Covariance matrix estimation using repeated measurements when data are incomplete

摘要

Modeling repeated measurements data has been studied extensively lately in the parametric situation. However, it is significant to study the effect of various treatments over a period of time where the repeated measurements on the same subject are expected to be correlated. The correlation among the repeated measurements for all the subjects will be studied via the covariance matrix. The positive definite constraint is one of obstacles that encounters modeling the covariance structure, however the Cholesky decomposition removes this constraint and allows modeling the components of the covariance matrix [M. Pourahmadi, Joint mean-covariance models with applications to longitudinal data: unconstrained parameterization, Biometrika 86 (1999) 677–690]. In this paper, we adopt the estimation procedures introduced by Diggle and Verbyla [Nonparametric estimation of covariance structure in longitudinal data, Biometrics 54 (1998) 401–415] where the variogram cloud as well as the squared residuals are used to estimate the variogram and the variances via the kernel smoothing. Selecting the appropriate bandwidth value is one of the important steps in the estimation process, thus in our data analysis we choose the bandwidth using one of the most simple straight forward methods which is the cross-validation method developed by Rice and Silverman [Estimating the mean and covariance structure nonparametrically when the data are curves, J. Roy. Statist. Soc. B 53 (1991) 233–243] and adapted by others. Finally we apply these nonparametric techniques as well as a graphical method [M. Pourahmadi, Joint mean-covariance models with applications to longitudinal data: unconstrained parameterization, Biometrika 86 (1999) 677–690] to a real life data and use the penalized likelihood criterion like AIC and BIC to compare models of our interest.