Proportion of Superiority (Sometimes Called Cohen's \(U_3\))

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.

Usage

proportion_of_superiority(
  smd,
  n_1 = NULL,
  n_2 = NULL,
  conf_level = 0.95,
  smd_lower = NULL,
  smd_upper = NULL
)

Arguments

smd: Sample standardized mean difference (Cohen's d). Numeric scalar.
n_1, n_2: Group sample sizes (required if a CI is wanted).
conf_level: Confidence level for the CI. Default 0.95.
smd_lower, smd_upper: Optional pre-computed CI limits on d; when supplied directly, the function skips the noncentral t step and just transforms these limits.

Value

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.

Details

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.

References

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.) 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.

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Proportion of superiority at three reference d values:
proportion_of_superiority(smd = 0.2)
#>  term                      value
#>  smd                       0.2  
#>  proportion_of_superiority 0.579
proportion_of_superiority(smd = 0.5)
#>  term                      value
#>  smd                       0.5  
#>  proportion_of_superiority 0.691
proportion_of_superiority(smd = 0.8)
#>  term                      value
#>  smd                       0.8  
#>  proportion_of_superiority 0.788

# 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)
#>  term                      value
#>  smd                       0.5  
#>  proportion_of_superiority 0.691
#>  smd_lower                 0.101
#>  smd_upper                 0.897
#>  lower_limit               0.54 
#>  upper_limit               0.815
#> 
#> Confidence level: 95%