Simultaneous lower confidence bounds on benchmark dose profiles for joint-action continuous data

Benchmark analysis is a widely used tool in quantitative risk analysis. A core operation therein is estimation of minimum exposure levels, called Benchmark Doses (BMDs), that induce a pre-specified, adverse, Benchmark Response (BMR) in the target population or system. This approach is well understood when a single hazardous stimulus is studied. Deutsch and Piegorsch (2013) expand the estimation paradigm from the single-stimulus setting to joint-action, two-agent studies with continuous response outcomes. They define the benchmark profile (BMP) as a twodimensional analog of the single-dose BMD at which both agents achieve the specified BMR. We extend their results to show how confidence statements for the BMP can be constructed by inverting simultaneous confidence bands on the observed risk function. A simulation study reveals that these bands provide conservative coverage, but improve on previously studied methods at higher BMR-levels.