Sample Size Planning for Power in Factorial ANCOVA

Power and sample size for any effect (a main effect or any interaction) in a between-subjects factorial design with covariates: the analysis of covariance generalization of ss_power_factorial_anova. Covariates earn their keep by absorbing error variance: with a joint squared multiple correlation \(R^2\) between the covariates and the outcome within cells, the error variance falls by the factor \(1 - R^2\), so an effect size \(f\) defined on the original (ANOVA) metric grows to \(f / \sqrt{1 - R^2}\) in the covariate-adjusted analysis, at the price of one error degree of freedom per covariate.

Usage

ss_power_factorial_ancova(
  factor_levels,
  effect_indices,
  f = NULL,
  partial_eta_squared = NULL,
  covariate_R2 = 0,
  n_covariates = 0,
  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, for example c(2, 4, 3) for a 2 x 4 x 3 design.
effect_indices: Integer vector identifying the factors that define the effect of interest: 1 for the first factor's main effect, c(2, 3) for the B x C interaction, and so on.
f: Cohen's \(f\) for the chosen effect on the unadjusted (ANOVA) metric, that is, with the within-cell standard deviation of the outcome in its denominator. Supply this or partial_eta_squared, not both.
partial_eta_squared: Partial eta squared for the chosen effect on the unadjusted metric.
covariate_R2: Joint squared multiple correlation between the covariates and the outcome within cells, in \([0, 1)\). 0 reproduces the ANOVA analysis (with the covariate degrees of freedom still spent if n_covariates > 0).
n_covariates: Number of covariates, a non-negative integer.
desired_power: Desired power; the per-cell sample size is solved when n_per_cell is NULL. Defaults to 0.85 to match ss_power_factorial_anova.
alpha_level: Type I error rate.
n_per_cell: Per-cell sample size; when supplied, the realized power at that size is returned instead of solving for size.

Value

A data.frame with the per-cell and total sample sizes (or the supplied ones), the realized actual_power, the numerator and error degrees of freedom, the unadjusted and covariate-adjusted effect sizes (f, f_adjusted), covariate_R2, n_covariates, the noncentrality parameter, and alpha_level. 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

The test of an effect with numerator degrees of freedom \(\mathit{df}_h\) (the product of the involved factors' levels each minus one) is a noncentral F with noncentrality \(\lambda = N f_{\mathrm{adj}}^2\), where \(N\) is the total sample size, \(f_{\mathrm{adj}} = f / \sqrt{1 - R^2}\), and error degrees of freedom \(N - (\prod \mathrm{levels}) - q\) for \(q\) covariates (the standard one-line ANCOVA adjustment; Maxwell, Delaney, & Kelley, 2027, Chapter 9). The covariate slopes are assumed homogeneous across cells and the covariates measured at baseline, so that adjusting does not bias the treatment effects in a randomized design.

The complete worked example for this function, a 2 x 4 x 3 ANCOVA with two baseline covariates, planned effect by effect and then analyzed with Type III sums of squares, interaction plots, and focused follow-up contrasts, is the “Power for factorial ANCOVA” 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 Chapter 9 on designs with covariates.)

Author

Ken Kelley kkelley@nd.edu

Examples

# A 2 x 4 x 3 design, planning the f = .10 main effect of the
# two-level factor A. Two baseline covariates with a modest joint
# R^2 = .25 cut the required total N by roughly a quarter:
ss_power_factorial_ancova(factor_levels = c(2, 4, 3), effect_indices = 1,
                          f = 0.10, covariate_R2 = 0,    n_covariates = 0,
                          desired_power = 0.80)
#>  term                 value
#>  necessary_n_per_cell 33   
#>  total_N              792  
#>  actual_power         0.803
#>  df_effect            1    
#>  df_error             768  
#>  f                    0.1  
#>  f_adjusted           0.1  
#>  covariate_R2         0    
#>  n_covariates         0    
#>  noncentrality        7.92 
#>  alpha_level          0.05 
ss_power_factorial_ancova(factor_levels = c(2, 4, 3), effect_indices = 1,
                          f = 0.10, covariate_R2 = 0.25, n_covariates = 2,
                          desired_power = 0.80)
#>  term                 value
#>  necessary_n_per_cell 25   
#>  total_N              600  
#>  actual_power         0.806
#>  df_effect            1    
#>  df_error             574  
#>  f                    0.1  
#>  f_adjusted           0.115
#>  covariate_R2         0.25 
#>  n_covariates         2    
#>  noncentrality        8    
#>  alpha_level          0.05 

# Realized power for the three-way interaction at 6 per cell.
ss_power_factorial_ancova(factor_levels = c(2, 4, 3),
                          effect_indices = c(1, 2, 3), f = 0.15,
                          covariate_R2 = 0.25, n_covariates = 2,
                          n_per_cell = 6)
#>  term                 value
#>  specified_n_per_cell 6    
#>  total_N              144  
#>  actual_power         0.276
#>  df_effect            6    
#>  df_error             118  
#>  f                    0.15 
#>  f_adjusted           0.173
#>  covariate_R2         0.25 
#>  n_covariates         2    
#>  noncentrality        4.32 
#>  alpha_level          0.05