Application Of Bootstrap Approach In The Linear Mixed Model

Mixed effects model is a kind of important statistical model, the model have been applied in the Biostatistics,Economic management, Medical case analysis and thus attracted many researcher’s extensive attention and application.The hypothesis test of fixed effects and random effects in the mixed effects model the focus of attention of statistician.The significance test of fixed effects and random effects in the mixed model has been the focus of attention in the study of statisticians.The nested design models are special mixed models which often appear in Social hierarchy, multistage sampling surveys and economic surveys with unit effects and time effects. The nested design models are widely used in Multilevel model, regional economic surveys and econometrics, etc. Therefore it is also important about the significance test of fixed effects and random effect variance component in the nested design models.In this paper, we mainly study the test of fixed effects and random effects in the nested design model and study significance of the random effect variance component in a generic mixed effect model under the heteroscedasticity.There are following two aspects in this dissertation.1. We consider the significance test of fixed effects and random effects in the two-factor unbalanced nested design model without the assumption of equal error variance.For the problem of testing main effects, we propose a parametric bootstrap approach and compare it with the existing the generalized p value test. The Type I error rates of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test performs better than the GF test. The PB test and GF test performs very satisfactorily even for small samples, while the generalized p value test exhibits poor Type I error properties when the number of factorial combinations or treatments goes up.It is also noted that the same tests can be used to test the significance of the random effect variance component in a two-factor mixed effects nested model under unequal error variances.2. We consider the testing the hypothesis whether the variances are smaller than a specified nonnegative value in the two-factor nested design model without the assumption of equal error variance.Using the theory of resampling,we propose a parametric bootstrap approach and compare it with the existing the generalized p value test. The Type I error rates of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test performs better. We based on the spectral decomposition of design matrics using the parametric bootstrap approach propose a new testing statistic and t he Type I error rates of the tests are evaluated using Monte Carlo simulation.