ON THE DE LA GARZA PHENOMENON
ON THE DE LA GARZA PHENOMENON
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DOI：
10.1214/09-aos787
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发表时间：
2010-08-01
影响因子：
4.5
通讯作者：
Yang, Min
中科院分区：
文献类型：
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作者：
Yang, Min
一般来说，推导非线性模型的最优设计具有挑战性。关键的一步是确定所需的支持点的数量。目前的工具是根据具体情况处理这一问题的。模型、最优性准则和目标的每种组合都需要自己的证明。著名的de la Garza现象指出，在（p - 1）次多项式回归模型下，任何最优设计都可以基于最多p个设计点，即所有参数都可估计的最小支持点数量。这个结论是否也适用于非线性模型？如果答案是肯定的，那么通过分析或数值计算得出任何最佳设计都相对容易。在本文中，开发了一种新的方法来解决这个问题。利用这种新方法，可以很容易地证明许多常用的非线性模型，如Emax模型、指数模型、三参数和四参数对数线性模型、Emax-PK 1模型以及许多经典的多项式回归模型都存在de la Garza现象。所提出的方法统一和扩展了许多著名的优化设计文献中的结果。它有四个优点比目前的工具：（i）它可以应用于许多形式的非线性模型;连续或离散数据;数据均匀或非均匀的错误;（ii）它可以应用于任何设计区域;（iii）它可以应用于多阶段优化设计和（iv）它可以很容易地实现。
Deriving optimal designs for nonlinear models is, in general, challenging. One crucial step is to determine the number of support points needed. Current tools handle this on a case-by-case basis. Each combination of model, optimality criterion and objective requires its own proof. The celebrated de la Garza Phenomenon states that under a (p - 1)th-degree polynomial regression model, any optimal design can be based on at most p design points, the minimum number of support points such that all parameters are estimable. Does this conclusion also hold for nonlinear models? If the answer is yes, it would be relatively easy to derive any optimal design, analytically or numerically. In this paper, a novel approach is developed to address this question. Using this new approach, it can be easily shown that the de la Garza phenomenon exists for many commonly studied nonlinear models, such as the Emax model, exponential model, three- and four-parameter log-linear models, Emax-PK1 model, as well as many classical polynomial regression models. The proposed approach unities and extends many well-known results in the optimal design literature. It has four advantages over current tools: (i) it can be applied to many forms of nonlinear models; to continuous or discrete data; to data with homogeneous or nonhomogeneous errors; (ii) it can be applied to any design region; (iii) it can be applied to multiple-stage optimal design and (iv) it can be easily implemented.