A New Transformation of Treated-Control Matched-Pair Differences for Graphical Display

A new transformation is proposed for treated-minus-control matched pair differences that leaves the center of their distribution untouched, but symmetrically and smoothly transforms and shortens the tails. In this way, the center of the distribution is interpretable, undistorted and uncompressed, yet outliers are clear and distinct along the periphery. The transformation of pair differences, y↦ϱ(y) ,is strictly increasing, continuous, differentiable and odd, ϱ(−y)=−ϱ(y) , so its action in the extreme upper tail mirrors its action in the extreme lower tail. Moreover, the center of the distribution—typically 90% or 95% of the distribution—is not transformed, with ϱ(y)=y for −β≤y≤β , yet the nonlinear transformation of the tails is barely perceptible as it begins at ±β , in the sense that 1=ϱ′(β)=ϱ′(−β) , where ϱ′(·) is the derivative of ϱ(·) . The transformation is applied to an observational study of the effect of light daily alcohol consumption on the level of HDL cholesterol. The study has three control groups intended to address specific unmeasured biases; so, several types of pair differences require coordinated depiction focused on unmeasured bias, not outliers. An R package tailTransform implements the method, contains the data, and reproduces aspects of the graphs and data analysis.