Statistical Inference under Subjective and Not-Fully-Quantifiable Information on Experimental Units

在许多科学调查中，有大量关于实验材料的主观信息--哪些农地更肥沃，哪些病人更健康等等--很难量化为正式的、数字的变量，以便进一步用于统计分析。目前，统计科学无法有效地利用这些信息。这项研究开发了一系列技术，使人们能够恢复这些信息的大部分，并以完全客观的方式使用它们。重要的是，应用于主观信息恢复的相同数学技术也允许人们利用可能受到重大测量误差或其他偏差影响的“较差质量”协变量。这些质量较差的协变量被用来在潜在的实验单位之间创建人工分层，然后再测量它们的响应。作为这些技术基础的工作主体被称为排序集合抽样。在非常强的假设下，例如完美排序或其他精确指定的概率和判断排序模型，排序集抽样的现状提供了理论基础。即使与这些假设有很小的偏离，也可能导致不一致的估计值，甚至是渐近偏差。当招募一个单位进行研究的成本很高，或者当可用的实验单位数量有限时，研究界对使用分级集合抽样持保留意见。在这些情况下，最好使用所有可用的实验装置来进行实验。为了缓解这些担忧，这项研究从不同的角度看待排序集抽样，并确定了当前最先进的排序集抽样要么不合适，要么表现不佳，或者不能以令人满意的一般方式使用的三个方面。这些领域是(I)在实验设计中使用排序集抽样，(Ii)在最小判断建模假设下开发低结构和中等结构参数估计，以及(Iii)为排序过程开发模型。这项研究将对统计分析、几个已建立的科学研究领域、美国民众的生活质量和国家的科学基础设施产生重大影响。这项研究通过上一段描述的机制，将使研究人员能够通过新的统计分析从他们的实验中挤出更多信息。从实验中挤出更多信息的另一面是，能够通过较小的实验获得给定数量的信息。这项研究将开发出更好的设计来进行实验，影响到各种各样的学科。作为一个最好的例子，有了新的设计，用于确定新药益处的临床试验将需要较少的受试者。审判成本将降低，审判将在更短的时间跨度内完成。一项规模更小、成本更低、速度更快的试验的三重好处将加快新药的开发和批准，包括那些治疗癌症等晚期疾病的药物。除了经济利益外，在不牺牲审批过程中目前的保障措施的情况下，尽快将有前景的新疗法通过开发过程还有道德上的好处。通过加强统计学家和医学研究人员之间的合作以及对研究生的严格培训，该国的科学基础设施将得到加强。将特别重视妇女和少数群体在数学科学方面的培训。

In many scientific investigations, there is a wealth of subjective information about experimental material--which agricultural plots are more fertile, which patients are healthier, etc.--that is difficult to quantify as a formal, numerical variate for further use in statistical analysis. Currently, statistical science is unable to effectively exploit this information. This research develops a body of techniques that allow one to recover the bulk of this information and to use it in a fully objective fashion. Importantly, the same mathematical techniques that apply to recovery of subjective information also allow one to exploit ``lesser quality'' covariates which may be subject to substantial measurement error or other biases. These lesser quality covariates are used to create an artificial stratification among the potential experimental units before their responses are measured. The body of work that underlies these techniques is known as ranked set sampling. The current status of ranked set sampling provides a theoretical foundation under very strong assumptions, such as perfect ranking or other precisely specified probability and judgment ranking models. Even a small departure from these assumptions may result in inconsistent estimators that contain substantial bias, even asymptotically. There are reservations within the research community regarding the use of ranked set sampling when either the cost recruiting a unit for a study is substantial or when the number of available experimental units is limited. In these situations, it is desirable to use all available experimental units to perform the experiment. To alleviate these concerns, this research looks at ranked set sampling from a different perspective and identifies three areas where the current state-of-the-art ranked set sampling is either not appropriate, performs poorly, or cannot be used in a satisfactory general fashion. These areas are (i) the use of ranked set sampling in the design of experiments, (ii) the development of low- and medium structure parametric estimation under minimal judgment modeling assumptions, and (iii)the development of models for the ranking process.This research will have a substantial impact on statistical analysis, on several established areas of scientific research, on the quality of life of the U.S. populace, and on the scientific infrastructure of the country. The research, through the mechanisms described in the preceding paragraph, will allow researchers to squeeze more information out of their experiments with novel statistical analyses. The flip side of squeezing more information out of an experiment is the ability to obtain a given amount of information with a smaller experiment. The research will develop better designs to perform experiments, impacting a wide variety of disciplines. As a prime example, with the new designs, a clinical trial, used to establish the benefits of a new drug, will require a smaller number of subjects. The cost of the trial will be reduced and the trial will be completed in a shorter time span. The triple benefit of a smaller, cheaper and quicker trial will expedite the development and approval of new drugs, including those for terminal diseases, such as cancer. In addition to economic benefits, there is the ethical benefit of moving promising new therapies through the development process as quickly as possible, without sacrificing current safeguards in the approval process. The scientific infrastructure of the country will be enhanced by the increased collaboration between statisticians and medical researchers, and by the rigorous training of graduate students. Particular emphasis will be given to the training of women and minorities in the mathematical sciences.