Mixed Effects Model For Longitudinal Data

Longitudinal data is referred to data in which individuals are measured repeatedly through time,so it combines elements of cross-sectional data and time-series data.Many of statisticians have been interested in longitudinal data at present.It combines the characteristics of cross-sectional data and time-series data.It is observed and got in the order of time by the same sample,so longitudinal data can analyze effectively the change of individuals and variation among individuals.It plays an important part in applications contrast to cross- sectional data or time-series data.In this paper,we study two models: linear mixed-effects model and semiparametric mixed-effects model. The main content of this paper is as follows:1.Chapter 1 introduces what is longitudinal data and its character.Then we compare it with time series data , cross section data and multivariate statistical data,so we find the advantage of longitudinal data.At last,we introduce the study situation at present and the main work of this paper;2.Chapter 2 covers the linear mixed-effects model of longitudinal data and discuss the estimation of regression parameters and covariance parameters.We use the Newton- Raphon formula to get the estimation of unknown covariance parametersθ;3.Chapter 3 describes the hypothesis tests of some parameters; (1) F test, solve the least squares estimation of unknown parameters, then we can get the F statistical test . If the null hypothesis is true, F1 ~ F ( q , N ? np ? q); If we give the significant levelα,its rejection region is as follows: F1 > F1 ?α( q , N ? np ? q); (2) Likelihood ratio test. We consider the maximum likelihood estimation of unknown parameters, then propose the relation betweenβ? andβ?λ. On the basis of the spectral representations of the LRT, we provide a simple algorithm to simulate the null finite sample distribution of LRTn . An important feature of the algorithm is that its speed depends on the number of random effects q.(3) Restricted likelihood ratio test , we derive the spectral representations of the RLRT statistics.At last,we extend the above results to the genernal form linear model.This algorithm is feasible. 4. We mianly discuss the semiparametric mixed models.We use density kernel estimates and multivariate adaptive splines to fit the unknown mean function f (? ) .At last,the model we choose can written as linear mixed model,then we can use the method discussed in chapter 3 to hypothesis test the existence of random effect.