Sample Size for AIPE on Omega Squared (ANOVA Effect Size)

Determines the sample size needed for the noncentral F confidence interval on the population omega squared (\(\omega^2\)) to have a desired width (Accuracy in Parameter Estimation; Kelley, 2008; Steiger, 2004). The function uses the same noncentral F machinery as ci_omega_squared: at each candidate \(N\) it computes the expected CI width by inverting the noncentral F distribution and stops at the smallest \(N\) that achieves the target width.

Usage

ss_aipe_omega_squared(
  population_omega_squared,
  df_effect,
  width,
  which_width = c("Full", "Lower", "Upper"),
  conf_level = 0.95,
  assurance = NULL
)

Arguments

population_omega_squared: Anticipated population \(\omega^2\). Must lie in \([0, 1)\).
df_effect: Numerator degrees of freedom for the effect (e.g., \(a - 1\) for an \(a\)-group one-way ANOVA, or the appropriate per-effect numerator df in a factorial design).
width: Desired full width of the CI on \(\omega^2\).
which_width: Whether width is the "Full" width of the interval (default) or a half-width: "Lower" and "Upper" both interpret width as half the full width, so they plan for a full width of twice width and return the same sample size. Because the interval is not generally symmetric about the estimate, its realized lower and upper half-widths can differ from each other and from half the full width; the planner does not target them separately. A genuinely one-sided width target is not currently offered.
conf_level: Desired confidence level (default 0.95).
assurance: Optional. Probability that the realized CI is no wider than width; when supplied, the sample size is inflated by the standard chi squared correction (Kelley, 2008).

Value

A data.frame with rows for the recommended total sample size necessary_N, the expected CI width at that sample size, and the inputs echoed back.

Details

Connection to noncentral F machinery. The CI on \(\omega^2\) is built by inverting the noncentral F sampling distribution of the observed F statistic, following Steiger (2004) and Kelley (2007); see ci_omega_squared. To plan a sample size, we iterate: for each candidate \(N\), compute the F the analyst would observe at the population effect size, build its CI on \(\omega^2\), and stop at the smallest \(N\) whose CI width is below the target.

Population-effect-to-F mapping. Given a target \(\omega^2\), the expected sample F that yields exactly that \(\omega^2\) as the point estimate from \(\hat\omega^2 = df_{\text{eff}}(F - 1) / [df_{\text{eff}}(F - 1) + N]\) is \(F = 1 + \omega^2 N / [df_{\text{eff}} (1 - \omega^2)]\). This is the F value used at each iteration of the search.

Tolerance behavior at small N. For small candidate N the noncentral F lower limit is often clamped to zero (see ?conf_limits_ncf). The search ignores these clamps in the iteration and reports the final clamp count, if any, as an informational message; this matches the convention in ss_aipe_R2.

References

Algina, J., Moulder, B. C., & Moser, B. K. (2002). Sample size requirements for accurate estimation of squared semi-partial correlation coefficients. Multivariate Behavioral Research, 37(1), 37–57. doi:10.1207/s15327906mbr3701_02

Kelley, K. (2007). Confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20(8), 1–24. doi:10.18637/jss.v020.i08

Kelley, K. (2008). Sample size planning for the squared multiple correlation coefficient: Accuracy in parameter estimation via narrow confidence intervals. Multivariate Behavioral Research, 43(4), 524–555. doi:10.1080/00273170802490632

Kelley, K., & Preacher, K. J. (2012). On effect size. Psychological Methods, 17, 137–152. doi:10.1037/a0028086

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 3 on \(\eta^2\), Chapter 7 on factorial designs, and Chapter 11 on generalized \(\eta^2\) for within-subjects designs.)

Steiger, J. H. (2004). Beyond the F test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis. Psychological Methods, 9(2), 164–182. doi:10.1037/1082-989X.9.2.164

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Plan total N so the 95% CI on omega^2 has full width <= 0.10
#        in a 3-group one-way ANOVA (df_effect = 2), anticipated
#        omega^2 = 0.10.
ss_aipe_omega_squared(population_omega_squared = 0.10,
                      df_effect = 2,
                      width = 0.10)
#> During the iterative sample size search, the noncentral F lower-limit clamp in conf_limits_ncf() fired in 10 intermediate evaluations.
#>  term                     value 
#>  necessary_N              473   
#>  expected_width           0.0999
#>  population_omega_squared 0.1   
#>  df_effect                2     
#>  width_target             0.1   
#>  conf_level               0.95  
#> 
#> Confidence level: 95%

# 2. Same problem with 80% assurance:
ss_aipe_omega_squared(population_omega_squared = 0.10,
                      df_effect = 2,
                      width = 0.10,
                      assurance = 0.80)
#> During the iterative sample size search, the noncentral F lower-limit clamp in conf_limits_ncf() fired in 10 intermediate evaluations.
#>  term                     value 
#>  necessary_N              499   
#>  expected_width           0.0973
#>  population_omega_squared 0.1   
#>  df_effect                2     
#>  width_target             0.1   
#>  conf_level               0.95  
#> 
#> Confidence level: 95%