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

优化已经成为理论和应用科学大多数领域的基本工具。优化问题是一个非常重要的问题，但还没有得到充分的研究。在本提案中，一个重要的主题是需要客观的方法来解决三个领域的优化问题，即最优设计，自适应设计和分类随机变量的潜在应用。我们从一些可以用最优设计理论解决的估计问题开始。其中一个问题是确定列联表中数据在边际均匀性假设下的单元概率的最大似然估计。另一个问题是在回归模型中构造符合给定参数效率的设计。目标是实现对两个或多个模型有效的设计，以区分并选择最佳模型。然后，我们建议参照一项比较前瞻性随机介入试验的实际数据集，对循环数据进行最佳响应自适应设计。我们的计划还包括使用代理端点中的应用程序进行响应自适应设计。当真实终点的可用性由于成本或时间限制而较少时，替代终点被越来越多地用于评估治疗或暴露的效果。提出的工作也扩展到其他应用，如在有序分类随机变量的情况下。重点将给出多元有序分类随机变量联合分布的理论公式。

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.