Multivariate Repeated Measures Models And Comparison Of Estimators

Repeated measurements analysis are widely used in many fields, for example, the health and life sciences, epidemiology, biomedical research and so on. Several situations arise that involve repeated measurements on the experimental units. This thesis is devoted to the study of multi-variate repeated measures models and comparison of estimators. Three aspects of work are considered:The first aspect: we consider the multivariate repeated measurements analysis of variance (MRM ANOVA) model for complete data, a complete MRM design means that measurements are available at each time point for each experimental unit. The MRM generalizes RM in the sense that it allows a vector of observations at each measurement, the response variables are measured on each p occasions which regarded as p levels of a within-unit factor, we consider the case of multivariate response. The terminology we use for the various MRM designs in this aspect is a one-way MRM ANOVA refers to the situation with only one within- units factor, which we label as "Time" for convenience, while we label as "Group" for a between-units factor, we assume that there are q levels of the group factor with nj units assigned to the jth level, j =1,..., q, and denote the total sample size as n = n1+...+nq. We consider a linear model and parametrization for the one-way MRM design with one between-units factor incorporating univariate random effects, where Yi]k = [Yijk1...Yijkr] is the response measurement at time k for unit i within group j, i = 1, ..., nj is an index for experimental unit within group j, j = 1, ..., q is an index for levels of the between-units factor (Group), k = 1, ... ,p is an index for levels of the within-units factor (Time). For such model, the observations are transformed by an orthogonal matrix. The ANOVA based on the first set of transformed observations provides the ANOVA for the between-units effects, the problems of interest are to test for the group effect, time effect, interaction between group effectand time effect, while ANOVA based on the kth set of transformed observations, for each k =2,... ,p provides the ANOVA for the time effects and interaction of group effects and time effects, the test statistics for all problems are given, also sphericity test is studied. The likelihood ratio criterion of sphericity test, its asymptotic expansion and limiting distribution are obtained.The second aspect: we consider testing problem which is generalized sphericity test, the criterion of the test will be found and the asymptotic expansion of generalized sphericity test will be given. Also in this aspect we study the sphericity test in two models as an application of the generalized sphericity test, the first model is the nested repeated measures model (NRMM). In nested repeated measures model, we have independent individuals each having the same number of sub-individuals, where each sub-individual receives the same number of combination of rows treatments and columns treatments. We assume that all measurements have the same variance, a2; every pair of measurements that comes from different sub-individuals but the same individuals has covariance, o2p1 and every pair of measurements that comes from the same individual and the same sub-individual with (i) different columns and different rows treatments, (ii) the same column but different rows treatments, (iii) different columns but the same row treatments, have as their respective covariances. The second model is a one-way multivariate repeated measurements analysis of variance model, which is studied in the first aspect of this thesis. For both models the likelihood ratio criteri-ons, their asymptotic expansions and limiting distributions are obtained respectively.The third aspect: we consider comparison of Minimum Norm Quadratic Unbiased Estimator (MINQUE) and ordinary least squares estimator (LSE) of in the multivariate normal linear model Y ~ N{XB, V), where the design matrix A' need not have full rank and the dispersionmatrix V can be singular. We consider three loss functions: Sq...