Crossover designs randomly assign each unit to receive a sequence of treatments. By comparing outcomes within the same unit, these designs can effectively eliminate between-unit variation and facilitate the identification of both instantaneous effects of current treatments and carryover effects from past treatments. They are widely used in traditional biomedical studies and are increasingly adopted in modern digital platforms. However, standard analyses of crossover designs often rely on strong parametric models, making inference vulnerable to model misspecification. This paper adopts a design-based framework to analyze general crossover designs. We make two main contributions. First, we use potential outcomes to formally define the causal estimands and assumptions on the data-generating process. For any given type of crossover design and assumptions on potential outcomes, we outline a procedure for identification and estimation, emphasizing the central role of the treatment assignment mechanism in design-based inference. Second, we unify the analysis of crossover designs using least squares, with restrictions on the coefficients and weights on the units. Based on the theory, we recommend the specification of the regression function, weighting scheme, and coefficient restrictions to assess identifiability, construct efficient estimators, and estimate variances in a unified fashion. Crucially, the least squares procedure is simple to implement, and yields not only consistent and efficient point estimates but also valid variance estimates even when the working regression model is misspecified.

翻译：交叉设计通过随机分配每个实验单元接受一系列治疗序列。通过比较同一单元内的结果，此类设计能有效消除单元间变异，并有助于识别当前治疗的即时效应与既往治疗的残留效应。该设计广泛应用于传统生物医学研究，并越来越多地被现代数字平台采纳。然而，交叉设计的标准分析常依赖于强参数模型，使得推断易受模型误设的影响。本文采用基于设计的框架分析一般性交叉设计。我们做出两项主要贡献。首先，我们利用潜在结果正式定义因果估计量及数据生成过程的假设。针对任意给定类型的交叉设计与潜在结果假设，我们概述了识别与估计的流程，强调处理分配机制在设计推断中的核心作用。其次，我们使用最小二乘法统一交叉设计的分析，并对系数与单元权重施加约束。基于该理论，我们建议通过指定回归函数、加权方案与系数约束，以统一方式评估可识别性、构建高效估计量并估计方差。关键在于，最小二乘法流程易于实施，不仅能得到一致且高效的点估计，还能在工作回归模型误设的情况下提供有效的方差估计。