Statistical Inference Of Longitudinal Data With Measurement Error

Longitudinal data arises when subjects are followed over a period time. For longitudinal data, observations from different subjects are independent, but those within the same subject are dependent. A common problem complicating the statistical analysis of data is the inability to directly measure or accurately measure important covariates. It is well known that naively ignoring the measurement errors, the resulting estimate is inconsistent. This dissertation focuses on the inference for longitudinal data which may contain error-prone covariates. Major results and contributions are described as follows.Firstly, we study statistical inference for longitudinal partially linear models when the response variable is sometimes missing with missingness probability de-pending on true covariates, and the true covariates are measured with error. Since the true covariates can not be observed directly, the missingness mechanism is not missing at random. The block empirical likelihood procedure is used to esti-mate the regression coefficients. This leads us to prove a nonparametric version of Wilks'theorem. We use bias correction to construct the empirical likelihood ratio functions for the baseline functions and the bias-correction-based statistic asymp-totically follows a chi-squared distribution. In comparison with methods based on normal approximations, our proposed methods do not require a consistent estima-tor for the asymptotic variance and bias, and need not undersmooth the estimator of baseline function. Simulation and an example illustrate our proposed methods.Secondly, we investigate the efficiently statistical inference for the regression coefficient in the partially linear model. Since the within-subject correlation, em-pirical likelihood can be not directly applied for longitudinal data. The block em-pirical likelihood is used to estimate the coefficients. In the estimating procedure the working within-subject correlation matrices are considered. For any working correlation matrix, an empirical log-likelihood ratio for the parametric compo-nents, which are of primary interest, is proposed, and the nonparametric version of the Wilks'theorem is derived. Numerical results show that performance can be substantially improved by correctly specifying the working correlation structure. The proposed method is used to analyze an AIDS dataset.Thirdly, we study linear model for longitudinal data with error-prone covari-ates via quadratic inference functions methods. The estimates of the coefficients are obtained by minimizing a quadratic objective function. Then the consistency and asymptotic normality of the parameter estimators are established. For hypoth-esis testing, the testing statistics we propose asymptotically follow a chi-squared distributions whether the working structure is correct specified or not. Simula-tion results show that the estimates are most efficient when the correct covariance correlation structure is used. An analysis of an real data are also presented.At last, we investigate a class of marginal model for longitudinal data with errors in covariates. The estimates of regression coefficients are obtained via sim-ulation extrapolation method. In the simulation step of SIMEX method, naive estimates of coefficients are obtained by QIF method. We also show that the SIMEX estimates follow approximately normal distribution asymptotically. A simulation study is conducted to investigate the finite sample properties of the proposed methods. The results suggest that the proposed methods are gener-ally more efficient than conditional score method which ignore the correlation structure, and the estimates are most efficient by correctly specifying the working correlation matrix. The proposed methods are applied to an AIDS datast.