论文标题
成功：改进的胜利比例
Success-Odds: An improved Win-Ratio
论文作者
论文摘要
在医学各个领域的许多临床试验中，多个涉及非正常结果的终点也出现了非正常结果。在某些情况下，只能在序数或二分法量表上观察到结果。然后，通过比较了两个疗法组中的两名随机选择患者的结果，以“更好”，“相等”或“差”来评估两种疗法的成功。这些结果可以用概率$ p^ - = p（x <y）$，$ p_0 = p（x = y）$和$ p^+ = p（x〜> 〜y）$来描述。但是，对于临床医生而言，这些数量不太直观。因此，Noether（1987）引入了$λ= p^+ / p^ - $假设连续分布。 Pocock等人使用了相同的数量。 （2012）和Wang and Pocock（2016）也用于一般的非正常结果，并被称为“ win-Ratio” $λ_{WR} $。与Noether（1987）不同，Wang and Pocock（2016）明确允许数据中的联系。本手稿的目的是研究$λ_{wr} $的属性。事实证明，如果数据的准确程度较低，即包含更多的纽带，它具有更大的奇怪属性。因此，在纽带的情况下，赢得比率失去了其吸引人的财产，以描述和量化直观且可解释的治疗效果。因此，建议对$λ_{wr} =θ/（1-θ）$进行轻微的修改，即所谓的“成功odds”，其中$θ= p^ + + + + + \ frac12 p_0 $，如果$θ> \ frac12 $，则称为疗法的成功。如果没有关系，$λ_{so} $与$λ_{wr} $相同。派生的假设$λ_{so} = 1 $的测试和范围保留置信区间$λ_{so} $。通过两个反例，证明了对多种处理或分层设计的赢得比率和成功选择的概括并不简单，并且需要更详细的考虑因素。
Multiple and combined endpoints involving also non-normal outcomes appear in many clinical trials in various areas in medicine. In some cases, the outcome can be observed only on an ordinal or dichotomous scale. Then the success of two therapies is assessed by comparing the outcome of two randomly selected patients from the two therapy groups by 'better', 'equal' or 'worse'. These outcomes can be described by the probabilities $p^-=P(X<Y)$, $p_0=P(X=Y)$, and $p^+ =P(X~>~Y)$. For a clinician, however, these quantities are less intuitive. Therefore, Noether (1987) introduced the quantity $λ=p^+ / p^-$ assuming continuous distributions. The same quantity was used by Pocock et al. (2012) and by Wang and Pocock (2016) also for general non-normal outcomes and has been called 'win-ratio' $λ_{WR}$. Unlike Noether (1987), Wang and Pocock (2016) explicitly allowed for ties in the data. It is the aim of this manuscript to investigate the properties of $λ_{WR}$ in case of ties. It turns out that it has the strange property of becoming larger if the data are observed less accurately, i.e. include more ties. Thus, in case of ties, the win-ratio looses its appealing property to describe and quantify an intuitive and well interpretable treatment effect. Therefore, a slight modification of $λ_{WR} = θ/ (1-θ)$ is suggested, namely the so-called 'success-odds' where $θ=p^+ + \frac12 p_0$ is called a success of a therapy if $θ>\frac12$. In the case of no ties, $λ_{SO}$ is identical to $λ_{WR}$. A test for the hypothesis $λ_{SO}=1$ and range preserving confidence intervals for $λ_{SO}$ are derived. By two counterexamples it is demonstrated that generalizations of both the win-ratio and the success-odds to more than two treatments or to stratified designs are not straightforward and need more detailed considerations.
