Optimization problems in optimal regression design, response-adaptive design and categorical random variables

最优化已成为大多数理论和应用科学领域的基本工具。优化问题具有重要意义，但尚未得到充分的研究。在这项提案中，一个重要的主题是需要为三个领域的优化问题提供客观的方法，即最优设计、适应性设计和具有潜在应用的分类随机变量。我们从一些可以用最优设计理论解决的估计问题开始。一个这样的问题是在列联表中数据的边际齐性假设下确定单元概率的最大似然估计。另一个问题是构造满足回归模型中某一参数给定效率的设计。目标是实现一种对两个或更多型号有效的设计，以区分和选择最好的一个。然后，我们建议参考一个比较前瞻性随机干预试验的实际数据集，对圆形数据进行最优反应自适应设计。我们的计划也是致力于响应自适应设计，并在代理终端中应用。当真正的终点由于成本或时间限制而可用性较低时，替代终点被越来越多的人用来评估治疗或暴露的效果。所提出的工作也扩展到其他应用，例如有序分类随机变量的情况。重点将集中在多变量有序分类随机变量联合分布的理论表述上。拟议的研究将对统计学和应用科学有重要贡献。该选型工作在工业和化工模型试验的相关问题中具有潜在的应用价值。利用圆形数据进行自适应设计的工作将为气象学和天文学的应用研究人员提供有效和实用的工具。为了有效地使用代理端点，需要提出用于代理响应的统计装置。有序分类随机变量的研究将为纵向研究和聚类数据的分析提供重要的贡献。

Optimization has been a basic tool in most of the areas of theoretical and applied sciences. The optimization problems are of great importance, and they have yet to be adequately studied. In this proposal, an important theme is the need for objective methodologies for optimization problems in three areas, namely, optimal design, adaptive design and categorical random variables with potential applications. We start with some estimation problems that can be solved using optimal design theory. One such problem is to determine the maximum likelihood estimates of the cell probabilities under the hypothesis of marginal homogeneity for data in contingency tables. Another problem is constructing designs subject to achieving a given efficiency of a parameter in a regression model. The goal is to implement a design that is efficient for two or more models, to discriminate and select the best one. Then we propose to work on optimal response-adaptive designs for circular data with reference to a practical data set on a comparative prospective randomized interventional trial. Our plan is also to work on response-adaptive designs with applications in surrogate endpoints. Surrogate endpoints are used with growing interest to evaluate the effects of treatments or exposures when availability of true endpoints is less due to cost or time constraint. The proposed work also extends to other applications such as in the case of ordinal categorical random variables. The focus will be given on theoretical formulation of the joint distribution of multivariate ordinal categorical random variables.The proposed research will have significant contribution to statistics and applied sciences. The work on model selection has potential application in problems related to industrial and chemical engineering for carrying out tests for models. The work on adaptive designs with circular data will provide efficient and practical tools to applied researchers in meteorology and astronomy. The statistical devices proposed for surrogate response are needed for efficient use of surrogate endpoints. The proposed research on ordinal categorical random variables will provide important contribution in longitudinal studies and in analyzing clustered data.