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例行公事地处理高维的大数据集，因此变量选择是对这些数据集进行建模的关键步骤。经典的变量选择方法通常是在评估所有可能的子模型之后实施的。它们是计算密集型的，对于高维数据变得不可行。一类通过正则化的方法已经被开发出来针对这类应用，它们的计算效率和稳定性得到了赞扬。我们提出了两个不同的有趣的研究成分，它们在统计方法上是相似的。*在第一个研究成分中，我们考虑了纵向过程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，纵向过程和生存过程往往以一种自然的方式联系在一起，由于单独分析可能导致有偏见的结果，联合建模是必要的。参数似然方法用于联合建模，但存在错误指定分布假设的风险。由于一些协变量对反应的影响很小或没有影响，因此我们倾向于有一个更简约的模型，以避免维度的诅咒。我们提出了一种惩罚经验似然法，这是一种基于非参数似然的方法，它将同时进行估计和变量选择。我们将首先开发在纵向数据中使用惩罚经验似然进行变量选择的方法，然后在联合建模中使用相同的方法。我们将为我们的方法建立理论上的证明，并进行大量的模拟来评估其性能。我们将在一个真实的案例中实施所提出的方法。*在第二个研究部分中，我们考虑了设计实验中的多响应建模，以得出因素水平的最佳组合。多个反应在任何实验中都是常见的，但单独分析每个反应而不考虑反应之间的相关性将导致错误的结论。在这项研究中，我们将考虑用多变量有序响应的一种特殊情况来建模连续类型和离散类型的响应。我们提出了一种惩罚多元广义估计方程(MGEE)来估计参数。在该方法中，我们有效地对响应之间的相关性进行建模，同时进行变量选择和参数估计。我们将发展达到因素水平的最佳组合的方法，并将所建议的方法扩展到最优设计。为了减少误指定的风险，我们探索了基于经验似然的方法。提出的方法在工业中有更广泛的应用，我们将在实际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.**