In a longitudinal study of dose-response, the presence of confounding or non-compliance compromises the estimation of the true effect of a treatment. Standard regression methods cannot remove the bias introduced by patient-selected treatment level, that is, they do not permit the estimation of the causal effect of dose. Using an approach based on the Generalized Propensity Score (GPS), a generalization of the classical, binary treatment propensity score, it is possible to construct a balancing score that provides a more meaningful estimation procedure for the true (unconfounded) effect of dose. Previously, the GPS has been applied only in a single interval setting. In this paper, we extend the GPS methodology to the longitudinal setting. The methodology is applied to simulated data and two real data sets; first, we study the Riesby depression data, and secondly we present analysis of a recent study, the Monitored Occlusion Treatment of Amblyopia Study (MOTAS), which investigated the dose-response relationship in an ophthalmological setting between occlusion and improvement in visual acuity. The MOTAS study was revolutionary as it was the first to accurately measure occlusion dose received by the child.
Moodie, Erica E M and Stephens, David A., "Estimation of Dose-Response Functions for Longitudinal Data" (November 2007). COBRA Preprint Series. Working Paper 32.