Longitudinal Modified Treatment Policies (LMTPs) provide a framework for defining a broad class of causal target parameters for continuous and categorical exposures. We propose Local LMTPs, a generalization of LMTPs to settings where the target parameter is conditional on subsets of units defined by the treatment or exposure. Such parameters have wide scientific relevance, with well-known parameters such as the Average Treatment Effect on the Treated (ATT) falling within the class. We provide a formal causal identification result that expresses the Local LMTP parameter in terms of sequential regressions, and derive the efficient influence function of the parameter which defines its semi-parametric and local asymptotic minimax efficiency bound. Efficient semi-parametric inference of Local LMTP parameters requires estimating the ratios of functions of complex conditional probabilities (or densities). We propose an estimator for Local LMTP parameters that directly estimates these required ratios via empirical loss minimization, drawing on the theory of Riesz representers. The estimator is implemented using a combination of ensemble machine learning algorithms and deep neural networks, and evaluated via simulation studies. We illustrate in simulation that estimation of the density ratios using Riesz representation might provide more stable estimators in finite samples in the presence of empirical violations of the overlap/positivity assumption.

翻译：纵向修正处理策略（LMTPs）为定义连续和分类暴露因素的广泛因果目标参数提供了框架。本文提出局部LMTPs，将LMTPs推广至目标参数以处理或暴露定义的子集单元为条件的情形。此类参数具有广泛科学意义，其中常见的处理组平均处理效应（ATT）即属于该类参数范畴。我们通过序贯回归形式表达局部LMTP参数，给出形式化因果识别结果，并推导该参数的有效影响函数，该函数定义了其半参数及局部渐近极小极大效率界。局部LMTP参数的有效半参数推断需估计复杂条件概率（或密度）函数的比值。我们提出一种基于经验损失最小化直接估计所需比值的局部LMTP参数估计器，其理论依据为Riesz表示理论。该估计器融合集成机器学习算法与深度神经网络实现，并通过模拟研究进行评估。模拟结果表明，在重叠/正定性假设经验性违背的情况下，采用Riesz表示估计密度比可在有限样本中提供更稳定的估计量。