Nonparametric Benchmark Dose Estimation with Continuous Dose-Response Data

Lizhen Lin, Walter W. Piegorsch, Rabi Bhattacharya

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


We propose a new method for risk-analytic benchmark dose (BMD) estimation in a dose-response setting when the responses are measured on a continuous scale. For each dose level d, the observation X(d) is assumed to follow a normal distribution: N(μ(d),σ2). No specific parametric form is imposed upon the mean μ(d), however. Instead, nonparametric maximum likelihood estimates of μ(d) and σ are obtained under a monotonicity constraint on μ(d). For purposes of quantitative risk assessment, a 'hybrid' form of risk function is defined for any dose d as R(d) = P[X(d) < c], where c > 0 is a constant independent of d. The BMD is then determined by inverting the additional risk functionRA(d) = R(d) - R(0) at some specified value of benchmark response. Asymptotic theory for the point estimators is derived, and a finite-sample study is conducted, using both real and simulated data. When a large number of doses are available, we propose an adaptive grouping method for estimating the BMD, which is shown to have optimal mean integrated squared error under appropriate designs.

Original languageEnglish (US)
Pages (from-to)713-731
Number of pages19
JournalScandinavian Journal of Statistics
Issue number3
StatePublished - Sep 1 2015


  • Benchmark analysis
  • Benchmark dose
  • Bootstrap confidence limits
  • Dose-response analysis
  • Isotonic regression
  • Model uncertainty
  • Pool-adjacent-violators algorithm
  • Quantitative responses
  • Quantitative risk assessment

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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