Estimating And Checking For Linear Mixed Effects Models With Longitudinal Data

For the analysis of continuous longitudinal data in biology, economics, society, and agriculture, linear mixed-effects models have been popularly used. This is because the random effects and errors in such models are usually assumed to be normally distributed. Then one can use Maximum Likelihood Estimation (MLE) and Restricted Maximum Likelihood Estimation (RMLE) to estimate the unknown parameters and analyze the asymptotic properties of estimation. Especially, one can directly use statistical software, such as SAS and R, to analyze data. However, in the study of these models, People find that the normally distributed assumption is not always true, especially the that of the random effects. How to check the normality of the distributions of the random effects and errors and how to estimate the unknown parameters when the normal assumption is rejected are the issues studied in this thesis.Firstly we will check the normally distributed assumption of the random effects in the linear mixed-effects models. In the literature, based on the empirical characteristic functions, Epps & Pulley (1983) proposed the goodness-of-fit test for the normality of univariate random variables, Baringhaus & Henze (1988) suggested the corresponding one for the multivariate case. Such tests are called BHEP ones. Here, we extend the BHEP test of Henze & Wanger (1997) to construct our testing statistic. Because the random effects are not observed, the best linear biased predictions (BLUP) are used. In the study of the asymptotic properties, the testing statistics converges to a zero mean Gaussian process and is very sensitive to the to the local alternatives converging to the null one at a parameter convergent rate. Furthermore, to overcome the problem that the limiting null distribution of the test is not tractable, we suggest a new method: conditional Monte Carlo test (CMCT) to approximate the null distribution, and then to simulate p-values. The test is compared with the existing methods, the power is examined, and several examples are applied to illustrate the usefulness of our test in the analysis of longitudinal data.In the analysis of the above test, we find that the normal assumption is really rejected. Secondly, then, we will study how to estimate the unknown parameters without normal distribution and the local properties of the random effects and errors, that is some higher moments. In this thesis, we mainly estimate the former fourth moments. We first study the models that the random effects are univariate and the corresponding designed matrices are the column vectors of ones. Based on the characteristic of such models, we constructed estimating equations and then obtain the corresponding nonparametric estimation. In the study of the asymptotic properties, we find that the limiting variance of our estimation is minimum when the replicated numbers of every group experiments. In this sense, our estimation is more optimal than that of Cox & Hall (2002) that firstly study these problems in the literature. Moreover, we can also obtain its estimation by our method. Moreover, The simple simulations again verify that our estimation is better, especially in the case of estimation of the higher moments. However, both methods are not easy to be extended to estimate more than fourth moments or to estimation in the general case of the multivariate random effects. Just as the comment of Jiang (2006), it is difficult to construct estimating equations in the general linear mixed-effects models because of the designed matrices of the random effects. In order to conquer this difficulty, we suggest a naive estimating method of moment based on matrix algorithms Kronecker tensor product, vec and mathematical expectation. We study the asymptotic properties of estimation and make a simple simulation. Throughout comparing the above estimating methods, we have the following results: the estimators of the moments of the errors does not depend on the random effects, and that of the random effects does not depend on the errors, and then the corresponding asymptotic variances are very simple and optimal; when the random effects are multivariate, we can not construct different estimating equations for the random effects and errors respectively, which results that the asymptotic covariances of estimation are very complex and then the estimating efficiency is bad. Then we suggest another estimating method—orthogonality-based estimation of moments. For any matrix A being not of full row rank, as is well known, there exists an orthogonal matrix B such that BA = 0. For example, one can use QR decomposition to find the orthogonal matrix B, or use the orthogonal projection matrix of A as B. By this property of matrix, we can firstly removed the random effects of the models and then construct estimating equations for the moments of the errors. As for estimation of the moments of the random effects, we can not remove the errors similarly and just use the estimating equation of the above second estimating method but plug in the orthogonal-based estimation of the moments of the errors.