Measuring effect size: a non-parametric analogue of omega 2

When comparing two groups of subjects, one of the many measures of effect size is omega 2. This paper suggests a non-parametric analogue of omega 2 based on the following point of view. Given an observation from one of two groups, but not knowing whether it came from the first or second group, how certain can we be that the observation came from the first group? This is in contrast to omega 2 where, given that an observation came from a specific group, say the first group, how much does this reduce our uncertainty about the dependent variable? One problem with omega 2 is that it is not robust--it is a function of the variances--so it can be misleading for reasons reviewed in the paper. Four estimators of the proposed measure of effect size are described and compared in a simulation study. Contrary to what was expected, the .632 bootstrap estimator performed best in terms of bias and mean squared error.