Multi-objective optimal design of experiments

实验与纯粹的观察性研究的不同之处在于，有意识地对被研究的系统进行干预，并观察这些干预的效果。在几乎所有的科学、工程、医学和商业领域，实验是得出因果结论的最可靠、最可靠的方法--如果我们想知道做出改变的效果，我们就必须做出改变并观察效果。除了最简单的实验外，解释所有实验的结果都需要使用统计模型分析实验响应(输出)的数据，统计模型的复杂程度取决于实验的复杂程度。从实验中得出的结论的有效性和稳健性取决于数据相对于所使用的统计模型的信息量有多大，以及数据的信息量有多大取决于实验设计的方式。在过去的100年里，为了处理不同的实验结构和从中收集的数据，实验的统计设计得到了发展。历史上有两种不同的方法。优化设计包括定义一个数学函数，该函数取决于实验中使用的特定干预措施(处理)集，然后选择处理方法来优化该函数。这具有易于理解的优点，即与数据分析的属性直接相关，例如选择一种设计来最小化某些重要数量的估计的方差。然而，它的缺点是过分简化了实验者在实践中实际拥有的多个目标。另一方面，经典设计选择具有吸引人的数学结构(通常基于对称性)的设计，这可以使设计对许多目标都很好。然而，对于某些实验结构，经典设计可能很难找到或不可能找到，并且不能保证它们对于任何特定实验的目标都是非常好的。本项目旨在开发和实施同时获得最优设计和经典设计的方法，即多目标优化设计(MODS)。Moods使用优化数学函数的想法，但该函数代表了实验者在实践中的许多不同目标之间的妥协。一些目标可以用来限制我们搜索最优设计的设计集合，例如，在某些情况下，我们可能会将搜索限制在允许我们获得因素主要影响的不相关估计的设计上。其他目标将被组合在一个复合最优标准中，该标准定义了几个单独的简单标准的加权几何平均值。由于情绪需要比标准设计更复杂的优化，我们将推导出理论结果来简化标准，例如通过证明两个目标实际上是互补的，因此只需要一个目标。我们还将开发寻找最优设计的算法，并在可供实验人员使用的程序中实现它们。本项目的重点将放在四种类型的实验上：多个处理因素同时变化的实验；在受试者网络上进行的实验；测量的响应是函数(或曲线)的实验；以及同一实验单元内的处理因素可以随时间变化的实验。这些结构的广度应该有助于其他研究人员在未来将这些方法适应于不同类型的实验。由于如此多的应用领域使用实验，因此本文开发的方法具有直接或在针对特定类型的实验进一步发展后应用于许多不同领域的潜力。实验者将受益于能够以尽可能经济的方式和尽可能无偏见地从他们的实验中准确地获得所需的信息。

An experiment differs from a purely observational study in that interventions are deliberately made to the system under study and the effects of these interventions observed. In almost all fields of science, engineering, medicine and business, experiments are the most robust and reliable way of drawing causal conclusions - if we want to know the effect of making a change, we have to make that change and observe the effect. Interpreting the results of all but the simplest experiments involves analysing data on responses (outputs) from the experiment, using statistical models whose complexity depends on the complexity of the experiment. The validity and robustness of conclusions that can be drawn from the experiment depend on how informative the data are with respect to the statistical models used, and how informative the data are depends on the way the experiment is designed. The statistical design of experiments has developed over the last 100 years to deal with different structures of experiments and data collected from them. Historically there have been two different approaches. Optimal design involves defining a mathematical function, which depends on the particular sets of interventions (treatments) used in the experiment, and then choosing the treatments to optimise this function. This has the advantage of being easily understood to be directly related to the properties of the data analysis, e.g. choose a design to minimise the variance of the estimate of some important quantity. However, it has the disadvantage of oversimplifying the multiple objectives that experimenters actually have in practice. Classical design, on the other hand, chooses designs with attractive mathematical structures (usually based on symmetries) which can make the designs fairly good for many objectives. However, classical designs can be difficult or impossible to find for some experimental structures and there is no guarantee that they will be very good for the objectives of any particular experiment.This project aims to develop and implement methods which will get the best of both optimal and classical designs, namely multi-objective optimal designs (MOODs). MOODs use the idea of optimising a mathematical function, but that function represents a compromise between the many different objectives that experimenters have in practice. Some of the objectives can be used to restrict the set of designs over which we search for an optimum, e.g. in some cases we might restrict the search to designs which allow us to obtain uncorrelated estimates of the main effects of factors. Other objectives will be combined in a compound optimality criterion, which defines a weighted geometric mean of several individual simple criteria. Since MOODs require a more complex optimisation than standard designs, we will derive theoretical results to allow simplification of the criterion, e.g. by showing that two objectives are actually complementary, so only one is needed. We will also develop algorithms for searching for optimal designs and implement them in programs that can be used by experimenters.The focus in this project will be on four types of experiment: those with many treatment factors being varied simultaneously; those where the experiments are carried out on a network of subjects; those in which the measured response is a function (or curve); and those in which the treatment factors can be varied over time within the same experimental unit. The breadth of these structures should help other researchers adapt the methods to different types of experiment in the future.Since so many areas of application use experiments, the methods developed here have the potential to be applied in many different fields, either directly or after further development for particular types of experiment. Experimenters will benefit from being able to get exactly the information required from their experiment as economically and as free from bias as possible.