There are limited options to estimate the treatment effects of variables which are continuous and measured at multiple time points, particularly if the true dose-response curve should be estimated as closely as possible. However, these situations may be of relevance: in pharmacology, one may be interested in how outcomes of people living with -- and treated for -- HIV, such as viral failure, would vary for time-varying interventions such as different drug concentration trajectories. A challenge for doing causal inference with continuous interventions is that the positivity assumption is typically violated. To address positivity violations, we develop projection functions, which reweigh and redefine the estimand of interest based on functions of the conditional support for the respective interventions. With these functions, we obtain the desired dose-response curve in areas of enough support, and otherwise a meaningful estimand that does not require the positivity assumption. We develop $g$-computation type plug-in estimators for this case. Those are contrasted with g-computation estimators which are applied to continuous interventions without specifically addressing positivity violations, which we propose to be presented with diagnostics. The ideas are illustrated with longitudinal data from HIV positive children treated with an efavirenz-based regimen as part of the CHAPAS-3 trial, which enrolled children $<13$ years in Zambia/Uganda. Simulations show in which situations a standard $g$-computation approach is appropriate, and in which it leads to bias and how the proposed weighted estimation approach then recovers the alternative estimand of interest.

翻译：针对连续且多重时间点测量的变量，现有估计其治疗效果的方法有限，尤其当需尽可能精确估计真实剂量-反应曲线时。然而此类情境具有现实意义：例如在药理学中，研究者可能关注接受HIV治疗的患者（如病毒学失败等结局）如何随时间变化的干预（如不同药物浓度轨迹）而改变。连续干预措施的因果推断面临的核心挑战在于：正向假设通常被违反。为处理正向性违反问题，我们开发了投影函数，该函数基于各干预条件支持域的函数重新加权并重新定义目标估计量。借助这些函数，我们可在支持域充足的区域获得所需剂量-反应曲线，而在其他区域则获得不依赖正向性假设的有意义估计量。针对此情形，我们开发了$g$计算型插件估计量，并将其与未专门处理正向性违反即应用于连续干预的$g$计算估计量进行对比——我们建议后者需附带诊断结果呈现。通过CHAPAS-3试验（在赞比亚/乌干达招募$<13$岁儿童）中接受依非韦伦方案治疗的HIV阳性患儿纵向数据，我们阐释了上述方法。模拟研究展示了标准$g$计算方法适用的情境，以及导致偏倚的情境，同时说明所提出的加权估计方法如何恢复替代目标估计量。