A random set approach to confidence regions with applications to the effective dose with combinations of agents

The effective dose (ED) is the pharmaceutical dosage required to produce a therapeutic response in a fixed proportion of the patients. When only one drug is considered, the problem is a univariate one and has been well‐studied. However, in the multidimensional setting, that is, in the presence of combinations of agents, estimation of the ED becomes more difficult. This study is focused on the plug‐in logistic regression estimator of the multidimensional ED. We discuss consistency of such estimators and focus on the problem of simultaneous confidence regions. We develop a bootstrap algorithm to estimate confidence regions for the multidimensional ED. Through simulation, we show that the proposed method gives 95% confidence regions, which have better empirical coverage than the previous method for moderate to large sample sizes. The novel approach is illustrated on a cytotoxicity study on the effect of two toxins in the leukemia cell line HL‐60 and a decompression sickness study of the effects of the duration and depth of the dive. Copyright © 2014 John Wiley & Sons, Ltd.