# Common language effect size- place of equality

In Common language effect size (CLES) (http://en.wikipedia.org/wiki/Effect_size#Common_language_effect_size) how are cases with equality managed. Suppose there are 10 cases in each of control and treatment groups. Hence 100 pairs can be compared. If in 30 of these controls are higher, 40 of these treatment group is higher and in 30 two groups are equal, what will be effect size? Will it be 40/100 or 40/70? It seems CLES may not be suitable for data where many pairs have equal value.

## 1 Answer

You may want to read the Wikipedia page for the Mann–Whitney U test (also called the Wilcoxon rank-sum test). In essence, the U statistic of the test is based on the same idea as the "common language effect size". And the issue of ties comes up in this context as well. The solution there is to assign a value of 0.5 to ties. So, in your example, the value would be $55/100 = 0.55$. This also has intuitive appeal, since this will give a value of $0.5$ for two groups with exactly the same data (e.g., $2,4,5$ in the first group and $2,4,5$ in the second group) and two groups with only ties (e.g., $4,4,4$ in the first group and $4,4,4$ in the second group).

• Thanks for your reply. Is there any reference that CLES also uses 0.5 for ties? Which formal package of CLES should we use? Can we used CLES for both numeric and categorical data? – rnso Apr 22 '15 at 11:15
• As a reference, you could cite pretty much any book on non-parametric statistics that describes the Mann-Whitney U test. For example, Hollander & Wolfe (1999), see page 118, or Sprent & Smeeton (2000), see section 5.2.4. I don't know what you mean by "formal package". And with categorical data, how would you determine which value is larger? So, no, the CLES isn't applicable then. – Wolfgang Apr 22 '15 at 15:37
• I should have written ordinal rather than categorical. – rnso Apr 22 '15 at 15:55
• Should be okay for an ordinal scale, since you can determine unambiguously whether one data point is larger than another data point -- after all, that is all that the CLES is based on. – Wolfgang Apr 22 '15 at 17:46