Estimating effects of longitudinal modified treatment policies (LMTPs) on rates of change in health outcomes

Longitudinal data often contains outcomes measured at multiple visits and scientific interest may lie in quantifying the effect of an intervention on an outcome's rate of change. For example, one may wish to study the progression (or trajectory) of a disease over time under different hypothetical interventions. We extend the longitudinal modified treatment policy (LMTP) methodology introduced in D\'iaz et al. (2023) to estimate effects of complex interventions on rates of change in an outcome over time. We exploit the theoretical properties of a nonparametric efficient influence function (EIF)-based estimator to introduce a novel inference framework that can be used to construct simultaneous confidence intervals for a variety of causal effects of interest and to formally test relevant global and local hypotheses about rates of change. We demonstrate the utility of our framework in investigating whether a longitudinal shift intervention affects an outcome's counterfactual trajectory, as compared with no intervention. We present results from a simulation study to illustrate the performance of our inference framework in a longitudinal setting with time-varying confounding and a continuous exposure. We also apply our inference framework to the Columbia Brain Health DataBank (CBDB) to examine the effect of shifting blood pressure on the progression of dementia.