Nonparametric assessment of regimen response curve estimators

In the framework of dynamic marginal structural models, regimen-response curve is a function that describes the relation between the mean outcome and the parameters in the class of decision rules. The modeling choice of the regimen-response curve is crucial in constructing an optimal regime, as a misspecified model can lead to a biased estimate with questionable causal interpretability. However, the existing literature lacks methods to evaluate and compare different working models. To address this problem, we will leverage risk to assess the "goodness-of-fit" of an imposed working model. We consider the counterfactual risk as our target parameter and derive inverse probability weighting and canonical gradients to map it to the observed data. We provide asymptotic properties of the resulting risk estimators, considering both fixed and data-dependent target parameters. We will show that the inverse probability weighting estimator can be efficient and asymptotic linear when the weight functions are estimated using a sieve-based estimator. The proposed method is implemented on the LS1 study to estimate a regimen-response curve for patients with Parkinson's disease.