Zhu Yiliang, Wang Tao, Jelsovsky Jenny Z H
Department of Epidemiology and Biostatistics, University of South Florida, Tampa, FL 33612, USA.
Risk Anal. 2007 Apr;27(2):447-65. doi: 10.1111/j.1539-6924.2007.00897.x.
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.
基准剂量（BMD）是指会导致风险在背景水平之上出现小幅增加（基准风险增加水平，BMR）的暴露水平。用于推导非致癌效应风险评估参考剂量的BMD方法具有优势，因为BMD的估计不限于实验剂量，并且利用了大多数可用的剂量反应信息。为了量化BMD估计值的统计不确定性，我们通常计算并报告其下限置信区间（即BMDL），甚至可能将其视为比BMD本身更保守的替代值。BMDL的计算可能涉及结合德尔塔法的BMD正态置信区间。因此，诸如样本量小和模型参数非线性等因素会影响德尔塔法BMDL的性能，替代方法会很有用。在本文中，我们提出一种自助法来估计BMDL，该方法采用一种方案，包括模型拟合后对残差进行重采样以及用于参数估计的一步公式。我们用发育毒性实验中的聚类二元数据说明了该方法。我们的分析表明，对于中度升高的剂量反应数据，BMD估计值的分布往往向左偏斜，自助法BMDL平均比德尔塔法BMDL小，因此能更保守地量化风险。从统计学上讲，自助法BMDL比德尔塔法BMDL更诚实地量化了真实BMD的不确定性，因为其覆盖概率比德尔塔法BMDL更接近名义水平。我们发现，只要模型在BMD区域附近对数据的拟合程度相当，BMD和BMDL估计值通常对模型选择不敏感。我们的分析还表明，在存在显著且中等强度的剂量反应关系时，标准方案下的发育毒性实验支持以5%的BMR评估BMD以及以95%的置信水平评估BMDL的剂量反应。
