  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. (2017)
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. We illustrate the
new method 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.
