Sample Size or Power for an Unstandardized Contrast in a One-Way Between-Subjects ANOVA

Determine the necessary per-group sample size to achieve a desired level of statistical power for the test of a single planned (unstandardized) contrast in a one-way between-subjects analysis of variance, or, given a per-group sample size, return the realized statistical power.

Usage

ss_power_c(
  psi,
  c_weights,
  sigma,
  desired_power = 0.85,
  alpha_level = 0.05,
  n = NULL,
  directional = FALSE
)

Arguments

psi

The population unstandardized contrast effect, \(\psi = \sum c_j \mu_j\)

c_weights

Vector of contrast weights (must sum to zero); use fractional weights so that the positive weights sum to 1 (e.g., c(0.5, 0.5, -0.5, -0.5))

sigma

Within-group population standard deviation

desired_power

Desired statistical power (default 0.85)

alpha_level

Type I error rate (default 0.05)

n

Per-group sample size (assumed balanced); if specified, returns the realized power

directional

Logical: TRUE for a one-sided test (in the same sign as psi), FALSE (default) for a two-sided test

Value

A data.frame with rows for necessary_n_per_group (or specified_n_per_group), actual_power, and noncentral_t_parm. The result carries the dmar_ss_power class, so tidy and glance summarize it in broom convention.

Details

Under the alternative hypothesis the contrast t-statistic follows a noncentral t-distribution with degrees of freedom \(N - J\) (where \(N = n J\) is the total sample size and \(J\) the number of groups, taken as length(c_weights)) and noncentrality parameter \(\lambda = \psi / (\sigma \sqrt{\sum c_j^2 / n})\).

The function searches over per-group sample sizes \(n\) until power first reaches desired_power; when n is supplied it instead returns the realized power.

References

Lai, K., & Kelley, K. (2012). Accuracy in parameter estimation for ANCOVA and ANOVA contrasts: Sample size planning via narrow confidence intervals. British Journal of Mathematical and Statistical Psychology, 65, 350–370. doi:10.1111/j.2044-8317.2011.02029.x

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_sc, ss_power_one_way_anova, ss_power_c_ancova, ci_c, ss_aipe_c

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_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_rm_anova(), ss_power_sc(), ss_power_sem(), ss_power_smd(), ss_power_split_plot_anova()

Author

Ken Kelley kkelley@nd.edu

Examples

# Power for the contrast (Group 1 + Group 2) / 2 vs (Group 3 + Group 4) / 2
# with population contrast = 0.5, within-group sigma = 1, desired power = .80
ss_power_c(psi = 0.5, c_weights = c(0.5, 0.5, -0.5, -0.5), sigma = 1,
           desired_power = 0.80)
#>  term                  value
#>  necessary_n_per_group 32   
#>  actual_power          0.801
#>  noncentral_t_parm     2.83 

# Realized power for n = 30 per group
ss_power_c(psi = 0.5, c_weights = c(0.5, 0.5, -0.5, -0.5), sigma = 1, n = 30)
#>  term                  value
#>  specified_n_per_group 30   
#>  actual_power          0.775
#>  noncentral_t_parm     2.74