Mathematical Sciences: Statistical Theory and Methods for Errors in Variables Regression and other Multivariate Inference Problems

数学科学：变量回归和其他多元推理问题中误差的统计理论和方法

• 批准号: 9504924
• 负责人: Leon Gleser
• 金额: $ 13.5万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-15 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504924&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Statistical; Theory; Methods

Proposal: DMS 9504924 PI: Leon Gleser Institution: U. of Pittsburgh Title: STATISTICAL THEORY AND METHODS FOR ERRORS-IN-VARIABLES REGRESSION AND OTHER MULTIVARIATE INFERENCE PROBLEMS Abstract: This research compares the mean squared error risks of various estimators of regression slopes that correct for bias due to measurement error in the predictor variables. Included in these comparisons is the naive (uncorrected) least squares estimator that ignores measurement error. Emphasis is placed on bias correction that makes use of knowledge about the repeatability (reliability) of the vector of measurements on the predictors to "correct the slope estimates for attenuation." Various ways of obtaining and analyzing information concerning the reliability of the vector of measured predictors are considered, including the use of reliability studies of individual components and/or subvectors of the predictor vector and the elicitation and use of expert opinion. These separate sources of information are combined to yield frequentist or Bayes point estimators of the reliability matrix of the measured predictors; this matrix is used to correct least squares regression slopes for attenuation. Methods for forming confidence (or credible) intervals for slopes, and also for components of the reliability matrix are derived and compared. Computational algorithms for finding likelihoods and posterior distributions in both these problems, and also in multivariate growth curve and seemingly unrelated regression models, are written and tested on real and simulated data. Much of scientific research concerning complex systems or organisms is concerned with relationships among quantities . For example, the number of successfully hatched eggs of a certain species of marshland wildfowl might be related to the concentrations of one or more pesticides in the marsh. When the rates of response or change of a variable (such as the number of successfully hatched eggs) to changes in other predictor variables (such as various pesticide concentrations) are estimated by conventional statistical methods, it is assumed that the predictor variables are measured exactly. Unfortunately, environmental, biological or psychological variables rarely are measured exactly. If predictor variables are measured with error, the conventional estimates of response rates are biased. To correct for such bias, errors-in-variables regression models assume that each measurement is the sums of the true value of the quantity being measured and a random error. In this case, the proportion of the unit-to-unit variability of a measurement that is due to the variability over units of the true value, called the reliability of the measurement, can be used to correct the bias of the conventional estimates of response rates. When several variables are simultaneously used as predictors of another, more than the reliability of each individual predictor is required for this purpose; it is the reliability of the measurements of the predictors as a whole, or ensemble, that must be ascertained. Because the particular collection of predictors may not have been used before, information about the reliability of the predictors as an ensemble must be pieced together from a variety of sources. The investigator's research is concerned with how best to obtain and combine information from data summaries of prior studies that used some (but not necessarily all) of the predictor variable, data from the current study and expert opinion to obtain the required reliability information and correct bias in the conventional estimates of response rates. Also studied are ways to determine and summarize the accuracy of the resulting estimates. Insights and methodology from this research has broad applicability to questions of combining information from several small studies concerning the interrelationships among variables, not all of which appear in every study. One of the products of the research is computer software that allows information to be combined from several studies on the same subjects (or environmental locations) and then displayed so as to give the relative likelihoods of various statistical models in the light of the evidence presented by the data itself.

建议：DMS 9504924 PI：Leon Gleser机构：匹兹堡大学标题：变量中误差回归和其他多元推理问题的统计理论和方法摘要：这项研究比较了回归斜率的各种估计器的均方误差风险，这些估计器因预测变量中的测量误差而校正偏差。在这些比较中包括忽略测量误差的朴素(未校正)最小二乘估计器。重点放在偏差校正上，它利用关于预测值上的测量向量的重复性(可靠性)的知识来“校正衰减的斜率估计”。考虑了获得和分析与测量的预测向量的可靠性有关的信息的各种方式，包括使用预测向量的各个分量和/或子向量的可靠性研究以及引出和使用专家意见。这些独立的信息源被组合在一起，以产生测量的预报器的可靠性矩阵的频率或贝叶斯点估计；该矩阵用于校正衰减的最小二乘回归斜率。推导并比较了斜率和可靠性矩阵各分量的置信度(或可信)区间的形成方法。在这两个问题中，以及在多元增长曲线和看似无关的回归模型中，都编写了计算算法，并在真实和模拟数据上进行了测试。许多关于复杂系统或生物体的科学研究都与量之间的关系有关。例如，某一种沼泽野禽成功孵化的蛋的数量可能与沼泽中一种或多种杀虫剂的浓度有关。当用传统的统计方法估计一个变量(如成功孵化的蛋数)对其他预测变量(如各种农药浓度)变化的反应或变化率时，假设预测变量是准确测量的。不幸的是，环境、生物或心理变量很少被准确测量。如果预测变量的测量有误差，则传统的响应率估计是有偏差的。为了纠正这种偏差，变量误差回归模型假定每个测量值都是被测量量的真值和随机误差的总和。在这种情况下，测量的单位对单位的变异性是由于相对于真实值的单位的变异性，称为测量的可靠性，可以用来校正对响应率的传统估计的偏差。当几个变量同时用作另一个变量的预测值时，为此目的需要的不仅仅是每个单独预测值的可靠性；必须确定的是预测值作为一个整体或整体的测量的可靠性。由于预报器的特定集合可能以前从未使用过，因此关于预报器作为一个集合的可靠性的信息必须从各种来源拼凑在一起。研究人员的研究涉及如何最好地从使用部分(但不一定是全部)预测变量的先前研究的数据摘要、当前研究的数据和专家意见中获取和结合信息，以获得所需的可靠性信息并纠正传统的应答率估计中的偏差。还研究了确定和总结结果估计的准确性的方法。这项研究的见解和方法对于综合几个关于变量之间相互关系的小型研究的信息具有广泛的适用性，并不是所有的研究都出现在每一项研究中。这项研究的产品之一是计算机软件，它允许将同一主题(或环境位置)的几项研究的信息组合在一起，然后进行显示，以便根据数据本身提供的证据给出各种统计模型的相对可能性。