A Boundary-Respecting Confidence Interval Method for a General Effect Size Measure in Paired Experiments

Abstract

The most general form of analysis in paired experiments investigates whether within-pair differences are stochastically positive, and the most natural measure of stochastic positivity is θ, the expected sign of Walsh averages, with values in [−1, + 1]. The estimate of this effect size measure is asymptotically equivalent to the generalized Wilcoxon signed rank test statistic. Its non null variance can be estimated, leading to formulation of Wald confidence intervals, but to find more accurate score-type, boundary-respecting intervals, some conceptual and technical difficulties must be overcome. This is achieved firstly by a transformation result stating that any continuous stochastically positive variable D can be transformed by a smooth odd function g so that g(D) and g(−D) come from a symmetric location shift model. In turn, this allows the rank method assumption that observed differences have a symmetric distribution. The further assumption that this is from an extended logistic family, covering a wide range of shapes ranging from heavy to light tailed, along with some accurate functional approximations, enables a simple iterative method to be developed for confidence intervals for the effect size measure θ. Simulation studies show that the proposed method performs better than other existing confidence interval methods.