Abstract

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, $y\mapsto \varrho (y)$,is strictly increasing, continuous, differentiable and odd, $\varrho (-y)=-\varrho (y)$, 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 $\varrho (y)=y$ for $-\beta \le y\le \beta$, yet the nonlinear transformation of the tails is barely perceptible as it begins at $\pm \beta$, in the sense that $1={\varrho }^{\prime }(\beta )={\varrho }^{\prime }(-\beta )$, where ${\varrho }^{\prime }(·)$ is the derivative of $\varrho (·)$. 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.

摘要

为处理-减去-控制匹配对差异提出了一种新的变换，它使它们的分布中心保持不变，但对称且平滑地变换并缩短了尾部。通过这种方式，分布的中心是可解释的、未失真的和未压缩的，而外围的异常值则清晰而明显。对差异的转换，$\mathrm{是的}\mapsto \varrho (\mathrm{是的})$, 是严格递增的、连续的、可微的和奇数的，$\varrho (-\mathrm{是的})=-\varrho (\mathrm{是的})$, 所以它在最上尾的动作反映了它在最下尾的动作。此外，分布的中心——通常是分布的 90% 或 95%——没有被变换，$\varrho (\mathrm{是的})=\mathrm{是的}$为了$-\beta \le \mathrm{是的}\le \beta$，但尾部的非线性变换几乎无法察觉，因为它开始于$\pm \beta$， 在某种意义上说$1={\varrho }^{\text{'}}(\beta )={\varrho }^{\text{'}}(-\beta )$， 在哪里${\varrho }^{\text{'}}(·)$是的导数$\varrho (·)$. 该转换应用于一项关于每日少量饮酒对 HDL 胆固醇水平影响的观察性研究。该研究有三个对照组，旨在解决特定的未测量偏差；因此，几种类型的配对差异需要协调描述，重点是未测量的偏差，而不是异常值。R 包 tailTransform 实现该方法，包含数据，并再现图形和数据分析的各个方面。