Abstract Textbooks on response surface methodology generally stress the importance of lack-of-fit tests and estimation of pure error. For lack-of-fit tests to be possible and other inference to be unbiased, experiments should allow for pure-error estimation. Therefore, they should involve replicated treatments. While most textbooks focus on lack-of-fit testing in the context of completely randomized designs, many response surface experiments are not completely randomized and require block or split-plot structures. The analysis of data from blocked or split-plot experiments is generally based on a mixed regression model with two variance components instead of one. In this article, we present a novel approach to designing blocked and split-plot experiments which ensures that the two variance components can be efficiently estimated from pure error and guarantees a precise estimation of the response surface model. Our novel approach involves a new Bayesian compound D-optimal design criterion which pays attention to both the variance components and the fixed treatment effects. One part of the compound criterion (the part concerned with the treatment effects) is based on the response surface model of interest, while the other part (which is concerned with pure-error estimates of the variance components) is based on the full treatment model. We demonstrate that our new criterion yields split-plot designs that outperform existing designs from the literature both in terms of the precision of the pure-error estimates and the precision of the estimates of the factor effects.

中文翻译：

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最优块和裂区设计确保方差分量的精确纯误差估计

摘要 关于响应面方法的教科书通常强调失拟检验和估计纯误差的重要性。为了使失配测试成为可能，并且其他推断是无偏的，实验应该允许纯误差估计。因此，它们应该涉及重复处理。虽然大多数教科书侧重于完全随机设计背景下的失拟检验，但许多响应面实验并不是完全随机的，需要块或裂区结构。对来自块区或裂区实验的数据的分析通常基于具有两个方差分量而不是一个方差分量的混合回归模型。在本文中，我们提出了一种设计块状和裂区实验的新方法，它确保可以从纯误差有效地估计两个方差分量，并保证对响应面模型的精确估计。我们的新方法涉及一种新的贝叶斯复合 D 最优设计标准，它关注方差分量和固定处理效果。复合标准的一部分（与处理效果有关的部分）基于感兴趣的响应面模型，而另一部分（与方差分量的纯误差估计有关）基于完整的处理模型.