Collaborative Research: Inference for Linear Model Parameters in Model-free Populations

通常的统计做法是采用线性最小二乘分析，即使从这种分析得出的推论的假设是不成立的。目前的目标是为在面对这一常见困境时合理地使用有用的推论陈述提供指南。为此，首先对数据生成过程进行几乎无模型的描述：观测数据是来自总体的随机样本，该总体由向量值协变量和伴随的真实响应组成。推理的形式是对响应和协变量之间的关系进行线性描述。推理的目标是对响应对协变量的总体依赖关系的最佳线性描述。第一个任务是提供一个数学框架，准确地描述这种情况，并进行严格的分析。在这个公式中，可以研究可能的推理形式。结果表明，线性系数的常规样本最小二乘估计具有一定的渐近最优性。但这些系数的常规标准误差和可信区间一般不是渐近正确的。正确的渐近推断是由Bootstrap或所谓的夹心估计的适当形式提供的。目前的研究将讨论这些推理过程的变化。它还将描述这些推理过程为实际样本量提供可靠结果的情况。无模型视角使我们对其他重要的统计环境有了更多的了解，可能涉及到信息丰富的协变量。其中之一涉及随机临床试验，其中关注的焦点是平均治疗效果，而不是安慰剂或替代的标准治疗。一般公式建议使用新的估计量和相关推理，并将研究其性质和变体。统计实践建立在相应的统计理论基础上，并得到相应的统计理论的证明。这一理论有一个共同的总体范式：观察统计数据。对该数据进行统计描述，然后根据该统计描述对数据进行分析。在这个范例中有一个假设，即分析模型与生成数据的实际模型非常吻合。在实践中，情况往往并非如此。目前的提议建立了一个新的、连贯的理论，它超越了共同的范式，因为它允许数据的统计模型和分析的模型有很大的不同。拟议研究的效果应该是首先警告从业者经常遇到但很少认识到的危险。当回归分析和方差分析等线性模型可能不能充分准确地反映统计样本的真实性质时，这些都是使用线性模型的常见做法所固有的。然后，它将提供有效并可在这种情况下负责任地使用的替代推理形式。为了补充其理论和方法取向，该研究通过几名高级调查人员在不同领域的应用活动，包括社会科学--特别是犯罪学--运筹学和保健，与应用保持密切联系。

It is common statistical practice to employ a linear, least squares analysis, even though the assumptions justifying the inference from such an analysis are not valid. The current goal is to provide a guide to useful inferential statements that can be justifiably used in the face of this common dilemma. For this purpose, begin with a nearly model-free description of the data generating process: the observations are a random sample from a population consisting of vector-valued covariates and accompanying real responses. Inference is in the form of a linear description of the relation between the response and the covariates. The target of inference is the, suitably defined, best linear description of the population dependence of the response on the covariates. A first task is to provide a mathematical framework to accurately describe such a situation and enable its rigorous analysis. Within this formulation possible forms of inference can then be investigated. It can be shown that the conventional sample least-squares estimate of the linear coefficients has certain asymptotic optimality properties. But the conventional standard errors and confidence intervals for these coefficients are in general not asymptotically correct. Correct asymptotic inference is provided by suitable forms of either the bootstrap or the so-called sandwich estimator. The current research will discuss variations of these inferential procedures. It will also describe situations in which these inferential procedures provide trustworthy results for realistic sample sizes. The model-free perspective leads to additional understanding of other important statistical settings involving possibly informative covariates. One of these relates to Randomized Clinical Trials in which interest centers on the Average Treatment Effect, compared to that of a placebo or alternate, standard treatment. The general formulation suggests use of a new estimator and related inference, and the properties and variants of this will be investigated.Statistical practice is built on and justified by corresponding statistical theory. That theory has a common overarching paradigm: Statistical data is observed. A statistical description is adopted for this data and the data is then analyzed according to this statistical description. There is a presumption in this paradigm that the analytic model agrees sufficiently well with the actual model that generated the data. This is often not the case in practice. The current proposal builds a new, coherent theory that goes beyond the common paradigm in that it allows the statistical model for the data and the model for the analysis to be very different. The effect of the proposed research should be to first warn practitioners of often encountered but rarely recognized dangers. These are inherent in the common practice of using linear models such as regression analysis and ANOVA when they may not sufficiently accurately represent the true nature of the statistical sample. It will then provide alternate forms of inference that are valid and can be responsibly utilized in such situations. To complement its theoretical, methodological orientation the research maintains close connections with applications through the applied activities of several of the senior investigators in diverse areas including social science - especially criminology -operations research and health care.