsilon_{ij},$$ where $u_j \sim N(0, \sigma_u^2)$ is the level-2 random intercept and $\epsilon_{ij} \sim N(0, \sigma_e^2)$ is the level-1 residual. The treatment indicator $T_j$ varies between level-2 units (i.e., entire clusters or entire subjects are assigned to treatment or control). The intra-class correlation is $\rho = \sigma_u^2 / (\sigma_u^2 + \sigma_e^2)$ and the total outcome variance is $\sigma_y^2 = \sigma_u^2 + \sigma_e^2$.

The standard error of the estimated treatment effect is $SE(\hat\beta_1) = \sigma_y \sqrt{2 (1 + (n - 1)\rho) / (J n)}$, giving a noncentrality parameter of $$\lambda = d \sqrt{J n / (2 (1 + (n - 1)\rho))}$$ under a two-sample t-test with $2J - 2$ degrees of freedom.

The factor $1 + (n - 1)\rho$ is the design effect: as the within-cluster correlation grows, the effective information per level-2 unit shrinks, so more level-2 units are needed for a given level of power.

References

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge.

Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2, 173–185. doi:10.1037/1082-989X.2.2.173

Other sample size for power: power_fisher_exact(), ss_aipe_mixed_effects(), 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_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(), ss_aipe_mixed_effects_sensitivity(), ss_power_split_plot_anova()

Author

Ken Kelley kkelley@nd.edu

Examples

# 30 individuals per cluster, ICC = 0.05, standardized effect d = 0.30, power = .80
ss_power_mixed_effects(d = 0.30, n = 30, rho = 0.05, desired_power = 0.80)
#>  term                value
#>  necessary_J_per_arm 16   
#>  total_N             960  
#>  noncentrality       2.97 
#>  actual_power        0.819

# Same effect but with much higher ICC (e.g., schools or therapists)
ss_power_mixed_effects(d = 0.30, n = 30, rho = 0.20, desired_power = 0.80)
#>  term                value
#>  necessary_J_per_arm 41   
#>  total_N             2460 
#>  noncentrality       2.85 
#>  actual_power        0.805

# Realized power with 25 level-2 units per arm
ss_power_mixed_effects(d = 0.30, n = 30, rho = 0.05, J = 25)
#>  term                value
#>  specified_J_per_arm 25   
#>  total_N             1500 
#>  noncentrality       3.71 
#>  actual_power        0.953

# directional test
ss_power_mixed_effects(d = 0.30, n = 30, rho = 0.05, desired_power = 0.80,
                         directional = TRUE)
#>  term                value
#>  necessary_J_per_arm 12   
#>  total_N             720  
#>  noncentrality       2.57 
#>  actual_power        0.801