A martingale approach to continuous-time marginal structural models

TL;DR

This paper introduces a martingale-based mathematical framework for continuous-time marginal structural models, linking causal inference in observational studies to Girsanov's change of measure, enabling simulation of randomized trial data.

Contribution

It provides a novel continuous-time interpretation of marginal structural models using martingale measures and Girsanov's theorem, enhancing causal inference methods.

Findings

Mathematical interpretation of marginal structural models via Girsanov's change of measure

Identification of conditions for absolute continuity between observational and randomized trial measures

Method for simulating randomized trial data using weighted observational data

Abstract

Marginal structural models were introduced in order to provide estimates of causal effects from interventions based on observational studies in epidemiological research. The key point is that this can be understood in terms of Girsanov's change of measure. This offers a mathematical interpretation of marginal structural models that has not been available before. We consider both a model of an observational study and a model of a hypothetical randomized trial. These models correspond to different martingale measures -- the observational measure and the randomized trial measure -- on some underlying space. We describe situations where the randomized trial measure is absolutely continuous with respect to the observational measure. The resulting continuous-time likelihood ratio process with respect to these two probability measures corresponds to the weights in discrete-time marginal structural models. In order to do inference for the hypothetical randomized trial, we can simulate samples using observational data weighted by this likelihood ratio.