Variable Selection in Joint Modelling of Longitudinal and Survival Data and Multi-Response Optimization in Designed Experiments

信息技术的进步使研究人员能够处理大量的高维数据集，因此，变量选择是建模这些数据集的一个重要步骤。经典的变量选择方法通常在评估所有可能的子模型之后实施。它们是计算密集型的，并且对于高维数据变得不可行。已经开发了一类通过正则化的方法来针对此类应用，并且它们因其计算效率和稳定性而受到称赞。我们提出了两个不同的有趣的研究组成部分，它们在统计方法上是相似的。在第一个研究部分中，我们考虑纵向和生存过程的联合建模中的估计和变量选择。在许多情况下，纵向和生存过程往往以自然的方式相关联，由于单独分析可能导致有偏见的结果，因此联合建模是必要的。参数似然法用于联合建模，但存在分布假设规格错误的风险。由于某些协变量对响应的影响很小或没有影响，因此我们更倾向于使用更简约的模型来避免维度灾难。我们提出了一个惩罚的经验似然，一个非参数似然为基础的方法，估计和变量选择将同时进行。我们将首先开发的变量选择方法，使用惩罚的经验似然纵向数据，然后使用相同的方法在联合建模。我们将建立我们的方法的理论依据，并进行大量的模拟，以评估其性能。 我们将在一个真实的案例中实现所提出的方法。*在第二个研究部分中，我们考虑在设计的实验中进行多响应建模，以达到最佳的因素水平组合。在任何实验中，多个响应都是常见的，但是对每个响应单独分析而不考虑响应之间的相关性将导致错误的结论。 在这项研究中，我们将考虑建模的连续和离散类型的响应与多元有序响应的特殊情况。我们提出了一个惩罚的多元版本的广义估计方程（MGEE）的参数估计。在这种方法中，我们有效地模拟响应之间的相关性和变量的选择和参数的估计同时进行。我们将开发方法来达到最佳组合的因素水平，并将所提出的方法扩展到最优设计。为了减少错误指定的风险，我们探索了经验似然为基础的方法。所提出的方法在工业中有更广泛的应用，我们将在真实的工业问题中实施它。

Advances in information technology have enabled researchers to deal with large data sets with high dimensions routinely and variable selection is hence an essential step in modeling these data sets. Classical variable selection approaches are often implemented after evaluating all possible sub-models. They are computationally intensive and become infeasible with high dimensional data. A class of methods via regularization has been developed to target such applications, and they have been lauded for their computational efficiency and stability. We propose two different interesting research components, which are similar in terms of statistical methodology.****In the first research component, we consider the estimation and variable selection in the joint modelling of longitudinal and survival processes. In many situations, longitudinal and survival processes are often associated in a natural way and joint modelling is necessary due to the fact that separate analysis may lead to biased results. Parametric likelihood approaches are used in joint modelling, but there is a risk of mis-specifications on the distributional assumptions. Since some of the covariates has little or no effect on the response,  we prefer to have a more parsimonious model to avoid the curse of dimensionality. We propose a penalized empirical likelihood, a non-parametric likelihood-based approach, by which estimation and variable selection will be carried out simultaneously. We will first develop methodologies for variable selection using penalized empirical likelihood in longitudinal data and then use the same approach in joint modelling. We will establish theoretical justification of our approach and conduct a large number of simulations to evaluate its performance. We will implement the proposed methodology in a real case example.***In the second research component, we consider multi-response modelling in designed experiments to arrive at an optimum combination of factor levels. Multiple responses are common in any experiment, but separate analysis for each response without considering the correlation among the responses will lead to wrong conclusions. In this research, we will consider modeling the responses of continuous and discrete types with a special case of multivariate ordinal responses. We propose a penalized multivariate version of generalized estimating equations (MGEE) to estimate the parameters. In this approach, we model the correlation among the responses effectively and variable selection and the estimation of parameters are carried out simultaneously. We will develop methods for arriving at the optimum combination of factor levels and extend the proposed method to optimal designs. To reduce the risk of mis-specification, we explore the empirical likelihood-based approach. The proposed methodology has wider applications in industries and we will implement it in real industrial problems.**