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Bootstrap Estimation of Benchmark Doses and Confidence Limits with Clustered Quantal Data

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  • Yiliang Zhu
  • Tao Wang
  • Jenny Z.H. Jelsovsky

Abstract

The benchmark dose (BMD) is an exposure level that would induce a small risk increase (BMR level) above the background. The BMD approach to deriving a reference dose for risk assessment of noncancer effects is advantageous in that the estimate of BMD is not restricted to experimental doses and utilizes most available dose‐response information. To quantify statistical uncertainty of a BMD estimate, we often calculate and report its lower confidence limit (i.e., BMDL), and may even consider it as a more conservative alternative to BMD itself. Computation of BMDL may involve normal confidence limits to BMD in conjunction with the delta method. Therefore, factors, such as small sample size and nonlinearity in model parameters, can affect the performance of the delta method BMDL, and alternative methods are useful. In this article, we propose a bootstrap method to estimate BMDL utilizing a scheme that consists of a resampling of residuals after model fitting and a one‐step formula for parameter estimation. We illustrate the method with clustered binary data from developmental toxicity experiments. Our analysis shows that with moderately elevated dose‐response data, the distribution of BMD estimator tends to be left‐skewed and bootstrap BMDL s are smaller than the delta method BMDL s on average, hence quantifying risk more conservatively. Statistically, the bootstrap BMDL quantifies the uncertainty of the true BMD more honestly than the delta method BMDL as its coverage probability is closer to the nominal level than that of delta method BMDL. We find that BMD and BMDL estimates are generally insensitive to model choices provided that the models fit the data comparably well near the region of BMD. Our analysis also suggests that, in the presence of a significant and moderately strong dose‐response relationship, the developmental toxicity experiments under the standard protocol support dose‐response assessment at 5% BMR for BMD and 95% confidence level for BMDL.

Suggested Citation

  • Yiliang Zhu & Tao Wang & Jenny Z.H. Jelsovsky, 2007. "Bootstrap Estimation of Benchmark Doses and Confidence Limits with Clustered Quantal Data," Risk Analysis, John Wiley & Sons, vol. 27(2), pages 447-465, April.
    • Handle: RePEc:wly:riskan:v:27:y:2007:i:2:p:447-465
    DOI: 10.1111/j.1539-6924.2007.00897.x
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    References listed on IDEAS

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    1. Mirjam Moerbeek & Aldert H. Piersma & Wout Slob, 2004. "A Comparison of Three Methods for Calculating Confidence Intervals for the Benchmark Dose," Risk Analysis, John Wiley & Sons, vol. 24(1), pages 31-40, 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.
    3. Esben Budtz-Jørgensen & Niels Keiding & Philippe Grandjean, 2001. "Benchmark Dose Calculation from Epidemiological Data," Biometrics, The International Biometric Society, vol. 57(3), pages 698-706, September.
    4. Moulton, Lawrence H. & Zeger, Scott L., 1991. "Bootstrapping generalized linear models," Computational Statistics & Data Analysis, Elsevier, vol. 11(1), pages 53-63, January.
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