  A new class of Marginal Structural Models (MSMs), History-Restricted MSMs
(HRMSMs), was recently introduced for longitudinal data for the purpose of
defining causal parameters which may often be better suited for public health
research or at least more practicable than MSMs \citejoffe,feldman. HRMSMs
allow investigators to analyze the causal effect of a treatment on an outcome
based on a fixed, shorter and user-specified history of exposure compared to
MSMs. By default, the latter represent the treatment causal effect of interest
based on a treatment history defined by the treatments assigned between the
study's start and outcome collection. We lay out in this article the formal
statistical framework behind HRMSMs. Beyond allowing a more flexible causal
analysis, HRMSMs improve computational tractability and mitigate statistical
power concerns when designing longitudinal studies. We also develop three
consistent estimators of HRMSM parameters under sufficient model assumptions:
the Inverse Probability of Treatment Weighted (IPTW), G-computation and Double
Robust (DR) estimators. In addition, we show that the assumptions commonly
adopted for identification and consistent estimation of MSM parameters
(existence of counterfactuals, consistency, time-ordering and sequential
randomization assumptions) also lead to identification and consistent
estimation of HRMSM parameters.
