Sample Size or Power for a One-Way Repeated Measures ANOVA Omnibus F Test

Determine the necessary number of subjects to achieve a desired level of statistical power for the omnibus F test of the within-subjects factor in a one-way repeated measures ANOVA, or, given a number of subjects, return the realized statistical power.

Usage

ss_power_rm_anova(
  a,
  f = NULL,
  eta_squared = NULL,
  rho = 0,
  epsilon = 1,
  desired_power = 0.85,
  alpha_level = 0.05,
  n = NULL
)

Arguments

a: Number of measurement occasions (levels of the within-subjects factor)
f: Cohen's f effect size for the within-subjects factor (the population value); supply this or eta_squared, but not both
eta_squared: Population eta squared (proportion of variance, on the relevant scale, accounted for by the within-subjects factor); supply this or f
rho: Average correlation among the repeated measures (default 0). With rho > 0, the within-subjects test gains efficiency relative to a between-subjects analogue
epsilon: Greenhouse-Geisser / Huynh-Feldt sphericity adjustment in (0, 1] (default 1, sphericity assumed). When epsilon < 1, both numerator and denominator degrees of freedom are multiplied by epsilon
desired_power: Desired statistical power (default 0.85)
alpha_level: Type I error rate (default 0.05)
n: Number of subjects (each measured at all a occasions); if specified, returns the realized power

Value

A data.frame with rows for necessary_n_subjects (or specified_n_subjects), a, 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 number of subjects).

Details

Under the alternative hypothesis with sphericity (epsilon = 1), the within-subjects F statistic follows a noncentral F distribution with numerator df \(a - 1\), denominator df \((n - 1)(a - 1)\), and noncentrality parameter \(\lambda = n a f^2 / (1 - \rho)\), where \(f\) is Cohen's f for the within-subjects effect and \(\rho\) is the average correlation across the repeated measures (Maxwell, Delaney, & Kelley, 2027). Setting rho = 0 reduces to the between-subjects expression.

When sphericity is violated, supplying epsilon (e.g., a Greenhouse-Geisser estimate) rescales the test using the Muller-Barton convention: both numerator and denominator degrees of freedom are multiplied by epsilon, and the noncentrality parameter is likewise multiplied by epsilon. Smaller epsilon therefore reduces power and increases the necessary sample size.

References

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

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.

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_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_sc(), ss_power_sem(), ss_power_smd(), ss_power_split_plot_anova()

Author

Ken Kelley kkelley@nd.edu

Examples

# 4 measurement occasions, f = 0.25, average within-subject correlation 0.5, power = .80
ss_power_rm_anova(a = 4, f = 0.25, rho = 0.5, desired_power = 0.80)
#>  term                 value
#>  necessary_n_subjects 24   
#>  a                    4    
#>  effect_df            3    
#>  error_df             69   
#>  noncentrality        12   
#>  actual_power         0.817

# Same but with Greenhouse-Geisser epsilon = 0.75
ss_power_rm_anova(a = 4, f = 0.25, rho = 0.5, epsilon = 0.75, desired_power = 0.80)
#>  term                 value
#>  necessary_n_subjects 29   
#>  a                    4    
#>  effect_df            2.25 
#>  error_df             63   
#>  noncentrality        10.9 
#>  actual_power         0.814

# Realized power at n = 20 subjects, a = 4
ss_power_rm_anova(a = 4, f = 0.25, rho = 0.5, n = 20)
#>  term                 value
#>  specified_n_subjects 20   
#>  a                    4    
#>  effect_df            3    
#>  error_df             57   
#>  noncentrality        10   
#>  actual_power         0.729