Causal effects are commonly defined as comparisons of the potential outcomes under treatment and control, but this definition is threatened by the possibility that either the treatment or the control condition is not well defined, existing instead in more than one version. This is often a real possibility in nonexperimental or observational studies of treatments because these treatments occur in the natural or social world without the laboratory control needed to ensure identically the same treatment or control condition occurs in every instance. We consider the simplest case: Either the treatment condition or the control condition exists in two versions that are easily recognized in the data but are of uncertain, perhaps doubtful, relevance, for example, branded Advil versus generic ibuprofen. Common practice does not address versions of treatment: Typically, the issue is either ignored or explicitly stated but assumed to be absent. Common practice is reluctant to address two versions of treatment because the obvious solution entails dividing the data into two parts with two analyses, thereby (a) reducing power to detect versions of treatment in each part, (b) creating problems of multiple inference in coordinating the two analyses, and (c) failing to report a single primary analysis that uses everyone. We propose and illustrate a new method of analysis that begins with a single primary analysis of everyone that would be correct if the two versions do not differ, adds a second analysis that would be correct were there two different effects for the two versions, controls the family-wise error rate in all assertions made by the several analyses, and yet pays no price in power to detect a constant treatment effect in the primary analysis of everyone. Our method can be applied to analyses of constant additive treatment effects on continuous outcomes. Unlike conventional simultaneous inferences, the new method is coordinating several analyses that are valid under different assumptions, so that one analysis would never be performed if one knew for certain that the assumptions of the other analysis are true. It is a multiple assumptions problem rather than a multiple hypotheses problem. We discuss the relative merits of the method with respect to more conventional approaches to analyzing multiple comparisons. The method is motivated and illustrated using a study of the possibility that repeated head trauma in high school football causes an increase in risk of early onset cognitive decline.

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两个版本的因果推论

通常将因果效应定义为对治疗和控制下潜在结果的比较，但是此定义受到以下可能性的威胁：治疗或控制条件未明确定义，而是存在多个版本。在治疗的非实验性或观察性研究中，这通常是真正的可能性，因为这些治疗发生在自然或社会世界中，而无需实验室控制以确保在每种情况下均出现相同的治疗或控制情况。我们考虑最简单的情况：治疗条件或控制条件以两种形式存在，它们很容易在数据中识别出来，但不确定性（可能是可疑的）相关性，例如Advil品牌与通用布洛芬品牌。常规做法未解决各种处理问题：通常，该问题被忽略或明确声明但假定不存在。常见的做法是不愿处理两个版本的治疗，因为明显的解决方案需要通过两次分析将数据分为两个部分，从而（a）降低检测每个部分中的版本的能力，（b）在协调中产生多重推论的问题两项分析，以及（c）无法报告使用所有人员的单一主要分析。我们提出并举例说明了一种新的分析方法，该方法从对每个人的单一主要分析开始，如果两个版本没有不同，则将是正确的；如果第二个版本具有两个不同的影响，则将添加第二个分析，将是正确的；控制由多个分析得出的所有断言中的家庭式错误率，但在对每个人的初步分析中检测不变的治疗效果均不付出任何代价。我们的方法可用于分析持续治疗对持续治疗效果的影响。与传统的同时推断不同，新方法将协调在不同假设下有效的多个分析，因此，如果一个确定地确定另一种分析的假设是正确的，则将永远不会执行一种分析。这是一个多重假设问题，而不是多重假设问题。我们讨论了该方法相对于分析多种比较的更常规方法的相对优点。