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.
ss_aipe_omega_squared(
population_omega_squared,
df_effect,
width,
which_width = c("Full", "Lower", "Upper"),
conf_level = 0.95,
assurance = NULL
)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.
Anticipated population \(\omega^2\). Must lie in \([0, 1)\).
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).
Desired full width of the CI on \(\omega^2\).
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.
Desired confidence level (default 0.95).
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).
Ken Kelley [email protected]
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 ?ci_nc_F). 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.
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. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1080/00273170802490632")
Kelley, K., & Preacher, K. J. (2012). On effect size. Psychological Methods, 17, 137--152. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1037/1082-989X.9.2.164")
ci_omega_squared, omega_squared,
omega_squared_partial, ss_aipe_R2
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 AIPE sample size planning:
ss_aipe_c_sensitivity(),
ss_aipe_cliff_delta(),
ss_aipe_cliff_delta_sensitivity(),
ss_aipe_composite_sem(),
ss_aipe_equivalence_r(),
ss_aipe_equivalence_r_sensitivity(),
ss_aipe_equivalence_smd(),
ss_aipe_equivalence_smd_sensitivity(),
ss_aipe_icc(),
ss_aipe_icc_sensitivity(),
ss_aipe_indirect_effect(),
ss_aipe_indirect_effect_sensitivity(),
ss_aipe_mixed_effects_sensitivity(),
ss_aipe_omega_squared_sensitivity(),
ss_aipe_partial_r(),
ss_aipe_partial_r_sensitivity(),
ss_aipe_pcm_sensitivity(),
ss_aipe_r(),
ss_aipe_r_sensitivity(),
ss_aipe_reliability_sensitivity(),
ss_aipe_semipartial_r(),
ss_aipe_semipartial_r_sensitivity()
# 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)
# 2. Same problem with 80% assurance:
ss_aipe_omega_squared(population_omega_squared = 0.10,
df_effect = 2,
width = 0.10,
assurance = 0.80)
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