Sample Size or Power for a Factorial Between-Subjects ANOVA Effect

Determine the necessary per-cell sample size to achieve a desired level of statistical power for a single F test (main effect or interaction) in a between-subjects factorial ANOVA, or, given a per-cell sample size, return the realized statistical power. The function handles two-way and higher-order factorial designs.

Usage

ss_power_factorial_anova(
  factor_levels,
  effect_indices,
  f = NULL,
  partial_eta_squared = NULL,
  desired_power = 0.85,
  alpha_level = 0.05,
  n_per_cell = NULL
)

Arguments

factor_levels

Integer vector giving the number of levels of each factor (e.g., c(2, 3) for a 2 x 3 design, c(2, 2, 4) for a 2 x 2 x 4 design)

effect_indices

Integer vector identifying the factors that define the effect of interest. For example, 1 requests the main effect of the first factor; c(1, 2) requests the AxB two-way interaction; c(1, 2, 3) requests the three-way interaction

f

Cohen's f effect size for the chosen effect; supply this or partial_eta_squared, but not both

partial_eta_squared

Partial eta squared for the chosen effect; supply this or f

desired_power

Desired statistical power (default 0.85)

alpha_level

Type I error rate (default 0.05)

n_per_cell

Per-cell sample size; if specified, returns the realized power

Value

A data.frame with rows for necessary_n_per_cell (or specified_n_per_cell), 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-cell count).

Details

For a between-subjects factorial design, the F statistic for the chosen effect follows a noncentral F distribution under the alternative with numerator degrees of freedom \(\prod_{i \in S} (k_i - 1)\) (where \(S\) is effect_indices and \(k_i\) is factor_levels[i]), denominator degrees of freedom \(N - K\) (where \(N = n_{cell} \prod k_i\) and \(K = \prod k_i\) is the number of cells), and noncentrality parameter \(\lambda = N f^2\). Cohen's f relates to partial eta squared via \(f = \sqrt{\eta_p^2 / (1 - \eta_p^2)}\).

The function searches over per-cell sample sizes until power reaches desired_power; when n_per_cell is supplied it returns the realized power.

For covariates, see ss_power_factorial_ancova. A complete worked three-factor example (a 2 x 4 x 3 design planned effect by effect, simulated, analyzed with Type III sums of squares, plotted, and followed up with focused contrasts) is the vignette vignette("ancova_2x4x3_power", package = "DMAR").

References

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

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

See also

ss_power_one_way_anova, ss_power_c, conf_limits_ncf

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_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_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()

Author

Ken Kelley kkelley@nd.edu

Examples

# 2 x 3 design, main effect of factor B (the 3-level factor), f = 0.25, power = .80
ss_power_factorial_anova(factor_levels = c(2, 3), effect_indices = 2,
                         f = 0.25, desired_power = 0.80)
#>  term                 value
#>  necessary_n_per_cell 27   
#>  total_N              162  
#>  effect_df            2    
#>  error_df             156  
#>  noncentrality        10.1 
#>  actual_power         0.813

# 2 x 2 design, AxB interaction, partial eta squared = 0.06, power = .80
ss_power_factorial_anova(factor_levels = c(2, 2), effect_indices = c(1, 2),
                         partial_eta_squared = 0.06, desired_power = 0.80)
#>  term                 value
#>  necessary_n_per_cell 32   
#>  total_N              128  
#>  effect_df            1    
#>  error_df             124  
#>  noncentrality        8.17 
#>  actual_power         0.81 

# 2 x 2 x 3 design, three-way interaction, f = 0.20
ss_power_factorial_anova(factor_levels = c(2, 2, 3), effect_indices = c(1, 2, 3),
                         f = 0.20, desired_power = 0.80)
#>  term                 value
#>  necessary_n_per_cell 21   
#>  total_N              252  
#>  effect_df            2    
#>  error_df             240  
#>  noncentrality        10.1 
#>  actual_power         0.814

# Realized power for n_per_cell = 20 in a 2x3 design, AxB interaction, f = 0.25
ss_power_factorial_anova(factor_levels = c(2, 3), effect_indices = c(1, 2),
                         f = 0.25, n_per_cell = 20)
#>  term                 value
#>  specified_n_per_cell 20   
#>  total_N              120  
#>  effect_df            2    
#>  error_df             114  
#>  noncentrality        7.5  
#>  actual_power         0.675