Computes the proportion of the treatment-group population that exceeds the control-group mean under bivariate normality with equal variances. This quantity is sometimes called Cohen's \(U_3\) (Cohen, 1988). Under those assumptions it equals \(\Phi(\delta)\), where \(\delta\) is the population standardized mean difference. When sample sizes are supplied, the CI on the proportion of superiority is constructed by transforming the noncentral t CI on Cohen's d via \(\Phi\), which is monotone and therefore preserves coverage exactly.
proportion_of_superiority(
smd,
n_1 = NULL,
n_2 = NULL,
conf_level = 0.95,
smd_lower = NULL,
smd_upper = NULL
)A data.frame with rows for d, the
proportion of superiority, and (when a CI is constructable) the
lower / upper limits on d and on the proportion of
superiority.
Sample standardized mean difference (Cohen's d). Numeric scalar.
Group sample sizes (required if a CI is wanted).
Confidence level for the CI. Default 0.95.
Optional pre-computed CI limits on d; when supplied directly, the function skips the noncentral t step and just transforms these limits.
Ken Kelley [email protected]
The proportion of superiority is one of three "U" indices Cohen (1988) defined; the other two (\(U_1\), the proportion of non-overlap, and \(U_2\), the proportion of either population that exceeds the same percentile in the other) can be derived from it directly: with \(U_3 = \Phi(\delta)\) for the proportion of superiority, Cohen's \(U_2 = \Phi(\delta/2)\) and \(U_1 = (2 \cdot U_2 - 1) / U_2\) (Cohen, 1988, Table 2.2.1).
Why this rather than cles. The proportion of
superiority answers the question "what fraction of the treatment
population exceeds the control-group mean,'' whereas
cles answers "what fraction of randomly drawn pairs
favor the treatment over the control.'' Both are unitless
probability-scale summaries of a Cohen's-d difference, but
the proportion of superiority is marginal while CLES is paired.
Specifically, \(\Phi(d)\) versus \(\Phi(d/\sqrt{2})\); for
\(d = 0.5\), the proportion of superiority is 0.69 and CLES is
0.64.
CI construction. Because \(\Phi(\cdot)\) is monotone,
the CI on the proportion of superiority is just
\([\Phi(d_L),\, \Phi(d_U)]\) where \([d_L,\, d_U]\) is the
noncentral t CI on d from ci_smd.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum. (See Section 2.2 for the \(U_1\), \(U_2\), and \(U_3\) indices.)
Hedges, L. V., & Olkin, I. (1985). Statistical methods for meta-analysis. Academic Press.
Kelley, K. (2007). Confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20(8), 1--24. (The noncentral t interval on the standardized mean difference that is transformed here.) tools:::Rd_expr_doi("10.18637/jss.v020.i08")
Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge.
cles, nnt_from_smd,
smd, ci_smd,
probability_of_superiority_paired
Other effect size estimates:
cles(),
cliff_delta(),
correction_for_attenuation(),
eta_squared(),
eta_squared_generalized(),
eta_squared_partial(),
expected_partial_r(),
expected_r(),
expected_smd(),
nnt_from_smd(),
omega_squared(),
omega_squared_partial(),
probability_of_superiority_paired(),
responder_analysis(),
smd_trimmed()
# 1. Proportion of superiority at three reference d values:
proportion_of_superiority(smd = 0.2)
proportion_of_superiority(smd = 0.5)
proportion_of_superiority(smd = 0.8)
# 2. With a noncentral t CI from sample sizes:
proportion_of_superiority(smd = 0.5, n_1 = 50, n_2 = 50, conf_level = 0.95)
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