  It is often of interest in observational studies to measure the causal effect
of a treatment on time-to-event outcomes. In a medical setting, observational
studies commonly involve patients who initiate medication therapy and others
who do not, and the goal is to infer the effect of medication therapy on time
until recovery, a pre-defined level of improvement, or some other time-to-event
outcome. A difficulty with such studies is that the notion of a medication
initiation time does not exist in the control group. We propose an approach to
infer causal effects of an intervention in longitudinal observational studies
when the time of treatment assignment is only observed for treated units and
where treatment is given by indication. We present a framework for
conceptualizing an underlying randomized experiment in this setting based on
separating the process that governs the time of study arm assignment from the
mechanism that determines the assignment. Our approach involves inferring the
missing times of assignment followed by estimating treatment effects. This
approach allows us to incorporate uncertainty about the missing times of study
arm assignment, which induces uncertainty in both the selection of the control
group and the measurement of time-to-event outcomes for these controls. We
demonstrate our approach to study the effects on mortality of inappropriately
prescribing phosphodiesterase type 5 inhibitors (PDE5Is), a medication
contraindicated for groups 2 and 3 pulmonary hypertension, using administrative
data from the Veterans Affairs (VA) health care system.
