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

采用线性最小二乘分析是常见的统计实践，即使证明这种分析推断的假设是无效的。当前的目标是提供一个有用的推理陈述的指南，这些陈述可以在面对这种常见的困境时合理地使用。为此目的，从一个几乎无模型的数据生成过程描述开始：观察值是由向量值协变量和伴随的真实响应组成的总体中的随机样本。推理的形式是响应和协变量之间关系的线性描述。推理的目标是对响应对协变量的总体依赖性进行适当定义的最佳线性描述。第一个任务是提供一个数学框架来准确地描述这种情况，并使其能够进行严格的分析。在这个公式之内，推理的可能形式就可以被研究了。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置信区间通常不是渐近正确的。正确的渐近推断是由适当形式的自举或所谓的三明治估计量提供的。目前的研究将讨论这些推理程序的变化。它还将描述这些推理程序为实际样本量提供可靠结果的情况。无模型的视角导致对其他重要统计设置的额外理解，这些设置可能涉及信息丰富的协变量。其中之一与随机临床试验有关，在随机临床试验中，人们关注的是与安慰剂或替代标准治疗相比的平均治疗效果。一般公式建议使用一个新的估计量和相关的推断，并将研究其性质和变体。统计实践建立在相应的统计理论的基础上，并得到相应的统计理论的证明。该理论有一个共同的总体范式：统计数据是观察到的。对该数据采用统计描述，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，超越了常见的范式，因为它允许数据的统计模型和分析模型非常不同。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。当使用线性模型（如回归分析和方差分析）可能不能充分准确地代表统计样本的真实性质时，这些是使用线性模型的常见做法所固有的。然后，它将提供另一种有效的推理形式，并且可以在这种情况下负责任地使用。为了补充其理论和方法取向，该研究通过若干高级调查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.