Simultaneous confidence bands for Abbott-adjusted quantal response models in benchmark analysis

Abstract

We study use of a Scheffé-style simultaneous confidence band as applied to low-dose risk estimation with quantal response data. We consider two formulations for the dose-response risk function, an Abbott-adjusted Weibull model and an Abbott-adjusted log-logistic model. Using the simultaneous construction, we derive methods for estimating upper confidence limits on predicted extra risk and, by inverting the upper bands on risk, lower bounds on the benchmark dose, or BMD, at which a specific level of 'benchmark risk' is attained. Monte Carlo evaluations explore the operating characteristics of the simultaneous limits.

Benchmark Dose Profiles for Joint‐Action Quantal Data in Quantitative Risk Assessment

“…The parameterizations for were then determined by specifying response rates at the points ( x 1 , x 2 )= (0,0), (0,1), (1,0) for () and ( x 1 , x 2 )= (0,0), (0,), (0,1), (,0), (1,0), (1,1) for (); see the upper portions of each sub‐table. These represent a variety of response surfaces combined from single‐regressor risk models given previously by Bailer and Smith (1994) and Buckley and Piegorsch (2008). All parameterizations were studied over the same constraint region , with the upper limits equal to the largest dose on each scale: Ω 1 =Ω 2 = 1.…”

Information‐theoretic model‐averaged benchmark dose analysis in environmental risk assessment

“…Under a Weibull model, R( x ; β ) = γ 0 + (1 − γ 0 )[1 − exp {− e β 0 x β 1 }] = γ 0 + (1−γ 0 )[1 − exp{−exp(β 0 + β 1 log[ x ])}], we again recover an Abbott-adjusted dose-response function, now with extra risk R E ( x ) = 1 − exp{−exp(β 0 + β 1 log[ x ])} (Buckley and Piegorsch, 2008). Thus the expression for BM̂D 100BMR is once again similar to (A.6): BM̂D 100BMR = exp{(W BMR − b 0 )/b 1 }, with W BMR = log{−log(1 − BMR)}.…”

“…This has the same structure as model M6—called an Abbott-adjusted model (Buckley and Piegorsch, 2008)—with the logistic c.d.f. in M6 replaced by the standard normal c.d.f.…”

Simultaneous confidence sets for several effective doses

“…Though the focus of this paper is on logistic regression, the methods described here require only that there exists a regression line of the form with f () being some link function and an approximately normal with some covariance matrix . The simultaneous confidence sets considered in this paper immediately extend to the probit model, log logistic or even the Weibull models such as those described in Buckley and Piegorsch () and Deutsch and Piegorsch (). Indeed these methods also extend to the normal linear regression model in which the unknown variance σ 2 needs to be estimated too.…”