  Identifying a biomarker or treatment-dose threshold that marks a specified
level of risk is an important problem, especially in clinical trials. This
risk, viewed as a function of thresholds and possibly adjusted for covariates,
we call the threshold-response function. Extending the work of Donovan, Hudgens
and Gilbert (2019), we propose a nonparametric efficient estimator for the
covariate-adjusted threshold-response function, which utilizes machine learning
and Targeted Minimum-Loss Estimation (TMLE). We additionally propose a more
general estimator, based on sequential regression, that also applies when there
is outcome missingness. We show that the threshold-response for a given
threshold may be viewed as the expected outcome under a stochastic intervention
where all participants are given a treatment dose above the threshold. We prove
the estimator is efficient and characterize its asymptotic distribution. A
method to construct simultaneous 95% confidence bands for the
threshold-response function and its inverse is given. Furthermore, we discuss
how to adjust our estimator when the treatment or biomarker is
missing-at-random, as is the case in clinical trials with biased sampling
designs, using inverse-probability-weighting. The methods are assessed in a
diverse set of simulation settings with rare outcomes and cumulative
case-control sampling. The methods are employed to estimate neutralizing
antibody thresholds for virologically confirmed dengue risk in the CYD14 and
CYD15 dengue vaccine trials.
