Defining and estimating causal effects for ordinal data is challenging. Standard average treatment effects are not appropriate for ordinal scales, and alternative estimands, such as the probabilities that the treatment outcome exceeds or does not worsen the control outcome, are generally not identifiable. Existing work provides sharp bounds for these quantities based only on marginal distributions. Motivated by a previous observation showing that bounds obtained under an independence working assumption can be substantially tighter, we investigate conditions under which such bounds are valid. We show that commonly used notions of positive dependence, including positive quadrant dependence and positive regression dependence, are not sufficient to justify these bounds. We then propose a new dependence condition, diagonal tail dominance (DTD), under which the independence-based bounds are guaranteed to hold. We explain why this condition is quite strong and may not be appropriate in many settings, limiting the justification for using the independence-based bounds. However, local DTD may be plausible in many applications, and we derive improved bounds that exploit an independence working assumption on selected parts of the probability table. Through theoretical results, numerical examples, and an analysis of data from a clinical trial of a new treatment for acute ischemic stroke, we illustrate the properties of the bounds and the role of the proposed conditions.

翻译：定义和估计有序数据的因果效应具有挑战性。标准的平均处理效应不适用于有序量表，而替代性估计量，例如处理结果优于或未劣于对照结果的概率，通常无法识别。现有研究仅基于边际分布为这些量提供了严格边界。受先前观察（即在独立工作假设下获得的边界可能显著更紧）的启发，我们研究了这些边界成立的条件。我们证明，常用的正相依概念（包括正象限相依和正回归相依）不足以验证这些边界。随后，我们提出了一种新的相依条件——对角尾部优势（DTD），在该条件下，基于独立的边界得以保证成立。我们解释了为何这一条件相当严格，可能在许多场景中不适用，从而限制了基于独立边界的使用合理性。然而，局部DTD在许多应用中可能是合理的，我们推导了改进的边界，该边界利用了概率表选定部分上的独立工作假设。通过理论结果、数值示例以及对一项急性缺血性卒中新疗法临床试验数据的分析，我们展示了这些边界的性质及所提条件的作用。