Optimal and Robust Designs for Regression

本研究计划的主要目的是了解试验设计的性质如何影响各种假设精度或最有可能没有精度的模型在回归中的估计、预测和外推的结果，并开发构建回归设计的理论和方法，以提供估计精度和防止可能的模型偏离之间的优化平衡的有效结果。*广义线性固定模型(GLM)、广义线性混合模型(GLMM)、加速寿命测试(ALT)和分位数回归已吸引了许多研究活动。在我目前的资助下，在GLM、GLMM和ALT的优化和健壮设计方面取得了重大成果。近年来，我们对加权分位数回归(WQR)的分析也取得了重要的成果，但还没有对其设计问题进行探讨。本方案的主要目的是在GLM、GLMM、ALT设计施工的基础上，创新性地发展水利枢纽设计施工的理论和方法。*对于GLM：*我将继续致力于构建稳健设计，使用新开发的方法，针对在假设的线性预测器、模型参数规范和假设的链接函数中可能存在过度分散和不准确的一般GLM。*对于GLMM：*在我为GLMM构建最优设计的早期工作的基础上，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的线性预测不准确。GLMM的模型参数估计和Fisher信息量的计算往往具有挑战性，特别是当涉及多个随机效应或多维设计空间时。因此，我还将研究相关的计算问题。*对于ALT：*以前关于ALT的优化和稳健设计的研究主要集中在参数模型上，而我计划为常用的半参数模型-比例风险模型(PHM)构建ALT的优化和稳健设计。由于ALT的外推性质，在PHM中假定的基线风险函数通常具有不确定性，所得到的最优设计通常取决于PHM参数。因此，我还将针对假设的基线风险函数和初始参数值中可能的错误指定构建稳健的设计。*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，很少有文献(都被认为是经典的分位数回归)解决分位数回归的最优和稳健设计问题。研究表明，WQR是改善分位数回归统计分析性能的一种重要方法，特别是在存在异方差的情况下。因此，我还将发展WQR的优化和稳健设计的理论和方法。**

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.**