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Parameters Of A Dose-response Model Are On The Boundary: What Happens With Bmdl?

机译：剂量反应模型的参数在边界上：Bmdl会发生什么？

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摘要

It is well known that, under appropriate regularity conditions, the asymptotic distribution for the likelihood ratio statistic is x~2- 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 l-2α) have coverage close to the nominal level when there are 50 animals in each group, the coverage of nominal 1 -α onesided intervals is bounded between roughly l-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.
机译：众所周知，在适当的规律性条件下，似然比统计量的渐近分布为x〜2-。此结果在EPA的基准剂量软件中用于通过以下方法获得基准剂量（BMD）的较低置信区间（BMDL）轮廓似然法。最近，根据Self和Liang的工作，已经证明，如果违反某些规律性条件，即当某些讨厌参数的真实值位于边界上时，似然比的渐近分布保持不变。 BMD分析癌症生物测定数据通常是这种情况。在本文中，我们通过模拟研究当某些模型参数在参数空间边界上具有真实值时，BMD的一侧和两侧置信区间的覆盖范围。幸运的是，当每组有50只动物时，两侧置信区间（l-2α尺寸）的覆盖率接近标称水平，因此标称1-α双面间隔的覆盖范围大约在1-2α和1之间。在许多模拟方案中，单边置信度为95％（即α= 0.05），BMDL的覆盖率接近1，但对于某些方案，对于一组组，其BMDL覆盖率都接近90％。 50只动物，渐近（每组100,000只）。另一个重要的观察结果是，当真实参数低于对数逻辑模型的形状参数时，在约束模型中BMDL​​的覆盖范围可能很小（在BMDS分析中出现模型错误指定的情况并不罕见）甚至渐近地接近0我们还讨论了，只要将轮廓似然用于单面检验，就需要使用Self和Liang方法来得出正确的渐近分布。

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