The Analysis Of Covariance Matrix For Longitudinal Data With Multiple Outcomes

With the development of science and technology,multivariate longitudinal data are frequently encountered in medicine,biology,economics and other fields in recent years.It often involves two or more outcomes of interest measured repeatedly across time for each subject.Compared to univariate longitudinal data,in multivariate longitudinal data analysis,the measurements from multiple response variables at different time points are likely to be correlated.A main challenge in the analysis of such data is the complex correlation structure.In particular,with the advent of the big data era,many statistical analysis of high-dimensional data are faced with the challenges due to the poor performance of classical methods.For multivariate longitudinal data,both the dimensions of the response variables and the repeated measurements for a fixed outcome increase rapidly,leading to more complex data structure.Thus,it is necessary to model the covariance matrix or the inverse of the covariance matrix(precision matrix)properly in order to capture the main correlation information of data.Based on the assumption that the structural information of the response variables and different time points is separable,for the dimensions of the responses and the repeated measurements,we consider two cases.That is,they are fixed size or increase proportionally with the sample size.We consider to establish a parsimonious and interpretable model for the covariance matrix or the precision matrix of multivariate longitudinal data respectively.In the first part of this paper,the correlation matrix of multivariate longitudinal data with fixed size is expressed as a mixture of finite working correlation matrices under the framework of exponential distribution family,and each working correlation matrix has the Kronecker structure.Based on the idea of the finite mixture model,the correlation matrix is estimated by the pseudo-likelihood method,and further the regression parameters are estimated by the generalized estimation equation.Under some mild regularization conditions,we prove the consistency and asymptotic normality of the regression parameter and the correlation matrix parameter estimators.In the second part,combined with the unique structural characteristics of univariate longitudinal data,an adaptively block-banded sparse structure model is established for the precision matrix of high-dimensional multivariate longitudinal data.Based on the pseudo-likelihood method,we construct the bi-convex optimization problem,and the precision matrix estimators can be obtained by alternating convex search(ACS)and ADMM algorithms.We prove the consistency of the precision matrix estimators and the bandwidth structure recovery.Simulation studies and real data analysis show that the above proposed models can effectively capture the structural information of the covariance matrix or the precision matrix.