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Parameters of a Dose‐Response Model Are on the Boundary: What Happens with BMDL?

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  • Leonid Kopylev
  • John Fox

Abstract

It is well known that, under appropriate regularity conditions, the asymptotic distribution for the likelihood ratio statistic is χ2. This result is used in EPA's benchmark dose software to obtain a lower confidence bound (BMDL) for the benchmark dose (BMD) by the profile likelihood method. Recently, based on work by Self and Liang, it has been demonstrated that the asymptotic distribution of the likelihood ratio remains the same if some of the regularity conditions are violated, that is, when true values of some nuisance parameters are on the boundary. That is often the situation for BMD analysis of cancer bioassay data. In this article, we study by simulation the coverage of one‐ and two‐sided confidence intervals for BMD when some of the model parameters have true values on the boundary of a parameter space. Fortunately, because two‐sided confidence intervals (size 1–2α) have coverage close to the nominal level when there are 50 animals in each group, the coverage of nominal 1−α one‐sided intervals is bounded between roughly 1–2α and 1. In many of the simulation scenarios with a nominal one‐sided confidence level of 95%, that is, α= 0.05, coverage of the BMDL was close to 1, but for some scenarios coverage was close to 90%, both for a group size of 50 animals and asymptotically (group size 100,000). Another important observation is that when the true parameter is below the boundary, as with the shape parameter of a log‐logistic model, the coverage of BMDL in a constrained model (a case of model misspecification not uncommon in BMDS analyses) may be very small and even approach 0 asymptotically. We also discuss that whenever profile likelihood is used for one‐sided tests, the Self and Liang methodology is needed to derive the correct asymptotic distribution.

Suggested Citation

  • Leonid Kopylev & John Fox, 2009. "Parameters of a Dose‐Response Model Are on the Boundary: What Happens with BMDL?," Risk Analysis, John Wiley & Sons, vol. 29(1), pages 18-25, January.
  • Handle: RePEc:wly:riskan:v:29:y:2009:i:1:p:18-25
    DOI: 10.1111/j.1539-6924.2008.01125.x
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    References listed on IDEAS

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    1. Molenberghs, Geert & Verbeke, Geert, 2007. "Likelihood Ratio, Score, and Wald Tests in a Constrained Parameter Space," The American Statistician, American Statistical Association, vol. 61, pages 22-27, February.
    2. A. John Bailer & Randall J. Smith, 1994. "Estimating Upper Confidence Limits for Extra Risk in Quantal Multistage Models," Risk Analysis, John Wiley & Sons, vol. 14(6), pages 1001-1010, December.
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    Cited by:

    1. Nilabja Guha & Anindya Roy & Leonid Kopylev & John Fox & Maria Spassova & Paul White, 2013. "Nonparametric Bayesian Methods for Benchmark Dose Estimation," Risk Analysis, John Wiley & Sons, vol. 33(9), pages 1608-1619, September.

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