A NONPARAMETRIC DOUBLY ROBUST TEST FOR A CONTINUOUS TREATMENT EFFECT

The vast majority of literature on evaluating the significance of a treatment effect based on observational data has been confined to discrete treatments. These methods are not applicable to drawing inference for a continuous treatment, which arises in many important applications. To adjust for confounders when evaluating a continuous treatment, existing inference methods often rely on discretizing the treatment or using (possibly misspecified) parametric models for the effect curve. Recently, Kennedy et al. (J. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 (2017) 1229-1245) proposed nonparametric doubly robust estimation for a continuous treatment effect in observational studies. However, inference for the continuous treatment effect is a harder problem. To the best of our knowledge, a completely nonparametric doubly robust approach for inference in this setting is not yet available. We develop such a nonparametric doubly robust procedure in this paper for making inference on the continuous treatment effect curve. Using empirical process techniques for local U- and V-processes, we establish the test statistic's asymptotic distribution. Furthermore, we propose a wild bootstrap procedure for implementing the test in practice. In addition, we define a version of the test procedure based on sample splitting. We illustrate the new method(s) via simulations and a study of a constructed dataset relating the effect of nurse staffing hours on hospital performance. We implement our doubly robust dose response test in the R package DRDRtest on CRAN.