Collaborative Research: Model diagnostics in regression and Tobit regression models with measurement error

一组预测因子和响应之间关系的统计建模通常通过回归分析实现。在经典的回归模型中，预测变量和响应变量都被假设为直接可观察的。在测量误差回归模型中，预测变量不能直接观测到，而需要观测到一些替代变量。在Tobit回归模型中，只有当响应变量高于某个阈值时才能观察到它。 在回归和Tobit回归模型中存在测量误差的情况下，开发有用的和最佳的推理程序是理论和应用统计学中的主要关注点。然而，对于测量误差回归模型和有无测量误差的Tobit回归模型的拟合优度检验和失拟检验的研究却相对滞后。在这个项目中，研究者分析了变量误差和Berkson测量误差回归模型的随机分量分布的拟合优度检验，以及在有或没有这些测量误差的情况下Tobit回归模型中回归函数的一些非参数估计。此外，研究人员开发和分析缺乏的拟合和拟合优度测试的Tobit回归模型与这些测量误差。研究人员在这些模型中提供了一些新的，有用的和最佳的推理程序，并深入了解了统计学和相关学科的广泛专业受众的理论特性。该项目是在回归和Tobit回归模型中预测变量存在测量误差的情况下进行模型检查的最前沿。它推进和丰富了统计理论和方法，从而有助于填补统计中存在的重大空白和公认的理论空白。测量误差在健康科学、物理科学、经济学和社会科学中非常普遍。例如，在研究饮食对乳腺癌的影响时，用于预测乳腺癌的预测变量之一是长期饱和脂肪摄入量，而这是无法精确测量的。相反，在这种类型的调查中，经常使用每个患者24小时饮食回忆的替代品。同样，在研究辐射暴露对人体的影响时，一个人所暴露的确切辐射量往往是错误的。 在劳动研究中，当调查妇女的工作状况与其背景信息（如年龄、教育和工作经验）之间的关系时，测量误差的影响存在于教育变量（如母亲和父亲的教育经验）中。 在这些类型的研究中使用的Tobit回归模型，经常遭受测量误差的问题。大多数使用Tobit回归模型的实证研究往往忽略了测量误差，这通常会导致有偏见和效率低下的统计结论。该项目的研究重点，这有助于评估回归模型或模型的测量误差的存在下，随机成分的分布的准确性，有助于开发更准确的统计推断这些和其他类似的例子。

Statistical modeling for relationships between a collection of predictors and a response is often implemented by regression analysis. In the classical regression model, both predictors and response variables are assumed to be directly observable. In measurement error regression models, predictors cannot be observed directly, instead, some surrogates are observed. In Tobit regression models, the response variable is observed only when it is above some threshold. The development of useful and optimal inference procedures in the presence of measurement errors in regression and Tobit regression models is of major concern in theoretical and applied statistics. Despite this need, the study of goodness-of-fit and lack-of-fit tests in the measurement error regression models and Tobit regression models with/without measurement errors has lagged behind. In this project, the investigators analyze goodness-of-fit tests for the distributions of the random components of errors-in-variables and Berkson measurement error regression models, and some nonparametric estimators of regression functions in Tobit regression models with or without these measurement errors. Furthermore, the investigators develop and analyze lack-of-fit and goodness-of-fit tests in Tobit regression models with these measurement errors. The investigators make available some new, useful, and optimal inference procedures in these models with an in-depth understanding of their theoretical properties to a wide professional audience in statistics and related disciplines. This project is at the cutting edge of model checking in the presence of measurement error in predictors in regression and Tobit regression models. It advances and enriches the statistical theory and methodology, thereby helping to fill a significant void and well recognized theoretical gap that exists in statistics. Measurement errors are very prevalent in the health sciences, physical sciences, economics, and the social sciences. For example, when investigating the effect of diet on breast cancer, one of the predictor variables studied for predicting breast cancer is the long-term saturated fat intake which cannot be measured precisely. Instead, the surrogate of a 24 hour diet recall for each patient is often used in this type of investigation. Similarly, the exact amount of radiation a person is exposed to when studying the effect of radiation exposure on humans is often measured with error. In labor studies, when investigating the relationship between women's working status and their background information, such as age, education and working experience, the effect of measurement errors is present in the education variables (such as mother's and father's education experience). Tobit regression models, which are used in these types of studies, often suffer from the measurement error problem. Most empirical studies involving Tobit regression models tend to ignore the measurement errors, which usually leads to biased and inefficient statistical conclusions. The research focus of this project, which helps in assessing the accuracy of a regression model or of a model for the distributions of random components in the presence of measurement errors, helps to develop more accurate statistical inference for these and other similar examples.