DMAR (version 1.0.0)

ss_aipe_cliff_delta: Sample Size for AIPE on Cliff's \(\delta\)

Description

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
)

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.

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.

Author

Ken Kelley [email protected]

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. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_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()

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)

# 2. Unbalanced: twice as many in group 1.
ss_aipe_cliff_delta(delta = 0.30, width = 0.20, ratio = 2)