Estimation of dose-response functions for longitudinal data using the generalised propensity score

In a longitudinal study of dose-response, it is often necessary to adjust for confounding or non-compliance, which may otherwise compromise the estimation of the true effect of a treatment. Using an approach based on the generalised propensity score (GPS)--a generalisation of the classical, binary treatment propensity score--it is possible to construct a balancing score that provides an estimation procedure for the true (unconfounded) direct effect of dose on response. Previously, the GPS has been applied only in a single interval setting; in this article, we extend the GPS methodology to the longitudinal setting to estimate the direct effect of a continuous dose on a longitudinal response. The methodology is applied to two simulated examples, and a real longitudinal dose-response investigation, the Monitored Occlusion Treatment of Amblyopia Study (MOTAS). In the treatment of childhood amblyopia, a common ophthalmological condition, occlusion therapy (patching) was for many decades the standard medical treatment, despite the fact that its efficacy was not quantified. MOTAS was revolutionary, as it was the first study to obtain precise measurements of the amount of occlusion each study participant received over the course of the study.