Determines the sample size needed for the confidence interval on Cliff's (1993) \(\delta\) (and equivalently Vargha-Delaney's \(A = (\delta + 1) / 2\)) to have a desired width, using the maximum-variance bound on \(\hat\delta\) (Feng & Cliff, 2004, Equation 6, p. 324).
Usage
ss_aipe_cliff_delta(
delta,
width,
which_width = c("Full", "Lower", "Upper"),
conf_level = 0.95,
ratio = 1,
assurance = NULL
)Arguments
- delta
Anticipated population Cliff's \(\delta\); numeric scalar in \((-1, 1)\).
- width
Desired full width of the CI on \(\delta\).
- which_width
"Full"(default),"Lower", or"Upper".- conf_level
Desired confidence level. Default
0.95.- ratio
Ratio \(n_1 / n_2\) of the two group sample sizes. Default
1(balanced).- assurance
Optional. Probability that the realized CI is no wider than
width.
Value
A data.frame with rows for the recommended group
sample sizes \(n_1, n_2\), the expected CI width, and the inputs
echoed back.
Details
Maximum-variance bound. The variance of \(\hat\delta\) at a given \(\delta\) is largest in the bimodal configuration, where it equals \((1 - \delta^2)/n_b\) with \(n_b\) the bimodal group's size; for unequal groups the smaller sample size is used conservatively (Feng & Cliff, 2004, Equation 6 and following text, p. 324): $$\mathrm{Var}(\hat\delta) \;\le\; \frac{(1 - \delta^2)}{\min(n_1, n_2)}.$$ Setting the half-width of a Wald-style CI \(z_{1-\alpha/2} \sqrt{\mathrm{Var}(\hat\delta)}\) equal to the target half-width and solving gives the recommended per-group sample size. The bound is conservative; the realized CI is generally narrower than the target.
Allocation. The bound is dominated by \(\min(n_1, n_2)\),
so balanced allocation (ratio = 1) is approximately optimal
under standard conditions; unbalanced allocations require the larger
total N to achieve the same precision.
References
Cliff, N. (1993). Dominance statistics: Ordinal analyses to answer ordinal questions. Psychological Bulletin, 114(3), 494–509. doi:10.1037/0033-2909.114.3.494
Feng, D., & Cliff, N. (2004). Monte Carlo evaluation of ordinal d with improved confidence interval. Journal of Modern Applied Statistical Methods, 3(2), 322–332. doi:10.22237/jmasm/1099267560
Vargha, A., & Delaney, H. D. (2000). A critique and improvement of the CL common language effect size statistics of McGraw and Wong. Journal of Educational and Behavioral Statistics, 25(2), 101–132. doi:10.3102/10769986025002101
See also
cliff_delta, ss_aipe_partial_r,
ss_aipe_semipartial_r
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_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(),
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()
Author
Ken Kelley kkelley@nd.edu
Examples
# 1. Plan a balanced design so the 95% CI on delta has full width
# <= 0.20 when anticipating delta = 0.30.
ss_aipe_cliff_delta(delta = 0.30, width = 0.20)
#> term value
#> n_1 350
#> n_2 350
#> necessary_N 700
#> expected_width 0.2
#> delta 0.3
#> ratio 1
#> width_target 0.2
#> conf_level 0.95
#>
#> Confidence level: 95%
# 2. Unbalanced: twice as many in group 1.
ss_aipe_cliff_delta(delta = 0.30, width = 0.20, ratio = 2)
#> term value
#> n_1 700
#> n_2 350
#> necessary_N 1050
#> expected_width 0.2
#> delta 0.3
#> ratio 2
#> width_target 0.2
#> conf_level 0.95
#>
#> Confidence level: 95%