Computes the common-language (CL) effect size for two independent groups, defined as the probability that a randomly drawn observation from group 1 exceeds a randomly drawn observation from group 2 under bivariate normality with equal variances: $$\mathrm{CL} \;=\; \Pr(Y_1 > Y_2) \;=\; \Phi\!\bigl(\delta / \sqrt{2}\bigr),$$ where \(\delta\) is the population standardized mean difference and \(\Phi\) is the standard normal cumulative distribution function. When sample sizes are supplied, the confidence interval on CL is constructed by transforming the noncentral t-based CI on Cohen's d (Steiger & Fouladi, 1997; Kelley, 2007) through \(\Phi(\cdot / \sqrt{2})\), which is monotone-increasing so the coverage probability is preserved exactly. This is preferred over the normal-approximation CI on CL commonly seen in applied work (Brooks, Dalal, & Nolan, 2014).
Arguments
- smd
Sample standardized mean difference (Cohen's d); a numeric scalar. Positive means group 1 exceeds group 2.
- n_1, n_2
Sample sizes in the two groups; both required when a confidence interval on CL is desired.
- conf_level
Confidence level for the CI. Default
0.95.- smd_lower, smd_upper
Optional pre-computed confidence limits on d. If supplied, these are used directly and the noncentral computation is skipped.
Value
A data.frame with rows for the point estimate
(cl) and, when sample sizes are supplied, the lower and upper
CI limits. The d-equivalent of each row is also reported for
transparency.
Details
The common-language idea extends to other effect sizes; the common language effect size for correlations is developed by Liu, Carlson, and Kelley (2019).
Background. McGraw & Wong (1992) introduced the CL effect size to make Cohen's d more interpretable: instead of "the means differ by 0.5 SD," one can say "in 64 treated person scores higher than the control person." Under bivariate normality with equal variances, the population probability \(\Pr(Y_1 > Y_2)\) equals \(\Phi(\delta/\sqrt{2})\), where \(\delta = (\mu_1 - \mu_2)/\sigma\) (McGraw & Wong, 1992).
Connection to other measures. CL is identical to the AUC
(Area Under the Curve) interpretation of d in receiver-operating
analysis. Vargha & Delaney (2000) generalized CL to the nonparametric
setting (their A measure) by replacing the population p
with its empirical Mann-Whitney estimate; under bivariate normality
the two coincide. The success-rate-difference and number-needed-to-
treat scales (Kraemer & Kupfer, 2006; see nnt_from_smd)
are linear transformations of CL: \(\mathrm{SRD} = 2 \mathrm{CL} - 1\).
Confidence interval construction. Because \(\Phi(\cdot/\sqrt{2})\) is monotone-increasing, the CI on CL is obtained by transforming the CI on d: \([\Phi(d_L/\sqrt 2),\, \Phi(d_U/\sqrt 2)]\). This is an exact-coverage interval (under the noncentral t sampling model) and is more accurate than the normal-approximation CI on CL that uses a Wald-style variance for \(\hat p\) (Brooks, Dalal, & Nolan, 2014).
References
Brooks, M. E., Dalal, D. K., & Nolan, K. P. (2014). Are common language effect sizes easier to understand than traditional effect sizes? Journal of Applied Psychology, 99(2), 332–340. doi:10.1037/a0034745
Kelley, K. (2007). Confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20(8), 1–24. doi:10.18637/jss.v020.i08
Kraemer, H. C., & Kupfer, D. J. (2006). Size of treatment effects and their importance to clinical research and practice. Biological Psychiatry, 59(11), 990–996. doi:10.1016/j.biopsych.2005.09.014
Liu, X. S., Carlson, R., & Kelley, K. (2019). Common language effect size for correlations. The Journal of General Psychology, 146(3), 325–338. doi:10.1080/00221309.2019.1585321
McGraw, K. O., & Wong, S. P. (1992). A common language effect size statistic. Psychological Bulletin, 111(2), 361–365. doi:10.1037/0033-2909.111.2.361
Steiger, J. H., & Fouladi, R. T. (1997). Noncentrality interval estimation and the evaluation of statistical methods. In L. L. Harlow, S. A. Mulaik, & J. H. Steiger (Eds.), What if there were no significance tests? (pp. 221–257). Mahwah, NJ: Lawrence Erlbaum.
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
Other effect size estimates:
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(),
proportion_of_superiority(),
responder_analysis(),
smd_trimmed()
Author
Ken Kelley kkelley@nd.edu
Examples
# 1. Point estimate only.
cles(smd = 0.5)
#> term value
#> smd 0.5
#> cl 0.638
# 2. With a noncentral t CI from sample sizes (preferred):
cles(smd = 0.5, n_1 = 50, n_2 = 50, conf_level = 0.95)
#> term value
#> smd 0.5
#> cl 0.638
#> smd_lower 0.101
#> smd_upper 0.897
#> cl_lower 0.528
#> cl_upper 0.737
#>
#> Confidence level: 95%
# 3. With a pre-computed CI on d:
cles(smd = 0.5, smd_lower = 0.20, smd_upper = 0.80)
#> term value
#> smd 0.5
#> cl 0.638
#> smd_lower 0.2
#> smd_upper 0.8
#> cl_lower 0.556
#> cl_upper 0.714
#>
#> Confidence level: 95%
# 4. CL at three reference d values:
cles(smd = 0.2)
#> term value
#> smd 0.2
#> cl 0.556
cles(smd = 0.5)
#> term value
#> smd 0.5
#> cl 0.638
cles(smd = 0.8)
#> term value
#> smd 0.8
#> cl 0.714