Variable Screening For Ultra-high Dimensional Gener Alized Varying Coefficient Models With Longitudinal Data

Longitudinal data with the characteristic of independence between subjects and within-subject correlation,can reflect the differences between the individuals and the change of indi-viduals,which has an irreplaceable role in many fields such as medicine,economics and so on.With the rapid development of computing science and technology,the acquired data dimension is higher and its structure is more complex.On the one hand,it will be confronted with many challenges including the complexity of longitudinal data,computational expediency,statistical accuracy,and algorithmic stability for high-dimensional data.These problems have become the hotspot and difficulty of statistical research recently.On the other hand,it requires more flexible models to fit complex data,such as generalized varying coefficient model,which is a generaliza-tion of generalized linear model.This model inherits the advantages of generalized linear model as well as non-parameter estimation robustness,and it is more flexible in practical application.Although there are many literatures on variable selection problem with high-dimensional lon-gitudinal data,there are few studies on the problem of generalized variable coefficient model,especially for ultra-high dimension.Therefore,it is of great theoretical significance and practi-cal value to study the variable screening problem for ultra-high dimensional generalized varying coefficient model with longitudinal data.In this paper,we propose a nonparametric independence screening method for sparse ultra-high dimensional generalized varying coefficient models with longitudinal data.Our methods combine the ideas of sure independence screening(SIS)and the marginal generalized estimat-ing equation(GEE)method,called as NIS-GEE,considering both the marginal correlation be-tween response and covariates,and the subject correlation for variable screening.Firstly,we set up marginal generalized varying coefficient models based on marginal correlation between covariates and response.Secondly,we use nonparametric method and generalized estimating equations to estimate the coefficient functions,then adopt the key idea of sure independence screening to select variables.Furthermore it is shown that,under some regularity conditions,the proposed NIS-GEE method enjoys the sure screening properties.Meanwhile,the corre-sponding iterative algorithm is introduced to reduce false selection rate and improve effective-ness of screening.Simulation studies and a real data analysis are also carried out to assess the performance of our proposed methods.