Variable Selection And Variable Screening In High Dimensional Data With Multivariate Responses

Variable selection and variable screening are fundamental problems in modern statistical inference.In this thesis,consider the problems of variable selection in multivariate linear regression and screening for variables with treatment interaction both under high dimensional covariates.In the first part of the thesis,we propose two variable selection methods in multivariate linear regression with high dimensional covariates.The first method uses multiple correlation coefficient to fast reduce the dimension of the relevant predictors to a moderate or low level.The second method extends the univariate forward regression in a unified way for multivariate responses such that the variable selection and model estimation can be obtained simultaneously.We establish the sure screening property for both methods.Simulation and real data applications are presented to show the finite sample performance of the proposed methods in comparison with some naive method.In the second part of the thesis,we develop variable screening for making decision of optimal treatment regime with univariate response.Precision medicine is a medical paradigm that focuses on making effective treatment decision based on individual patient characteristics.When there are a large amount of patient information,such as patient’s genetic information,medical records and clinical measurements,available,it is of interest to select the covariates which have interactions with the treatment,for example,in determining the individualized treatment regime where only a subset of covariates with treatment interactions involves in decision making.We propose a marginal feature ranking and screening procedure for measuring interactions between the treatment and covariates.The method does not require imposing a specific model structure on the regression model and is applicable in a high dimensional setting.Theoretical properties of the proposed method is obtained.We evaluate the finite sample properties of our proposed approach via simulation studies.We also illustrate the proposed method by applying it to a data set from an AIDS clinical trials group study.Furthermore,we extend the method of proposed model-free variable screening for individualized treatment regimes when the treatment response is multivariate so that it can be applied for a broad set of models.Finite sample performances of model-free variable selection methods are examined through numerical studies...