Identification and responses to positivity violations in longitudinal studies: an illustration based on invasively mechanically ventilated ICU patients

Identification and estimation of causal effects relies on the positivity assumption, which states that there should be some positive probability of having each treatment level of interest, regardless of covariate levels. Violations of positivity can lead to both non-identified target causal parameters, estimator bias and inflated variance. We discuss reasons for and responses to positivity violations when the causal effect of interest involves treatments or exposures at multiple time points. For illustration, we use a data example involving the cumulative effect of controlling arterial oxygen tension over time on death among ICU patients receiving invasive mechanical ventilation. We distinguish between practical (by chance) violations that relate to sample size and structural violations in which certain treatment levels occur with zero probability for certain covariate levels. We focus on responses that either redefines the target population, e.g. via trimming or redefines the intervention making it more dynamic or more stochastic. Supported by a simulation study, we illustrate how these responses help restore needed positivity but also modify the target causal parameter. We further introduce an inability to intervene variable and show how such a variable will often be a time-dependent confounder and essential when addressing structural positivity violations in longitudinal settings.