A Principled Approach to Adjust for Unmeasured Time-Stable Confounding of Supervised Treatment

We propose a novel method to adjust for unmeasured time-stable confounding when the time between consecutive treatment administrations is fixed. We achieve this by focusing on a new-user cohort. Furthermore, we envisage that all time-stable confounding goes through the potential time on treatment as dictated by the disease condition at the initiation of treatment. Following this logic, we may eliminate all unmeasured time-stable confounding by adjusting for the potential time on treatment. A challenge with this approach is that right censoring of the potential time on treatment occurs when treatment is terminated at the time of the event of interest, for example, if the event of interest is death. We show how this challenge may be solved by means of the expectation-maximization algorithm without imposing any further assumptions on the distribution of the potential time on treatment. The usefulness of the methodology is illustrated in a simulation study. We also apply the methodology to investigate the effect of depression/anxiety drugs on subsequent poisoning by other medications in the Danish population by means of national registries. We find a protective effect of treatment with selective serotonin reuptake inhibitors on the risk of poisoning by various medications (1- year risk difference of approximately $-3\%$) and a standard Cox model analysis shows a harming effect (1-year risk difference of approximately $2\%$), which is consistent with what we would expect due to confounding by indication. Unmeasured time-stable confounding can be entirely adjusted for when the time between consecutive treatment administrations is fixed.