Optimal and Robust Designs for Regression

本研究计划的主要目标是了解实验设计的性质如何影响各种模型的估计，预测和回归外推的结果，假设或最有可能没有准确性，并开发理论和方法，构建回归设计，提供有效的结果，在估计精度和保护之间的优化平衡可能的模型偏离。 广义线性固定模型（GLM）、广义线性混合模型（GLMM）、加速寿命试验（ALT）和分位数回归等领域吸引了大量的研究活动。在我目前的资助下，已经获得了关于GLM、GLMM和ALT的最优和稳健设计的重要结果。最近，我们也取得了重要的结果，分析加权分位数回归（WQR），但尚未探讨其设计问题。本方案主要是在GLM、GLMM、ALT的基础上，创新性地发展了WQR的设计构造理论和方法。 对于GLM： 我将继续努力构建稳健的设计，使用新开发的方法，一般GLM可能过度分散和不准确的假设线性预测，在模型参数规格，并在假设的链接功能。 对于GLMM： 基于我早期为GLMM构建最优设计的工作，我将探索GLMM的稳健设计方法，其中可能存在错误指定的随机效应分布和假设线性预测值的不准确性。GLMM的模型参数估计和Fisher信息计算通常具有挑战性，特别是当涉及多个随机效应或多维设计空间时。因此，我也将研究有关的计算问题。 对于ALT： 鉴于以前的研究ALT的最优和稳健设计主要集中在参数模型，我计划构建ALT的最优和稳健设计的常用的半参数模型-比例风险模型（PHM）。由于ALT的外推性质，PHM中假设的基线风险函数往往具有不确定性，由此产生的最优设计往往依赖于PHM参数。因此，我还将构建稳健的设计，以防止在假设的基线风险函数和初始参数值中可能出现的错误。 对于WQR： 分位数回归最重要的优点是它能够对整个条件响应分布进行推断。然而，有很少的文献（都被认为是经典的分位数回归）解决最佳和稳健的设计问题的分位数回归。研究表明，WQR是提高分位数回归统计分析性能的一种重要方法，特别是在异方差存在的情况下。因此，我也将发展的理论和方法的最佳和稳健设计的WQR。

The main objective of this research program is to understand how the nature of designs of experiments affects the results of estimation, prediction, and extrapolation in regression for various models assumed with or most likely without accuracy and to develop theory and methods of constructing regression designs that provide efficient results with optimized balance between precision in estimates and protection from possible model departures. The areas of generalized linear fixed models (GLM), generalized linear mixed models (GLMM), accelerated life testing (ALT), and quantile regression have attracted much research activity. Significant results on optimal and robust designs for GLM, GLMM, and ALT have been obtained under my current grant. Recently, we have also achieved important results for analysis of weighted quantile regression (WQR) but not yet explored its design issue. This proposal aims to primarily build upon design construction for GLM, GLMM, ALT, and innovatively develop theory and methods of design construction for WQR. For GLM: I will continue working on constructing robust design, using newly developed methods, for general GLM with possible overdispersion and inaccuracies in the assumed linear predictor, in model parameter specification, and in the assumed link function. For GLMM: Building on my earlier work for constructing optimal design for GLMM, I will explore robust design methods for GLMM with possible misspecified random effects distribution and inaccuracy in the assumed linear predictor. Both model parameter estimation and the Fisher information calculation for GLMM are often challenging, especially when multiple random effects or a multidimensional design space is involved. Hence, I will also investigate the computational issues concerned. For ALT: Whereas previous research on optimal and robust design for ALT has mainly focused on parametric models, I plan to construct optimal and robust design for ALT for a commonly used semi-parametric model - a proportional hazard model (PHM). The assumed baseline hazard function in a PHM is often with uncertainty due to the extrapolation nature of ALT and the resulting optimal deigns often depend on PHM parameters. Therefore, I will also construct robust deigns against possible misspecification in the assumed baseline hazard function and in the initial parameter values. For WQR: The most important virtue of quantile regression is its capability to make inferences on the entire conditional response distribution. However, there is minimal literature (all considered classical quantile regression) addressing optimal and robust design problems for quantile regression. It has been shown that WQR is an important method to improve the performance of statistical analysis for quantile regression, especially when heteroscedasticity is present. Hence, I will also develop theory and methods of optimal and robust designs for WQR.