A Martingale Approach to Continuous Time Marginal Structural Models

Marginal structural models were introduced in order to provide estimates of causal effects from interventions based on observational studies in epidemiological research. We present a variant of the marginal structural strategy in continuous time using martingale theory and marked point processes. This offers a mathematical interpretation of marginal structural models that has not been available before. Our approach starts with a characterization of reasonable models of randomized trials in terms of local independence. Such a model gives a martingale measure that is equivalent to the observational measure. The continuous time likelihood ratio process with respect to these two probability measures corresponds to the weights in a discrete time marginal structural model. In order to do inference for the new measure, we can simulate sampling using the observed data weighted by this likelihood ratio.