Bias Correction in Non-Differentiable Estimating Equations for Optimal Dynamic Regimes
A dynamic regime is a function that takes treatment and covariate history and baseline covariates as inputs and returns a decision to be made. Robins (2004) proposed g-estimation using structural nested mean models for making inference about the optimal regime in a multi-interval trial. The method provides clear advantages over traditional parametric approaches.
Robins’ g-estimation method always yields consistent estimators, but these can be asymptotically biased under a given structural nested mean model for certain longitudinal distributions of the treatments and covariates, termed exceptional laws. In fact, under the null hypothesis of no treatment effect, every distribution constitutes an exceptional law under structural nested mean models which allow for interaction of current treatment with past treatments or covariates. This paper provides an explanation of exceptional laws and describes a new approach to g-estimation which we call Zeroing Instead of Plugging In (ZIPI). ZIPI shares all of the asymptotic properties of recursive g-estimates at non-exceptional laws while providing substantial reduction in the bias at exceptional laws when decision rule parameters are not shared across intervals.