Sample Size or Power for a Mixed-Effects ANOVA (Between X Within Design)

Determine the necessary per-group sample size to achieve a desired level of statistical power for one of the three F tests in a mixed-effects ANOVA with one between-subjects factor and one within-subjects factor – between-subjects main effect, within-subjects main effect, or the between x within interaction – or, given a per-group sample size, return the realized statistical power. (This design is also commonly called a split-plot factorial.)

Usage

ss_power_split_plot_anova(
  a,
  b,
  effect,
  f = NULL,
  partial_eta_squared = NULL,
  rho,
  epsilon = 1,
  desired_power = 0.85,
  alpha_level = 0.05,
  n = NULL
)

Arguments

a: Number of levels of the between-subjects factor (i.e., number of groups)
b: Number of levels of the within-subjects factor (i.e., number of measurement occasions)
effect: Which F test to compute power for: "between", "within", or "interaction"
f: Cohen's f effect size for the chosen effect (the population value); supply this or partial_eta_squared, but not both
partial_eta_squared: Partial eta squared for the chosen effect; supply this or f
rho: Average correlation among the repeated measures within a subject (must lie in (-1, 1)). Higher rho reduces power for the between-subjects test (because subject means contain more redundant information) and increases power for the within-subjects and interaction tests
epsilon: Greenhouse-Geisser / Huynh-Feldt sphericity adjustment in (0, 1] (default 1, sphericity assumed). Applied to the within-subjects and interaction tests but not the between-subjects test. Both numerator and denominator df, and the noncentrality, are multiplied by epsilon (Muller-Barton convention)
desired_power: Desired statistical power (default 0.85)
alpha_level: Type I error rate (default 0.05)
n: Per-group (between-subjects) sample size; if specified, returns the realized power

Value

A data.frame with rows for necessary_n_per_group (or specified_n_per_group), total_N, effect_df, error_df, 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 per-group count).

Details

This is a two-factor mixed-effects design: one between-subjects factor with $a$ levels and one within-subjects factor with $b$ levels; $n$ subjects are randomly assigned to each between-subjects level and each subject is measured at all $b$ within-subjects levels, for $N = na$ subjects total and $Nb$ observations. The covariance among the $b$ within-subject observations is summarized by rho, the average pairwise correlation.

The three F tests have noncentrality parameters $$\lambda_{B} = N b f^2 / (1 + (b - 1) \rho)$$ for the between-subjects test (numerator df $a - 1$, denominator df $N - a$), $$\lambda_{W} = N b f^2 \, \epsilon / (1 - \rho)$$ for the within-subjects test (numerator df $(b - 1)\epsilon$, denominator df $(N - a)(b - 1)\epsilon$), and the same form as $\lambda_W$ for the interaction (numerator df $(a - 1)(b - 1)\epsilon$, same denominator df). Cohen's f relates to partial eta squared via $f = \sqrt{\eta_p^2 / (1 - \eta_p^2)}$.

This design is the compound-symmetry (random intercept) special case of the two-level linear mixed-effects model: rho is the intraclass correlation and the $b$ occasions are the level-1 units of a subject. For two between-subjects groups the between-subjects F(1, .) test is therefore the two-level treatment t test of ss_power_mixed_effects squared, so the two planners agree on that shared case.

References

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

Muller, K. E., & Barton, C. N. (1989). Approximate power for repeated measures ANOVA lacking sphericity. Journal of the American Statistical Association, 84, 549–555.

Author

Ken Kelley kkelley@nd.edu

Examples

# 2 groups, 4 occasions, between-subjects effect, f = 0.25,
# average within-subject correlation 0.5, power = .80
ss_power_split_plot_anova(a = 2, b = 4, effect = "between", f = 0.25,
                    rho = 0.5, desired_power = 0.80)
#>  term                  value
#>  necessary_n_per_group 41   
#>  total_N               82   
#>  effect_df             1    
#>  error_df              80   
#>  noncentrality         8.2  
#>  actual_power          0.808

# Same design, within-subjects (occasion) main effect, f = 0.25
ss_power_split_plot_anova(a = 2, b = 4, effect = "within", f = 0.25,
                    rho = 0.5, desired_power = 0.80)
#>  term                  value
#>  necessary_n_per_group 12   
#>  total_N               24   
#>  effect_df             3    
#>  error_df              66   
#>  noncentrality         12   
#>  actual_power          0.816

# Same design, between x within interaction, f = 0.25
ss_power_split_plot_anova(a = 2, b = 4, effect = "interaction", f = 0.25,
                    rho = 0.5, desired_power = 0.80)
#>  term                  value
#>  necessary_n_per_group 12   
#>  total_N               24   
#>  effect_df             3    
#>  error_df              66   
#>  noncentrality         12   
#>  actual_power          0.816

# Realized power for n = 25 per group on the interaction test, partial eta^2 = 0.06
ss_power_split_plot_anova(a = 2, b = 4, effect = "interaction",
                    partial_eta_squared = 0.06, rho = 0.5, n = 25)
#>  term                  value
#>  specified_n_per_group 25   
#>  total_N               50   
#>  effect_df             3    
#>  error_df              144  
#>  noncentrality         25.5 
#>  actual_power          0.993

# Greenhouse-Geisser correction with epsilon = 0.7 on the within-subjects test
ss_power_split_plot_anova(a = 2, b = 4, effect = "within", f = 0.25,
                    rho = 0.5, epsilon = 0.7, desired_power = 0.80)
#>  term                  value
#>  necessary_n_per_group 15   
#>  total_N               30   
#>  effect_df             2.1  
#>  error_df              58.8 
#>  noncentrality         10.5 
#>  actual_power          0.808