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

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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.