Effects of interventions and optimal strategies in the stochastic system approach to causality

We consider the problem of defining the effect of an intervention on a time-varying risk factor or treatment for a disease or a physiological marker; we develop here the latter case. So, the system considered is $(Y,A,C)$, where $Y=(Y_t)$, is the marker process of interest, $A=A_t$ the treatment. A realistic case is that the treatment can be changed only at discrete times. In an observational study the treatment attribution law is unknown; however, the physical law can be estimated without knowing the treatment attribution law, provided a well-specified model is available. An intervention is specified by the treatment attribution law, which is thus known. Simple interventions will simply randomize the attribution of the treatment; interventions that take into account the past history will be called "strategies". The effect of interventions can be defined by a risk function $R^{\intr}=\Ee_{\intr}[L(\bar Y_{t_J}, \bar A_{t_{J}},C)]$, where $L(\bar Y_{t_J}, \bar A_{t_{J}},C)$ is a loss function, and contrasts between risk functions for different strategies can be formed. Once we can compute effects for any strategy, we can search for optimal or sub-optimal strategies; in particular we can find optimal parametric strategies. We present several ways for designing strategies. As an illustration, we consider the choice of a strategy for containing the HIV load below a certain level while limiting the treatment burden. A simulation study demonstrates the possibility of finding optimal parametric strategies.