Computes the asymptotic variance of the Algina-Keselman-Penfield (2005) robust standardized mean difference under the Yuen (1974) trimmed-mean framework, suitable for AIPE sample size planning for robust effect sizes (Keselman, Algina, Lix, Wilcox, & Deering, 2008).
var_smd_trimmed(population_smd_trimmed, n_1, n_2, trim = 0.2)A 1-row data.frame with columns term
("var_smd_trimmed") and value (the variance).
Anticipated population value of the robust trimmed SMD \(\delta_R\).
Per-group sample sizes.
Proportion to trim and Winsorize. Default 0.20.
Ken Kelley [email protected]
Variance formula. Under random sampling with trimming proportion \(\gamma\) from each tail, the variance of the trimmed-mean difference scales by \(1 / h_j\) (where \(h_j = n_j - 2 \lfloor \gamma n_j \rfloor\) is the number of retained observations in group \(j\)) rather than \(1 / n_j\). The large-sample variance of the standardized version, written on the \(d_R\) scale, is $$\mathrm{Var}(\hat d_R) \;\approx\; \frac{h_1 + h_2}{h_1 h_2} + \frac{\delta_R^2}{2 (h_1 + h_2)}.$$ For \(\gamma = 0\) this reduces to the standard Hedges-Olkin (1985) variance of Cohen's d.
When to use. For AIPE planning of a robust effect size
study, use var_smd_trimmed() in place of
var_smd(). Pair with smd_trimmed() for
the point estimate and noncentral t CI.
Algina, J., Keselman, H. J., & Penfield, R. D. (2005). An alternative to Cohen's standardized mean difference effect size: A robust parameter and confidence interval in the two independent groups case. Psychological Methods, 10(3), 317--328. tools:::Rd_expr_doi("10.1037/1082-989X.10.3.317")
Kelley, K., & Rausch, J. R. (2006). Sample size planning for the standardized mean difference: Accuracy in parameter estimation via narrow confidence intervals. Psychological Methods, 11(4), 363--385. tools:::Rd_expr_doi("10.1037/1082-989X.11.4.363")
Keselman, H. J., Algina, J., Lix, L. M., Wilcox, R. R., & Deering, K. N. (2008). A generally robust approach for testing hypotheses and setting confidence intervals for effect sizes. Psychological Methods, 13(2), 110--129. tools:::Rd_expr_doi("10.1037/1082-989X.13.2.110")
Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 4 on individual comparisons and Chapter 3 on one-way ANOVA.)
Yuen, K. K. (1974). The two-sample trimmed t for unequal population variances. Biometrika, 61(1), 165--170.
smd_trimmed, var_smd,
ss_aipe_smd
Other variance utilities:
var_alpha(),
var_cv(),
var_ete(),
var_indirect_effect(),
var_omega_squared(),
var_r(),
var_smd()
# 1. Population delta_R = 0.5, n = 30 per group, 20% trim:
var_smd_trimmed(population_smd_trimmed = 0.5, n_1 = 30, n_2 = 30)
# 2. Variance scales by h / n via the trimming proportion:
var_smd_trimmed(0.5, 30, 30, trim = 0.00)$value
var_smd_trimmed(0.5, 30, 30, trim = 0.20)$value
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