Non-parametric estimators of a monotonic dose-response curve and bootstrap confidence intervals

In this paper we consider study designs which include a placebo and an active control group as well as several dose groups of a new drug. A monotonically increasing dose-response function is assumed, and the objective is to estimate a dose with equivalent response to the active control group, including a confidence interval for this dose. We present different non-parametric methods to estimate the monotonic dose-response curve. These are derived from the isotonic regression estimator, a non-negative least squares estimator, and a bias adjusted non-negative least squares estimator using linear interpolation. The different confidence intervals are based upon an approach described by Korn, and upon two different bootstrap approaches. One of these bootstrap approaches is standard, and the second ensures that resampling is done from empiric distributions which comply with the order restrictions imposed. In our simulations we did not find any differences between the two bootstrap methods, and both clearly outperform Korn's confidence intervals. The non-negative least squares estimator yields biased results for moderate sample sizes. The bias adjustment for this estimator works well, even for small and moderate sample sizes, and surprisingly outperforms the isotonic regression method in certain situations.