Causal Effect Models for Intention to Treat and Realistic Individualized Treatment Rules

An important class of models in causal inference are the so-called marginal structural models which model the comparison between counterfactual outcome distributions corresponding with a static treatment intervention, conditional on user supplied baseline covariates, based on observing a longitudinal data structure on a sample of n independent and identically distributed experimental units. Identification of a static treatment regimen specific outcome distribution based on observational data requires beyond the so-called sequential randomization assumption that each experimental unit has positive probability of following the static treatment regimen. The latter assumption is called the experimental treatment assignment assumption (ETA) (which is parameter specific). In most studies the ETA is violated for the static treatment interventions to be compared because some of the static treatment interventions cannot be followed by all experimental units due to baseline characteristics or due to the occurrence of certain events over time. For example, the development of side effects to the prescribed drug dose in a cancer patient, or the development of drug-resistance of an HIV-virus in an HIV-infected patient following the prescribed drug, describe situations in which a physician would be forced to stop the assigned treatment regimen. In this article a generalization of marginal structural models is proposed – called intention to treat causal effect models – which does not rely on the ETA. The definition of an intention to treat causal effect requires a user-supplied definition of a time-dependent process keeping track of the possible treatment options for an experimental unit, and, if that is not available, it may be derived from a fitted treatment mechanism. The proposed intention to treat intervention enforces the static intervention until the time point at which next treatment does not belong to the set of possible treatment options, at which point the intervention is stopped. Locally efficient estimators of the desired intention to treat causal effects are provided. In addition causal effect models for realistic individualized treatment rules are presented which always map in the set of possible treatment options and are thereby also fully identifiable from the data; in particular it is shown that these models can be chosen to generalize marginal structural models. Analogous to Murphy et al. (2001), the corresponding locally efficient double robust inverse probability of treatment weighted estimator is presented.