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Keywords:

  • effect size;
  • confidence intervals;
  • kernel density

For a continuous outcome in a two-arm trial that satisfies normal distribution assumptions, we can transform the standardized mean difference with the use of the cumulative distribution function to be the effect size measure P(X < Y ).

This measure is already established within engineering as the reliability parameter in stress–strength models, where Y represents the strength of a component and X represents the stress the component undergoes. If X is greater than Y, then the component will fail.

In this paper, we consider the closely related effect size measure,

  • display math

This measure is also known as Somer's d, which was introduced by Somers in 1962 as an ordinal measure of association. In this paper, we explore this measure as a treatment effect size for a continuous outcome. Although the point estimates for λ are easily calculated, the interval is not so readily obtained. We compare kernel density estimation and use of bootstrap and jackknife methods to estimate confidence intervals against two further methods for estimating P(X < Y ) and their respective intervals, one of which makes no assumption about the underlying distribution and the other assumes a normal distribution. Simulations show that the choice of the best estimator depends on the value of λ, the variability within the data, and the underlying distribution of the data. Copyright © 2012 John Wiley & Sons, Ltd.