Estimation of the effect of interventions that modify the received treatment.

Motivated by a study of surgical operating time and post-operative outcomes for lung cancer, we consider the estimation of causal effects of continuous point-exposure treatments. To investigate causality, the standard paradigm postulates a series of treatment-specific counterfactual outcomes and establishes conditions under which we may learn about them from observational study data. While many choices are possible, causal effects are typically defined in terms of variation of the mean of counterfactual outcomes in hypothetical worlds in which specific treatment strategies are 'applied' to all individuals. For example, one might compare two worlds: one where each individual receives some specific dose and a second where each individual receives some other dose. For our motivating study, defining causal effects in this way corresponds to (hypothetical) interventions that could not conceivably be implemented in the real world. In this work, we consider an alternative, complimentary framework that investigates variation in the mean of counterfactual outcomes under hypothetical treatment strategies where each individual receives a treatment dose corresponding to that actually received but modified in some pre-specified way. Quantification of this variation is defined in terms of contrasts for specific interventions as well as in terms of the parameters of a new class of marginal structural mean models. Within this framework, we propose three estimators: an outcome regression estimator, an inverse probability of treatment weighted estimator and a doubly robust estimator. We illustrate the methods with an analysis of the motivating data.