## Abstract

Robins et al. (2004, Comparative Quantification of Health Risks: Global and Regional Burden of Disease Attributable to Selected Major Risk Factors. Geneva: World Health Organization) introduced the extended g-formula to estimate from observational data the risk of failure under hypothetical interventions wherein a subject’s treatment at time *k* is assigned based on the natural value of treatment at *k*; that is, the value of treatment that would have been observed at *k* were the intervention discontinued right before *k*. Several authors have parametrically applied the extended g-formula to estimate long-term failure risk under hypothetical interventions on time-varying behaviors in observational studies. For example, Taubman et al. (2009, Intervening on risk factors for coronary heart disease: an application of the parametric g-formula. International Journal of Epidemiology, 380(6):1599–1611) used this approach to estimate the 20-year risk of coronary heart disease in the Nurses’ Health Study under the hypothetical intervention “If a subject’s natural value of exercise by the end of day *k* is less than 30 minutes, set her exercise on day *k* to exactly 30 minutes; otherwise, do not intervene on her on that day”. Non-parametrically, the extended g-formula differs from the (non-extended) g-formula of Robins (1986, A new approach to causal inference in mortality studies with a sustained exposure period: application to the healthy worker survivor effect. Mathematical Modelling, 7:1393–1512) in that it is a function of (i) a user-specified intervention depending on the natural value of treatment and (ii) the distribution of natural treatment itself. Richardson and Robins (2013, http://www.csss.washington.edu/Papers/) recently defined a sufficient condition such that the extended g-formula may identify risk under an intervention that depends on the natural value of treatment, provided this expression is well-defined. In this paper, we complement this result by showing that the extended g-formula associated with an intervention depending on the natural value of treatment is algebraically equivalent to the (non-extended) g-formula associated with a particular random dynamic regime that does not depend on this value. Using previous results for random dynamic regimes, we show that this equivalence immediately gives a sufficient positivity condition that guarantees the extended g-formula is well-defined as well as semi-parametric alternatives to the parametric extended g-formula for estimation. Finally, given a hypothetical intervention that depends on the natural value of treatment, we define a plausible (implementable) approximation to this hypothetical intervention along with an untestable assumption that gives exact equivalence.

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