Assessing dynamic treatment strategies

Of interest here are sequential data-gathering and decision-making processes, started in the preceding chapter in this volume. The archetypical context is that of a sequence of medical decisions, taken at different time points during the follow-up of the patient, each decision involving choice of a treatment in the light of any interim responses or adverse reactions to earlier treatments. The problem consists of predicting the consequences that a (possibly new and possibly hypothetical) treatment plan will have on a future patient, on the basis of what we have learnt from the performances of past medical decision makers on past patients. While we make constant reference to a medical application context, the scope of the method is of much broader relevance. Our approach to this problem owes immensely to the seminal work of James Robins. In a rich series of papers, Robins (1986, 1987, 1988, 1989, 1992, 1997, 2000, 2004), Robins and Wasserman (1997) and Robins et al. (1999) introduce the idea of different treatment strategies applied in different hypothetical studies (these studies being analogous to our notion of ?regimes'). These papers examine conditions on the relationships between these studies, under which those treatment strategies can be compared. Among these, the sequential ran-domization condition is closely related to the ?stability' assumption used in this chapter. Furthermore, Robins (1986) introduced the G-computation algorithm (a special version of dynamic programming) to evaluate a sequential treatment strategy, which works where stan-dard regression ? even under the sequential randomization assumption ? fails. Most of the concepts expounded in this chapter have a counterpart in Robins' work, although, in the interest of readability, we shall frequently abstain from making the relationships with his work explicit. We shall assume that we have an idea of which variables informed past decisions, without assuming that these variables have all been observed in the past. Nor shall we assume that the rules that governed past decisions are discernible or similar to those that will guide future actions. In contrast with the previous chapter, we do use (fully stochastic) causal diagrams (more precisely, influence diagrams). In the context of sequential plan identification, graphical rep-resentations of causality were first advocated by Pearl and Robins (1995). In accord with these authors, we assume that the relevant causal knowledge is encoded in the form of a diagram with a completely specified topology, whose associated numerical conditional probabilities are given only for a subset of its variables, so-called observed variables. The remaining variables in the diagram are ?unobserved' or ?latent'. We shall, however, introduce a form of graphical representation of causality that differs from Pearl and Robins's diagrams in some respects. Part of the problem will be to characterize situations in which the estimation of the causal effects of the treatment plan of interest is not invalidated by unobserved confounding introduced by the latent variables. Throughout this chapter we use previous results by Dawid and Didelez (2010). Elja Arjas, both in his chapter (Chapter 7) in this volume and in the joint paper Arjas and Parner (2004), considers two probability models. One (called the ?obs' model) models the observational study that has generated the data. The other (called the ?ex' model) models the consequences of the future (perhaps hypothetical) application of the treatment plan of interest. In our approach, this distinction is embodied in a decision parameter, called the ?regime indicator', which indicates the particular regime, ?obs' or ?ex', in operation. Supplementing the set of domain variables with the mentioned regime indicator results in an ?augmented' set of variables. In our approach, conditions for the equivalence of Arjas's ?obs' and ?ex' distributions are phrased in terms of conditional independence conditions on this augmented set. Both chapters, ours and that of Arjas, avoid formulations based on ideas of potential outcomes or counterfactuals. We illustrate the methods and their motivations with the aid of two examples. Following Robins, we choose the first example in the area of the treatment of HIV infection. The second example will be in the treatment of abdominal aortic aneurysm. This chapter is about the identifiability of treatment plan effects, rather than about the methods of estimation and computation one is supposed to use if the problem turns out to be identifiable. The latter problems are discussed in greater detail in Chapter 17 in this volume, by Daniel, De Stavola and Cousens, who illustrate their method with the aid of the same HIV example we use here.